{{begin-eqn}} {{eqn | o =        | r = \int x^2 \sqrt {a x^2 + b x + c} \rd x }} {{eqn | r = \int \frac {2 a x^2} {2 a} \sqrt {a x^2 + b x + c} \rd x       | c = multiplying top and bottom by $2 a$ }} {{eqn | r = \int \frac {x \paren {2 a x + b - b} } {2 a} \sqrt {a x^2 + b x + c} \rd x }} {{eqn | n = 1       | r = \frac 1 {2 a} \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x - \frac b {2 a} \int x \sqrt {a x^2 + b x + c} \rd x       | c = Linear Combination of Integrals }} {{end-eqn}} Let: {{begin-eqn}} {{eqn | l = z       | r = a x^2 + b x + c }} {{eqn | ll= \leadsto       | l = \frac {\d z} {\rd x}       | r = 2 a x + b       | c = Derivative of Power }} {{eqn | ll= \leadsto       | l = \int \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x       | r = \int \sqrt z \rd z       | c = Integration by Substitution }} {{eqn | r = \frac {2 \paren {\sqrt z}^3} 3       | c = Primitive of Power }} {{eqn | n = 2       | r = \frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3       | c = substituting for $z$ }} {{end-eqn}} With a view to expressing the primitive $\displaystyle \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x$ in the form: :$\displaystyle \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$ let: {{begin-eqn}} {{eqn | l = u       | r = x       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d u} {\d x}       | r = 1       | c = Derivative of Identity Function }} {{end-eqn}} and let: {{begin-eqn}} {{eqn | l = \frac {\d v} {\d x}       | r = \int \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | ll= \leadsto       | l = v       | r = \frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3       | c = from $(2)$ above }} {{end-eqn}} Then: {{begin-eqn}} {{eqn | r = \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x       | o =        | c =  }} {{eqn | r = x \paren {\frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3} - \int \paren {\frac {2 \paren {\sqrt {a x^2 + b x + c} }^3} 3} \times 1 \rd x + C       | c = Integration by Parts }} {{eqn | n = 3       | r = \frac {2 x \paren {\sqrt {a x^2 + b x + c} }^3} 3 - \frac 2 3 \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x + C       | c = simplifying }} {{end-eqn}} Now consider: {{begin-eqn}} {{eqn | r = \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x       | o =        | c =  }} {{eqn | r = \int \paren {a x^2 + b x + c} \sqrt {a x^2 + b x + c} \rd x       | c = factorising }} {{eqn | n = 4       | r = a \int x^2 \sqrt {a x^2 + b x + c} \rd x + b \int x \sqrt {a x^2 + b x + c} \rd x + c \int \sqrt {a x^2 + b x + c} \rd x       | c = Linear Combination of Integrals }} {{end-eqn}} Thus: {{begin-eqn}} {{eqn | o =        | r = \int x^2 \sqrt {a x^2 + b x + c} \rd x }} {{eqn | r = \frac 1 {2 a} \int x \paren {2 a x + b} \sqrt {a x^2 + b x + c} \rd x - \frac b {2 a} \int x \sqrt {a x^2 + b x + c} \rd x       | c = from $(1)$ }} {{eqn | r = \frac 1 {2 a} \paren {\frac {2 x \paren {\sqrt {a x^2 + b x + c} }^3} 3 - \frac 2 3 \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x}       | c = from $(3)$ }} {{eqn | o =        | ro= -       | r = \frac b {2 a} \paren {\frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x}       | c = Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$ }} {{eqn | r = \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac 1 {3 a} \int \paren {\sqrt {a x^2 + b x + c} }^3 \rd x       | c = simplifying }} {{eqn | o =        | ro= -       | r = \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | r = \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 {3 a} \paren {a \int x^2 \sqrt {a x^2 + b x + c} \rd x + b \int x \sqrt {a x^2 + b x + c} \rd x + c \int \sqrt {a x^2 + b x + c} \rd x}       | c = from $(4)$ }} {{eqn | r = \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c = simplifying }} {{eqn | o =        | ro= -       | r = \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac b {3 a} \int x \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | r = \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \ rd x       | c =  }} {{eqn | o =        | ro= -       | r = \frac b {3 a} \paren {\frac {\paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {2 a} \int \sqrt {a x^2 + b x + c} \rd x}       | c = Lemma for Primitive of $x \sqrt {a x^2 + b x + c}$ }} {{eqn | r = \frac {x \paren {\sqrt {a x^2 + b x + c} }^3} {3 a} - \frac b {6 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {b^2} {4 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c = simplifying }} {{eqn | o =        | ro= -       | r = \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x - \frac c {3 a} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | o =        | ro= -       | r = \frac {b \paren {\sqrt {a x^2 + b x + c} }^3} {9 a^2} + \frac {b^2} {6 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | r = \paren {\frac x {3 a} - \frac b {6 a^2} - \frac b {9 a^2} } \paren {\sqrt {a x^2 + b x + c} }^3       | c = gathering terms }} {{eqn | o =        | ro= +       | r = \paren {\frac {b^2} {4 a^2} - \frac c {3 a} + \frac {b^2} {6 a^2} } \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | o =        | ro= -       | r = \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{end-eqn}} Hence: {{begin-eqn}} {{eqn | r = \int x^2 \sqrt {a x^2 + b x + c} \rd x + \frac 1 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x       | o =        | c =  }} {{eqn | r = \paren {\frac {6 a x} {18 a^2} - \frac {3 b} {18 a^2} - \frac {2 b} {18 a^2} } \paren {\sqrt {a x^2 + b x + c} }^3       | c =  }} {{eqn | o =        | ro= +       | r = \paren {\frac {3 b^2} {12 a^2} - \frac {4 a c} {12 a^2} + \frac {2 b^2} {12 a^2} } \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | ll= \leadsto       | l = \frac 4 3 \int x^2 \sqrt {a x^2 + b x + c} \rd x       | r = \frac {6 a x - 5 b} {18 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {12 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c =  }} {{eqn | ll= \leadsto       | l = \int x^2 \sqrt {a x^2 + b x + c} \rd x       | r = \frac {6 a x - 5 b} {24 a^2} \paren {\sqrt {a x^2 + b x + c} }^3 + \frac {5 b^2 - 4 a c} {16 a^2} \int \sqrt {a x^2 + b x + c} \rd x       | c = multiplying by $\dfrac 3 4$ }} {{end-eqn}} {{qed}}