Optimal. Leaf size=87 \[ \frac{a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac{(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac{a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac{B (a+b x)^8}{8 b^4} \]
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Rubi [A] time = 0.0529759, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {27, 76} \[ \frac{a^2 (a+b x)^5 (A b-a B)}{5 b^4}+\frac{(a+b x)^7 (A b-3 a B)}{7 b^4}-\frac{a (a+b x)^6 (2 A b-3 a B)}{6 b^4}+\frac{B (a+b x)^8}{8 b^4} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^2 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^2 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (-\frac{a^2 (-A b+a B) (a+b x)^4}{b^3}+\frac{a (-2 A b+3 a B) (a+b x)^5}{b^3}+\frac{(A b-3 a B) (a+b x)^6}{b^3}+\frac{B (a+b x)^7}{b^3}\right ) \, dx\\ &=\frac{a^2 (A b-a B) (a+b x)^5}{5 b^4}-\frac{a (2 A b-3 a B) (a+b x)^6}{6 b^4}+\frac{(A b-3 a B) (a+b x)^7}{7 b^4}+\frac{B (a+b x)^8}{8 b^4}\\ \end{align*}
Mathematica [A] time = 0.0257239, size = 88, normalized size = 1.01 \[ \frac{1}{840} x^3 \left (168 a^2 b^2 x^2 (6 A+5 B x)+168 a^3 b x (5 A+4 B x)+70 a^4 (4 A+3 B x)+80 a b^3 x^3 (7 A+6 B x)+15 b^4 x^4 (8 A+7 B x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 100, normalized size = 1.2 \begin{align*}{\frac{{b}^{4}B{x}^{8}}{8}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{7}}{7}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{4}}{4}}+{\frac{{a}^{4}A{x}^{3}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02124, size = 134, normalized size = 1.54 \begin{align*} \frac{1}{8} \, B b^{4} x^{8} + \frac{1}{3} \, A a^{4} x^{3} + \frac{1}{7} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{7} + \frac{1}{3} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{6} + \frac{2}{5} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{5} + \frac{1}{4} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11358, size = 225, normalized size = 2.59 \begin{align*} \frac{1}{8} x^{8} b^{4} B + \frac{4}{7} x^{7} b^{3} a B + \frac{1}{7} x^{7} b^{4} A + x^{6} b^{2} a^{2} B + \frac{2}{3} x^{6} b^{3} a A + \frac{4}{5} x^{5} b a^{3} B + \frac{6}{5} x^{5} b^{2} a^{2} A + \frac{1}{4} x^{4} a^{4} B + x^{4} b a^{3} A + \frac{1}{3} x^{3} a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081985, size = 104, normalized size = 1.2 \begin{align*} \frac{A a^{4} x^{3}}{3} + \frac{B b^{4} x^{8}}{8} + x^{7} \left (\frac{A b^{4}}{7} + \frac{4 B a b^{3}}{7}\right ) + x^{6} \left (\frac{2 A a b^{3}}{3} + B a^{2} b^{2}\right ) + x^{5} \left (\frac{6 A a^{2} b^{2}}{5} + \frac{4 B a^{3} b}{5}\right ) + x^{4} \left (A a^{3} b + \frac{B a^{4}}{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13566, size = 134, normalized size = 1.54 \begin{align*} \frac{1}{8} \, B b^{4} x^{8} + \frac{4}{7} \, B a b^{3} x^{7} + \frac{1}{7} \, A b^{4} x^{7} + B a^{2} b^{2} x^{6} + \frac{2}{3} \, A a b^{3} x^{6} + \frac{4}{5} \, B a^{3} b x^{5} + \frac{6}{5} \, A a^{2} b^{2} x^{5} + \frac{1}{4} \, B a^{4} x^{4} + A a^{3} b x^{4} + \frac{1}{3} \, A a^{4} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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