Optimal. Leaf size=99 \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]
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Rubi [A] time = 0.0570259, antiderivative size = 99, 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} \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin{align*} \int x^3 (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int x^3 (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A x^3+a^3 (4 A b+a B) x^4+2 a^2 b (3 A b+2 a B) x^5+2 a b^2 (2 A b+3 a B) x^6+b^3 (A b+4 a B) x^7+b^4 B x^8\right ) \, dx\\ &=\frac{1}{4} a^4 A x^4+\frac{1}{5} a^3 (4 A b+a B) x^5+\frac{1}{3} a^2 b (3 A b+2 a B) x^6+\frac{2}{7} a b^2 (2 A b+3 a B) x^7+\frac{1}{8} b^3 (A b+4 a B) x^8+\frac{1}{9} b^4 B x^9\\ \end{align*}
Mathematica [A] time = 0.0124008, size = 99, normalized size = 1. \[ \frac{1}{3} a^2 b x^6 (2 a B+3 A b)+\frac{1}{5} a^3 x^5 (a B+4 A b)+\frac{1}{4} a^4 A x^4+\frac{1}{8} b^3 x^8 (4 a B+A b)+\frac{2}{7} a b^2 x^7 (3 a B+2 A b)+\frac{1}{9} b^4 B x^9 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 100, normalized size = 1. \begin{align*}{\frac{{b}^{4}B{x}^{9}}{9}}+{\frac{ \left ( A{b}^{4}+4\,Ba{b}^{3} \right ){x}^{8}}{8}}+{\frac{ \left ( 4\,Aa{b}^{3}+6\,B{a}^{2}{b}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,A{a}^{2}{b}^{2}+4\,B{a}^{3}b \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,A{a}^{3}b+B{a}^{4} \right ){x}^{5}}{5}}+{\frac{{a}^{4}A{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.960282, size = 134, normalized size = 1.35 \begin{align*} \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{4} \, A a^{4} x^{4} + \frac{1}{8} \,{\left (4 \, B a b^{3} + A b^{4}\right )} x^{8} + \frac{2}{7} \,{\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{7} + \frac{1}{3} \,{\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{6} + \frac{1}{5} \,{\left (B a^{4} + 4 \, A a^{3} b\right )} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17465, size = 231, normalized size = 2.33 \begin{align*} \frac{1}{9} x^{9} b^{4} B + \frac{1}{2} x^{8} b^{3} a B + \frac{1}{8} x^{8} b^{4} A + \frac{6}{7} x^{7} b^{2} a^{2} B + \frac{4}{7} x^{7} b^{3} a A + \frac{2}{3} x^{6} b a^{3} B + x^{6} b^{2} a^{2} A + \frac{1}{5} x^{5} a^{4} B + \frac{4}{5} x^{5} b a^{3} A + \frac{1}{4} x^{4} a^{4} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086743, size = 105, normalized size = 1.06 \begin{align*} \frac{A a^{4} x^{4}}{4} + \frac{B b^{4} x^{9}}{9} + x^{8} \left (\frac{A b^{4}}{8} + \frac{B a b^{3}}{2}\right ) + x^{7} \left (\frac{4 A a b^{3}}{7} + \frac{6 B a^{2} b^{2}}{7}\right ) + x^{6} \left (A a^{2} b^{2} + \frac{2 B a^{3} b}{3}\right ) + x^{5} \left (\frac{4 A a^{3} b}{5} + \frac{B a^{4}}{5}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14638, size = 135, normalized size = 1.36 \begin{align*} \frac{1}{9} \, B b^{4} x^{9} + \frac{1}{2} \, B a b^{3} x^{8} + \frac{1}{8} \, A b^{4} x^{8} + \frac{6}{7} \, B a^{2} b^{2} x^{7} + \frac{4}{7} \, A a b^{3} x^{7} + \frac{2}{3} \, B a^{3} b x^{6} + A a^{2} b^{2} x^{6} + \frac{1}{5} \, B a^{4} x^{5} + \frac{4}{5} \, A a^{3} b x^{5} + \frac{1}{4} \, A a^{4} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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