Optimal. Leaf size=149 \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]
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Rubi [A] time = 0.186161, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1802} \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]
Antiderivative was successfully verified.
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Rule 1802
Rubi steps
\begin{align*} \int x^3 \left (a+b x^2\right )^3 \left (A+B x+C x^2+D x^3\right ) \, dx &=\int \left (a^3 A x^3+a^3 B x^4+a^2 (3 A b+a C) x^5+a^2 (3 b B+a D) x^6+3 a b (A b+a C) x^7+3 a b (b B+a D) x^8+b^2 (A b+3 a C) x^9+b^2 (b B+3 a D) x^{10}+b^3 C x^{11}+b^3 D x^{12}\right ) \, dx\\ &=\frac{1}{4} a^3 A x^4+\frac{1}{5} a^3 B x^5+\frac{1}{6} a^2 (3 A b+a C) x^6+\frac{1}{7} a^2 (3 b B+a D) x^7+\frac{3}{8} a b (A b+a C) x^8+\frac{1}{3} a b (b B+a D) x^9+\frac{1}{10} b^2 (A b+3 a C) x^{10}+\frac{1}{11} b^2 (b B+3 a D) x^{11}+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13}\\ \end{align*}
Mathematica [A] time = 0.0271811, size = 149, normalized size = 1. \[ \frac{1}{6} a^2 x^6 (a C+3 A b)+\frac{1}{4} a^3 A x^4+\frac{1}{7} a^2 x^7 (a D+3 b B)+\frac{1}{5} a^3 B x^5+\frac{1}{10} b^2 x^{10} (3 a C+A b)+\frac{3}{8} a b x^8 (a C+A b)+\frac{1}{11} b^2 x^{11} (3 a D+b B)+\frac{1}{3} a b x^9 (a D+b B)+\frac{1}{12} b^3 C x^{12}+\frac{1}{13} b^3 D x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 150, normalized size = 1. \begin{align*}{\frac{{b}^{3}D{x}^{13}}{13}}+{\frac{{b}^{3}C{x}^{12}}{12}}+{\frac{ \left ({b}^{3}B+3\,a{b}^{2}D \right ){x}^{11}}{11}}+{\frac{ \left ( A{b}^{3}+3\,a{b}^{2}C \right ){x}^{10}}{10}}+{\frac{ \left ( 3\,a{b}^{2}B+3\,{a}^{2}bD \right ){x}^{9}}{9}}+{\frac{ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bC \right ){x}^{8}}{8}}+{\frac{ \left ( 3\,{a}^{2}bB+{a}^{3}D \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,A{a}^{2}b+{a}^{3}C \right ){x}^{6}}{6}}+{\frac{{a}^{3}B{x}^{5}}{5}}+{\frac{{a}^{3}A{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01083, size = 196, normalized size = 1.32 \begin{align*} \frac{1}{13} \, D b^{3} x^{13} + \frac{1}{12} \, C b^{3} x^{12} + \frac{1}{11} \,{\left (3 \, D a b^{2} + B b^{3}\right )} x^{11} + \frac{1}{10} \,{\left (3 \, C a b^{2} + A b^{3}\right )} x^{10} + \frac{1}{3} \,{\left (D a^{2} b + B a b^{2}\right )} x^{9} + \frac{1}{5} \, B a^{3} x^{5} + \frac{3}{8} \,{\left (C a^{2} b + A a b^{2}\right )} x^{8} + \frac{1}{4} \, A a^{3} x^{4} + \frac{1}{7} \,{\left (D a^{3} + 3 \, B a^{2} b\right )} x^{7} + \frac{1}{6} \,{\left (C a^{3} + 3 \, A a^{2} b\right )} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37057, size = 382, normalized size = 2.56 \begin{align*} \frac{1}{13} x^{13} b^{3} D + \frac{1}{12} x^{12} b^{3} C + \frac{3}{11} x^{11} b^{2} a D + \frac{1}{11} x^{11} b^{3} B + \frac{3}{10} x^{10} b^{2} a C + \frac{1}{10} x^{10} b^{3} A + \frac{1}{3} x^{9} b a^{2} D + \frac{1}{3} x^{9} b^{2} a B + \frac{3}{8} x^{8} b a^{2} C + \frac{3}{8} x^{8} b^{2} a A + \frac{1}{7} x^{7} a^{3} D + \frac{3}{7} x^{7} b a^{2} B + \frac{1}{6} x^{6} a^{3} C + \frac{1}{2} x^{6} b a^{2} A + \frac{1}{5} x^{5} a^{3} B + \frac{1}{4} x^{4} a^{3} A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.091382, size = 163, normalized size = 1.09 \begin{align*} \frac{A a^{3} x^{4}}{4} + \frac{B a^{3} x^{5}}{5} + \frac{C b^{3} x^{12}}{12} + \frac{D b^{3} x^{13}}{13} + x^{11} \left (\frac{B b^{3}}{11} + \frac{3 D a b^{2}}{11}\right ) + x^{10} \left (\frac{A b^{3}}{10} + \frac{3 C a b^{2}}{10}\right ) + x^{9} \left (\frac{B a b^{2}}{3} + \frac{D a^{2} b}{3}\right ) + x^{8} \left (\frac{3 A a b^{2}}{8} + \frac{3 C a^{2} b}{8}\right ) + x^{7} \left (\frac{3 B a^{2} b}{7} + \frac{D a^{3}}{7}\right ) + x^{6} \left (\frac{A a^{2} b}{2} + \frac{C a^{3}}{6}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1554, size = 207, normalized size = 1.39 \begin{align*} \frac{1}{13} \, D b^{3} x^{13} + \frac{1}{12} \, C b^{3} x^{12} + \frac{3}{11} \, D a b^{2} x^{11} + \frac{1}{11} \, B b^{3} x^{11} + \frac{3}{10} \, C a b^{2} x^{10} + \frac{1}{10} \, A b^{3} x^{10} + \frac{1}{3} \, D a^{2} b x^{9} + \frac{1}{3} \, B a b^{2} x^{9} + \frac{3}{8} \, C a^{2} b x^{8} + \frac{3}{8} \, A a b^{2} x^{8} + \frac{1}{7} \, D a^{3} x^{7} + \frac{3}{7} \, B a^{2} b x^{7} + \frac{1}{6} \, C a^{3} x^{6} + \frac{1}{2} \, A a^{2} b x^{6} + \frac{1}{5} \, B a^{3} x^{5} + \frac{1}{4} \, A a^{3} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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