Optimal. Leaf size=121 \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]
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Rubi [A] time = 0.141367, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {766} \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int x^3 (A+B x) \left (a+c x^2\right )^4 \, dx &=\int \left (a^4 A x^3+a^4 B x^4+4 a^3 A c x^5+4 a^3 B c x^6+6 a^2 A c^2 x^7+6 a^2 B c^2 x^8+4 a A c^3 x^9+4 a B c^3 x^{10}+A c^4 x^{11}+B c^4 x^{12}\right ) \, dx\\ &=\frac{1}{4} a^4 A x^4+\frac{1}{5} a^4 B x^5+\frac{2}{3} a^3 A c x^6+\frac{4}{7} a^3 B c x^7+\frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^2 B c^2 x^9+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13}\\ \end{align*}
Mathematica [A] time = 0.0038559, size = 121, normalized size = 1. \[ \frac{3}{4} a^2 A c^2 x^8+\frac{2}{3} a^3 A c x^6+\frac{1}{4} a^4 A x^4+\frac{2}{3} a^2 B c^2 x^9+\frac{4}{7} a^3 B c x^7+\frac{1}{5} a^4 B x^5+\frac{2}{5} a A c^3 x^{10}+\frac{4}{11} a B c^3 x^{11}+\frac{1}{12} A c^4 x^{12}+\frac{1}{13} B c^4 x^{13} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 102, normalized size = 0.8 \begin{align*}{\frac{{a}^{4}A{x}^{4}}{4}}+{\frac{{a}^{4}B{x}^{5}}{5}}+{\frac{2\,{a}^{3}Ac{x}^{6}}{3}}+{\frac{4\,{a}^{3}Bc{x}^{7}}{7}}+{\frac{3\,{a}^{2}A{c}^{2}{x}^{8}}{4}}+{\frac{2\,{a}^{2}B{c}^{2}{x}^{9}}{3}}+{\frac{2\,aA{c}^{3}{x}^{10}}{5}}+{\frac{4\,aB{c}^{3}{x}^{11}}{11}}+{\frac{A{c}^{4}{x}^{12}}{12}}+{\frac{B{c}^{4}{x}^{13}}{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03292, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{13} \, B c^{4} x^{13} + \frac{1}{12} \, A c^{4} x^{12} + \frac{4}{11} \, B a c^{3} x^{11} + \frac{2}{5} \, A a c^{3} x^{10} + \frac{2}{3} \, B a^{2} c^{2} x^{9} + \frac{3}{4} \, A a^{2} c^{2} x^{8} + \frac{4}{7} \, B a^{3} c x^{7} + \frac{2}{3} \, A a^{3} c x^{6} + \frac{1}{5} \, B a^{4} 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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Fricas [A] time = 1.26367, size = 246, normalized size = 2.03 \begin{align*} \frac{1}{13} x^{13} c^{4} B + \frac{1}{12} x^{12} c^{4} A + \frac{4}{11} x^{11} c^{3} a B + \frac{2}{5} x^{10} c^{3} a A + \frac{2}{3} x^{9} c^{2} a^{2} B + \frac{3}{4} x^{8} c^{2} a^{2} A + \frac{4}{7} x^{7} c a^{3} B + \frac{2}{3} x^{6} c a^{3} A + \frac{1}{5} x^{5} a^{4} B + \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 = 0.082148, size = 124, normalized size = 1.02 \begin{align*} \frac{A a^{4} x^{4}}{4} + \frac{2 A a^{3} c x^{6}}{3} + \frac{3 A a^{2} c^{2} x^{8}}{4} + \frac{2 A a c^{3} x^{10}}{5} + \frac{A c^{4} x^{12}}{12} + \frac{B a^{4} x^{5}}{5} + \frac{4 B a^{3} c x^{7}}{7} + \frac{2 B a^{2} c^{2} x^{9}}{3} + \frac{4 B a c^{3} x^{11}}{11} + \frac{B c^{4} x^{13}}{13} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13318, size = 136, normalized size = 1.12 \begin{align*} \frac{1}{13} \, B c^{4} x^{13} + \frac{1}{12} \, A c^{4} x^{12} + \frac{4}{11} \, B a c^{3} x^{11} + \frac{2}{5} \, A a c^{3} x^{10} + \frac{2}{3} \, B a^{2} c^{2} x^{9} + \frac{3}{4} \, A a^{2} c^{2} x^{8} + \frac{4}{7} \, B a^{3} c x^{7} + \frac{2}{3} \, A a^{3} c x^{6} + \frac{1}{5} \, B a^{4} 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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