Optimal. Leaf size=81 \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]
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Rubi [A] time = 0.0361952, antiderivative size = 81, 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} \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Rule 766
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (a+c x^2\right )^3}{x} \, dx &=\int \left (a^3 B+\frac{a^3 A}{x}+3 a^2 A c x+3 a^2 B c x^2+3 a A c^2 x^3+3 a B c^2 x^4+A c^3 x^5+B c^3 x^6\right ) \, dx\\ &=a^3 B x+\frac{3}{2} a^2 A c x^2+a^2 B c x^3+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7+a^3 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0066101, size = 81, normalized size = 1. \[ \frac{3}{2} a^2 A c x^2+a^3 A \log (x)+a^2 B c x^3+a^3 B x+\frac{3}{4} a A c^2 x^4+\frac{3}{5} a B c^2 x^5+\frac{1}{6} A c^3 x^6+\frac{1}{7} B c^3 x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 72, normalized size = 0.9 \begin{align*}{a}^{3}Bx+{\frac{3\,{a}^{2}Ac{x}^{2}}{2}}+{a}^{2}Bc{x}^{3}+{\frac{3\,aA{c}^{2}{x}^{4}}{4}}+{\frac{3\,aB{c}^{2}{x}^{5}}{5}}+{\frac{A{c}^{3}{x}^{6}}{6}}+{\frac{B{c}^{3}{x}^{7}}{7}}+{a}^{3}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03982, size = 96, normalized size = 1.19 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5257, size = 167, normalized size = 2.06 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.385519, size = 85, normalized size = 1.05 \begin{align*} A a^{3} \log{\left (x \right )} + \frac{3 A a^{2} c x^{2}}{2} + \frac{3 A a c^{2} x^{4}}{4} + \frac{A c^{3} x^{6}}{6} + B a^{3} x + B a^{2} c x^{3} + \frac{3 B a c^{2} x^{5}}{5} + \frac{B c^{3} x^{7}}{7} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12998, size = 97, normalized size = 1.2 \begin{align*} \frac{1}{7} \, B c^{3} x^{7} + \frac{1}{6} \, A c^{3} x^{6} + \frac{3}{5} \, B a c^{2} x^{5} + \frac{3}{4} \, A a c^{2} x^{4} + B a^{2} c x^{3} + \frac{3}{2} \, A a^{2} c x^{2} + B a^{3} x + A a^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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