Optimal. Leaf size=53 \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]
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Rubi [A] time = 0.0209783, antiderivative size = 53, 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} \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]
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 )^2}{x} \, dx &=\int \left (a^2 B+\frac{a^2 A}{x}+2 a A c x+2 a B c x^2+A c^2 x^3+B c^2 x^4\right ) \, dx\\ &=a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5+a^2 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0063978, size = 53, normalized size = 1. \[ a^2 A \log (x)+a^2 B x+a A c x^2+\frac{2}{3} a B c x^3+\frac{1}{4} A c^2 x^4+\frac{1}{5} B c^2 x^5 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 48, normalized size = 0.9 \begin{align*}{a}^{2}Bx+aAc{x}^{2}+{\frac{2\,aBc{x}^{3}}{3}}+{\frac{A{c}^{2}{x}^{4}}{4}}+{\frac{B{c}^{2}{x}^{5}}{5}}+{a}^{2}A\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03605, size = 63, normalized size = 1.19 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48841, size = 113, normalized size = 2.13 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.32886, size = 54, normalized size = 1.02 \begin{align*} A a^{2} \log{\left (x \right )} + A a c x^{2} + \frac{A c^{2} x^{4}}{4} + B a^{2} x + \frac{2 B a c x^{3}}{3} + \frac{B c^{2} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14094, size = 65, normalized size = 1.23 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{4} \, A c^{2} x^{4} + \frac{2}{3} \, B a c x^{3} + A a c x^{2} + B a^{2} x + A a^{2} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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