Optimal. Leaf size=43 \[ a^2 A \log (x)+a A b x^2+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4 \]
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Rubi [A] time = 0.0315544, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 80, 43} \[ a^2 A \log (x)+a A b x^2+\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} A b^2 x^4 \]
Antiderivative was successfully verified.
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Rule 446
Rule 80
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (A+B x)}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{2} A \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,x^2\right )\\ &=\frac{B \left (a+b x^2\right )^3}{6 b}+\frac{1}{2} A \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,x^2\right )\\ &=a A b x^2+\frac{1}{4} A b^2 x^4+\frac{B \left (a+b x^2\right )^3}{6 b}+a^2 A \log (x)\\ \end{align*}
Mathematica [A] time = 0.0151837, size = 51, normalized size = 1.19 \[ a^2 A \log (x)+\frac{1}{4} b x^4 (2 a B+A b)+\frac{1}{2} a x^2 (a B+2 A b)+\frac{1}{6} b^2 B x^6 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 51, normalized size = 1.2 \begin{align*}{\frac{B{b}^{2}{x}^{6}}{6}}+{\frac{A{b}^{2}{x}^{4}}{4}}+{\frac{B{x}^{4}ab}{2}}+aAb{x}^{2}+{\frac{B{x}^{2}{a}^{2}}{2}}+{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 = 0.969851, size = 70, normalized size = 1.63 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + \frac{1}{2} \, A a^{2} \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42842, size = 116, normalized size = 2.7 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{4} \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + A a^{2} \log \left (x\right ) + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.27906, size = 49, normalized size = 1.14 \begin{align*} A a^{2} \log{\left (x \right )} + \frac{B b^{2} x^{6}}{6} + x^{4} \left (\frac{A b^{2}}{4} + \frac{B a b}{2}\right ) + x^{2} \left (A a b + \frac{B a^{2}}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12293, size = 72, normalized size = 1.67 \begin{align*} \frac{1}{6} \, B b^{2} x^{6} + \frac{1}{2} \, B a b x^{4} + \frac{1}{4} \, A b^{2} x^{4} + \frac{1}{2} \, B a^{2} x^{2} + A a b x^{2} + \frac{1}{2} \, A a^{2} \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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