Optimal. Leaf size=40 \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]
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Rubi [A] time = 0.0186338, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^3}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^3+\frac{a^3}{x^3}+\frac{3 a^2 b}{x^2}+\frac{3 a b^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3}{4 x^4}-\frac{3 a^2 b}{2 x^2}+\frac{b^3 x^2}{2}+3 a b^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0039348, size = 40, normalized size = 1. \[ -\frac{3 a^2 b}{2 x^2}-\frac{a^3}{4 x^4}+3 a b^2 \log (x)+\frac{b^3 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{2\,{x}^{2}}}+{\frac{{b}^{3}{x}^{2}}{2}}+3\,a{b}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.36099, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28901, size = 85, normalized size = 2.12 \begin{align*} \frac{2 \, b^{3} x^{6} + 12 \, a b^{2} x^{4} \log \left (x\right ) - 6 \, a^{2} b x^{2} - a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.339119, size = 36, normalized size = 0.9 \begin{align*} 3 a b^{2} \log{\left (x \right )} + \frac{b^{3} x^{2}}{2} - \frac{a^{3} + 6 a^{2} b x^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.3507, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + \frac{3}{2} \, a b^{2} \log \left (x^{2}\right ) - \frac{9 \, a b^{2} x^{4} + 6 \, a^{2} b x^{2} + a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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