Optimal. Leaf size=39 \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{2 x^2}+b^3 \log (x) \]
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Rubi [A] time = 0.0211947, antiderivative size = 39, 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} \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{2 x^2}+b^3 \log (x) \]
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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^3}{x^4}+\frac{3 a^2 b}{x^3}+\frac{3 a b^2}{x^2}+\frac{b^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3}{6 x^6}-\frac{3 a^2 b}{4 x^4}-\frac{3 a b^2}{2 x^2}+b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0038464, size = 39, normalized size = 1. \[ -\frac{3 a^2 b}{4 x^4}-\frac{a^3}{6 x^6}-\frac{3 a b^2}{2 x^2}+b^3 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}}{6\,{x}^{6}}}-{\frac{3\,{a}^{2}b}{4\,{x}^{4}}}-{\frac{3\,a{b}^{2}}{2\,{x}^{2}}}+{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.42442, size = 53, normalized size = 1.36 \begin{align*} \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) - \frac{18 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23594, size = 90, normalized size = 2.31 \begin{align*} \frac{12 \, b^{3} x^{6} \log \left (x\right ) - 18 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.377119, size = 36, normalized size = 0.92 \begin{align*} b^{3} \log{\left (x \right )} - \frac{2 a^{3} + 9 a^{2} b x^{2} + 18 a b^{2} x^{4}}{12 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.56305, size = 63, normalized size = 1.62 \begin{align*} \frac{1}{2} \, b^{3} \log \left (x^{2}\right ) - \frac{11 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 9 \, a^{2} b x^{2} + 2 \, a^{3}}{12 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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