Optimal. Leaf size=30 \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]
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Rubi [A] time = 0.0149898, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^2}{x^5} \, dx &=\int \frac{\left (b+a x^2\right )^2}{x^9} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+a x)^2}{x^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{x^5}+\frac{2 a b}{x^4}+\frac{a^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{8 x^8}-\frac{a b}{3 x^6}-\frac{a^2}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0008541, size = 30, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{a b}{3 x^6}-\frac{b^2}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{b}^{2}}{8\,{x}^{8}}}-{\frac{ab}{3\,{x}^{6}}}-{\frac{{a}^{2}}{4\,{x}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971239, size = 35, normalized size = 1.17 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41854, size = 58, normalized size = 1.93 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.345534, size = 27, normalized size = 0.9 \begin{align*} - \frac{6 a^{2} x^{4} + 8 a b x^{2} + 3 b^{2}}{24 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1697, size = 35, normalized size = 1.17 \begin{align*} -\frac{6 \, a^{2} x^{4} + 8 \, a b x^{2} + 3 \, b^{2}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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