Optimal. Leaf size=49 \[ -\frac{a}{2 b^2 \left (a x^2+b\right )}+\frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{1}{2 b^2 x^2} \]
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Rubi [A] time = 0.0326947, antiderivative size = 49, 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, 44} \[ -\frac{a}{2 b^2 \left (a x^2+b\right )}+\frac{a \log \left (a x^2+b\right )}{b^3}-\frac{2 a \log (x)}{b^3}-\frac{1}{2 b^2 x^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^2 x^7} \, dx &=\int \frac{1}{x^3 \left (b+a x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^2}-\frac{2 a}{b^3 x}+\frac{a^2}{b^2 (b+a x)^2}+\frac{2 a^2}{b^3 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 b^2 x^2}-\frac{a}{2 b^2 \left (b+a x^2\right )}-\frac{2 a \log (x)}{b^3}+\frac{a \log \left (b+a x^2\right )}{b^3}\\ \end{align*}
Mathematica [A] time = 0.036883, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{a}{a x^2+b}+\frac{1}{x^2}\right )-2 a \log \left (a x^2+b\right )+4 a \log (x)}{2 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 46, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{a}{2\,{b}^{2} \left ( a{x}^{2}+b \right ) }}-2\,{\frac{a\ln \left ( x \right ) }{{b}^{3}}}+{\frac{a\ln \left ( a{x}^{2}+b \right ) }{{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966648, size = 70, normalized size = 1.43 \begin{align*} -\frac{2 \, a x^{2} + b}{2 \,{\left (a b^{2} x^{4} + b^{3} x^{2}\right )}} + \frac{a \log \left (a x^{2} + b\right )}{b^{3}} - \frac{a \log \left (x^{2}\right )}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49994, size = 157, normalized size = 3.2 \begin{align*} -\frac{2 \, a b x^{2} + b^{2} - 2 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (a x^{2} + b\right ) + 4 \,{\left (a^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a b^{3} x^{4} + b^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.746399, size = 49, normalized size = 1. \begin{align*} - \frac{2 a \log{\left (x \right )}}{b^{3}} + \frac{a \log{\left (x^{2} + \frac{b}{a} \right )}}{b^{3}} - \frac{2 a x^{2} + b}{2 a b^{2} x^{4} + 2 b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14958, size = 69, normalized size = 1.41 \begin{align*} -\frac{a \log \left (x^{2}\right )}{b^{3}} + \frac{a \log \left ({\left | a x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, a x^{2} + b}{2 \,{\left (a x^{4} + b x^{2}\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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