Optimal. Leaf size=52 \[ \frac{b}{2 a^2 \left (a-b x^2\right )}-\frac{b \log \left (a-b x^2\right )}{a^3}+\frac{2 b \log (x)}{a^3}-\frac{1}{2 a^2 x^2} \]
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Rubi [A] time = 0.0382507, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {266, 44} \[ \frac{b}{2 a^2 \left (a-b x^2\right )}-\frac{b \log \left (a-b x^2\right )}{a^3}+\frac{2 b \log (x)}{a^3}-\frac{1}{2 a^2 x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a-b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a-b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}+\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a-b x)^2}+\frac{2 b^2}{a^3 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^2 x^2}+\frac{b}{2 a^2 \left (a-b x^2\right )}+\frac{2 b \log (x)}{a^3}-\frac{b \log \left (a-b x^2\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.030803, size = 44, normalized size = 0.85 \[ \frac{\frac{a b}{a-b x^2}-2 b \log \left (a-b x^2\right )-\frac{a}{x^2}+4 b \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 51, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}-a \right ) }}-{\frac{b\ln \left ( b{x}^{2}-a \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.9092, size = 77, normalized size = 1.48 \begin{align*} -\frac{2 \, b x^{2} - a}{2 \,{\left (a^{2} b x^{4} - a^{3} x^{2}\right )}} - \frac{b \log \left (b x^{2} - a\right )}{a^{3}} + \frac{b \log \left (x^{2}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23887, size = 157, normalized size = 3.02 \begin{align*} -\frac{2 \, a b x^{2} - a^{2} + 2 \,{\left (b^{2} x^{4} - a b x^{2}\right )} \log \left (b x^{2} - a\right ) - 4 \,{\left (b^{2} x^{4} - a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} - a^{4} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.592225, size = 49, normalized size = 0.94 \begin{align*} - \frac{- a + 2 b x^{2}}{- 2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac{2 b \log{\left (x \right )}}{a^{3}} - \frac{b \log{\left (- \frac{a}{b} + x^{2} \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.86492, size = 76, normalized size = 1.46 \begin{align*} \frac{b \log \left (x^{2}\right )}{a^{3}} - \frac{b \log \left ({\left | b x^{2} - a \right |}\right )}{a^{3}} - \frac{2 \, b x^{2} - a}{2 \,{\left (b x^{4} - a x^{2}\right )} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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