Optimal. Leaf size=86 \[ \frac{3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac{b^2}{4 a^3 \left (a+b x^2\right )^2}-\frac{3 b^2 \log \left (a+b x^2\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b}{2 a^4 x^2}-\frac{1}{4 a^3 x^4} \]
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Rubi [A] time = 0.0600754, antiderivative size = 86, 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, 44} \[ \frac{3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac{b^2}{4 a^3 \left (a+b x^2\right )^2}-\frac{3 b^2 \log \left (a+b x^2\right )}{a^5}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b}{2 a^4 x^2}-\frac{1}{4 a^3 x^4} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^3}-\frac{3 b}{a^4 x^2}+\frac{6 b^2}{a^5 x}-\frac{b^3}{a^3 (a+b x)^3}-\frac{3 b^3}{a^4 (a+b x)^2}-\frac{6 b^3}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a^3 x^4}+\frac{3 b}{2 a^4 x^2}+\frac{b^2}{4 a^3 \left (a+b x^2\right )^2}+\frac{3 b^2}{2 a^4 \left (a+b x^2\right )}+\frac{6 b^2 \log (x)}{a^5}-\frac{3 b^2 \log \left (a+b x^2\right )}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0470941, size = 74, normalized size = 0.86 \[ \frac{\frac{a \left (4 a^2 b x^2-a^3+18 a b^2 x^4+12 b^3 x^6\right )}{x^4 \left (a+b x^2\right )^2}-12 b^2 \log \left (a+b x^2\right )+24 b^2 \log (x)}{4 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 79, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{3}{x}^{4}}}+{\frac{3\,b}{2\,{a}^{4}{x}^{2}}}+{\frac{{b}^{2}}{4\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,{b}^{2}}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+6\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{5}}}-3\,{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.15005, size = 124, normalized size = 1.44 \begin{align*} \frac{12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \,{\left (a^{4} b^{2} x^{8} + 2 \, a^{5} b x^{6} + a^{6} x^{4}\right )}} - \frac{3 \, b^{2} \log \left (b x^{2} + a\right )}{a^{5}} + \frac{3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27323, size = 274, normalized size = 3.19 \begin{align*} \frac{12 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} - a^{4} - 12 \,{\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left (b^{4} x^{8} + 2 \, a b^{3} x^{6} + a^{2} b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{5} b^{2} x^{8} + 2 \, a^{6} b x^{6} + a^{7} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00023, size = 90, normalized size = 1.05 \begin{align*} \frac{- a^{3} + 4 a^{2} b x^{2} + 18 a b^{2} x^{4} + 12 b^{3} x^{6}}{4 a^{6} x^{4} + 8 a^{5} b x^{6} + 4 a^{4} b^{2} x^{8}} + \frac{6 b^{2} \log{\left (x \right )}}{a^{5}} - \frac{3 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.20303, size = 108, normalized size = 1.26 \begin{align*} \frac{3 \, b^{2} \log \left (x^{2}\right )}{a^{5}} - \frac{3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac{12 \, b^{3} x^{6} + 18 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - a^{3}}{4 \,{\left (b x^{4} + a x^{2}\right )}^{2} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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