Optimal. Leaf size=49 \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]
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Rubi [A] time = 0.0352087, 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, 43} \[ -\frac{b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac{b}{a^3 \left (a x^2+b\right )}+\frac{\log \left (a x^2+b\right )}{2 a^3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3 x} \, dx &=\int \frac{x^5}{\left (b+a x^2\right )^3} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(b+a x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{a^2 (b+a x)^3}-\frac{2 b}{a^2 (b+a x)^2}+\frac{1}{a^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{4 a^3 \left (b+a x^2\right )^2}+\frac{b}{a^3 \left (b+a x^2\right )}+\frac{\log \left (b+a x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0148355, size = 39, normalized size = 0.8 \[ \frac{\frac{b \left (4 a x^2+3 b\right )}{\left (a x^2+b\right )^2}+2 \log \left (a x^2+b\right )}{4 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{b}^{2}}{4\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{b}{{a}^{3} \left ( a{x}^{2}+b \right ) }}+{\frac{\ln \left ( a{x}^{2}+b \right ) }{2\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02658, size = 74, normalized size = 1.51 \begin{align*} \frac{4 \, a b x^{2} + 3 \, b^{2}}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} + \frac{\log \left (a x^{2} + b\right )}{2 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41498, size = 143, normalized size = 2.92 \begin{align*} \frac{4 \, a b x^{2} + 3 \, b^{2} + 2 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{4 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.59982, size = 53, normalized size = 1.08 \begin{align*} \frac{4 a b x^{2} + 3 b^{2}}{4 a^{5} x^{4} + 8 a^{4} b x^{2} + 4 a^{3} b^{2}} + \frac{\log{\left (a x^{2} + b \right )}}{2 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15909, size = 57, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} - \frac{3 \, a x^{4} + 2 \, b x^{2}}{4 \,{\left (a x^{2} + b\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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