Optimal. Leaf size=66 \[ \frac{b^2}{2 a^3 \left (a+b x^2\right )}-\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4} \]
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Rubi [A] time = 0.0537842, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 44} \[ \frac{b^2}{2 a^3 \left (a+b x^2\right )}-\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b}{a^3 x^2}-\frac{1}{4 a^2 x^4} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^5 \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx &=b^2 \int \frac{1}{x^5 \left (a b+b^2 x^2\right )^2} \, dx\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \frac{1}{x^3 \left (a b+b^2 x\right )^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 b^2 x^3}-\frac{2}{a^3 b x^2}+\frac{3}{a^4 x}-\frac{b}{a^3 (a+b x)^2}-\frac{3 b}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a^2 x^4}+\frac{b}{a^3 x^2}+\frac{b^2}{2 a^3 \left (a+b x^2\right )}+\frac{3 b^2 \log (x)}{a^4}-\frac{3 b^2 \log \left (a+b x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0497941, size = 57, normalized size = 0.86 \[ \frac{a \left (\frac{2 b^2}{a+b x^2}-\frac{a}{x^4}+\frac{4 b}{x^2}\right )-6 b^2 \log \left (a+b x^2\right )+12 b^2 \log (x)}{4 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.053, size = 61, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,{a}^{2}{x}^{4}}}+{\frac{b}{{x}^{2}{a}^{3}}}+{\frac{{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.984186, size = 95, normalized size = 1.44 \begin{align*} \frac{6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac{3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77421, size = 184, normalized size = 2.79 \begin{align*} \frac{6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.673053, size = 68, normalized size = 1.03 \begin{align*} \frac{- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac{3 b^{2} \log{\left (x \right )}}{a^{4}} - \frac{3 b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12036, size = 116, normalized size = 1.76 \begin{align*} \frac{3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \,{\left (b x^{2} + a\right )} a^{4}} - \frac{9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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