Optimal. Leaf size=70 \[ \frac{1}{2 a^3 \left (a+b x^2\right )}+\frac{1}{4 a^2 \left (a+b x^2\right )^2}-\frac{\log \left (a+b x^2\right )}{2 a^4}+\frac{\log (x)}{a^4}+\frac{1}{6 a \left (a+b x^2\right )^3} \]
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Rubi [A] time = 0.0758043, antiderivative size = 70, 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} \[ \frac{1}{2 a^3 \left (a+b x^2\right )}+\frac{1}{4 a^2 \left (a+b x^2\right )^2}-\frac{\log \left (a+b x^2\right )}{2 a^4}+\frac{\log (x)}{a^4}+\frac{1}{6 a \left (a+b x^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{x \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \frac{1}{x \left (a b+b^2 x\right )^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \left (\frac{1}{a^4 b^4 x}-\frac{1}{a b^3 (a+b x)^4}-\frac{1}{a^2 b^3 (a+b x)^3}-\frac{1}{a^3 b^3 (a+b x)^2}-\frac{1}{a^4 b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{6 a \left (a+b x^2\right )^3}+\frac{1}{4 a^2 \left (a+b x^2\right )^2}+\frac{1}{2 a^3 \left (a+b x^2\right )}+\frac{\log (x)}{a^4}-\frac{\log \left (a+b x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0355674, size = 54, normalized size = 0.77 \[ \frac{\frac{a \left (11 a^2+15 a b x^2+6 b^2 x^4\right )}{\left (a+b x^2\right )^3}-6 \log \left (a+b x^2\right )+12 \log (x)}{12 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.054, size = 63, normalized size = 0.9 \begin{align*}{\frac{1}{6\,a \left ( b{x}^{2}+a \right ) ^{3}}}+{\frac{1}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{1}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( x \right ) }{{a}^{4}}}-{\frac{\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 = 0.997209, size = 111, normalized size = 1.59 \begin{align*} \frac{6 \, b^{2} x^{4} + 15 \, a b x^{2} + 11 \, a^{2}}{12 \,{\left (a^{3} b^{3} x^{6} + 3 \, a^{4} b^{2} x^{4} + 3 \, a^{5} b x^{2} + a^{6}\right )}} - \frac{\log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{\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 [B] time = 1.92034, size = 288, normalized size = 4.11 \begin{align*} \frac{6 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 11 \, a^{3} - 6 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right ) + 12 \,{\left (b^{3} x^{6} + 3 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} + a^{3}\right )} \log \left (x\right )}{12 \,{\left (a^{4} b^{3} x^{6} + 3 \, a^{5} b^{2} x^{4} + 3 \, a^{6} b x^{2} + a^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.811277, size = 80, normalized size = 1.14 \begin{align*} \frac{11 a^{2} + 15 a b x^{2} + 6 b^{2} x^{4}}{12 a^{6} + 36 a^{5} b x^{2} + 36 a^{4} b^{2} x^{4} + 12 a^{3} b^{3} x^{6}} + \frac{\log{\left (x \right )}}{a^{4}} - \frac{\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 = 1.14283, size = 95, normalized size = 1.36 \begin{align*} \frac{\log \left (x^{2}\right )}{2 \, a^{4}} - \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{11 \, b^{3} x^{6} + 39 \, a b^{2} x^{4} + 48 \, a^{2} b x^{2} + 22 \, a^{3}}{12 \,{\left (b x^{2} + a\right )}^{3} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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