Optimal. Leaf size=84 \[ -\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3}+\frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{1}{2 a^4 x^2} \]
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Rubi [A] time = 0.0856393, antiderivative size = 84, 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} \[ -\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3}+\frac{2 b \log \left (a+b x^2\right )}{a^5}-\frac{4 b \log (x)}{a^5}-\frac{1}{2 a^4 x^2} \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx &=b^4 \int \frac{1}{x^3 \left (a b+b^2 x^2\right )^4} \, dx\\ &=\frac{1}{2} b^4 \operatorname{Subst}\left (\int \frac{1}{x^2 \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^2}-\frac{4}{a^5 b^3 x}+\frac{1}{a^2 b^2 (a+b x)^4}+\frac{2}{a^3 b^2 (a+b x)^3}+\frac{3}{a^4 b^2 (a+b x)^2}+\frac{4}{a^5 b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^4 x^2}-\frac{b}{6 a^2 \left (a+b x^2\right )^3}-\frac{b}{2 a^3 \left (a+b x^2\right )^2}-\frac{3 b}{2 a^4 \left (a+b x^2\right )}-\frac{4 b \log (x)}{a^5}+\frac{2 b \log \left (a+b x^2\right )}{a^5}\\ \end{align*}
Mathematica [A] time = 0.0642186, size = 70, normalized size = 0.83 \[ -\frac{\frac{a \left (22 a^2 b x^2+3 a^3+30 a b^2 x^4+12 b^3 x^6\right )}{x^2 \left (a+b x^2\right )^3}-12 b \log \left (a+b x^2\right )+24 b \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 77, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{a}^{4}{x}^{2}}}-{\frac{b}{6\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{3}}}-{\frac{b}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,b}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}-4\,{\frac{b\ln \left ( x \right ) }{{a}^{5}}}+2\,{\frac{b\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 = 1.01153, size = 134, normalized size = 1.6 \begin{align*} -\frac{12 \, b^{3} x^{6} + 30 \, a b^{2} x^{4} + 22 \, a^{2} b x^{2} + 3 \, a^{3}}{6 \,{\left (a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{6} + 3 \, a^{6} b x^{4} + a^{7} x^{2}\right )}} + \frac{2 \, b \log \left (b x^{2} + a\right )}{a^{5}} - \frac{2 \, b \log \left (x^{2}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.78155, size = 339, normalized size = 4.04 \begin{align*} -\frac{12 \, a b^{3} x^{6} + 30 \, a^{2} b^{2} x^{4} + 22 \, a^{3} b x^{2} + 3 \, a^{4} - 12 \,{\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left (b^{4} x^{8} + 3 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )} \log \left (x\right )}{6 \,{\left (a^{5} b^{3} x^{8} + 3 \, a^{6} b^{2} x^{6} + 3 \, a^{7} b x^{4} + a^{8} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27347, size = 100, normalized size = 1.19 \begin{align*} - \frac{3 a^{3} + 22 a^{2} b x^{2} + 30 a b^{2} x^{4} + 12 b^{3} x^{6}}{6 a^{7} x^{2} + 18 a^{6} b x^{4} + 18 a^{5} b^{2} x^{6} + 6 a^{4} b^{3} x^{8}} - \frac{4 b \log{\left (x \right )}}{a^{5}} + \frac{2 b \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 = 1.13287, size = 126, normalized size = 1.5 \begin{align*} -\frac{2 \, b \log \left (x^{2}\right )}{a^{5}} + \frac{2 \, b \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} + \frac{4 \, b x^{2} - a}{2 \, a^{5} x^{2}} - \frac{22 \, b^{4} x^{6} + 75 \, a b^{3} x^{4} + 87 \, a^{2} b^{2} x^{2} + 35 \, a^{3} b}{6 \,{\left (b x^{2} + a\right )}^{3} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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