Optimal. Leaf size=63 \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]
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Rubi [A] time = 0.0346818, antiderivative size = 63, 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} \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^7 \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a x^4}-\frac{b}{a^2 x^3}+\frac{b^2}{a^3 x^2}-\frac{b^3}{a^4 x}+\frac{b^4}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{6 a x^6}+\frac{b}{4 a^2 x^4}-\frac{b^2}{2 a^3 x^2}-\frac{b^3 \log (x)}{a^4}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0068349, size = 63, normalized size = 1. \[ -\frac{b^2}{2 a^3 x^2}+\frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 56, normalized size = 0.9 \begin{align*} -{\frac{1}{6\,a{x}^{6}}}+{\frac{b}{4\,{a}^{2}{x}^{4}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\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 = 2.08265, size = 78, normalized size = 1.24 \begin{align*} \frac{b^{3} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac{b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{6 \, b^{2} x^{4} - 3 \, a b x^{2} + 2 \, a^{2}}{12 \, a^{3} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27312, size = 134, normalized size = 2.13 \begin{align*} \frac{6 \, b^{3} x^{6} \log \left (b x^{2} + a\right ) - 12 \, b^{3} x^{6} \log \left (x\right ) - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.538223, size = 56, normalized size = 0.89 \begin{align*} - \frac{2 a^{2} - 3 a b x^{2} + 6 b^{2} x^{4}}{12 a^{3} x^{6}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \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 = 2.87401, size = 95, normalized size = 1.51 \begin{align*} -\frac{b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{11 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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