Optimal. Leaf size=49 \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]
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Rubi [A] time = 0.0329381, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1584, 266, 44} \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a x+b x^3\right )} \, dx &=\int \frac{1}{x^5 \left (a+b x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{4 a x^4}+\frac{b}{2 a^2 x^2}+\frac{b^2 \log (x)}{a^3}-\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}\\ \end{align*}
Mathematica [A] time = 0.0066015, size = 49, normalized size = 1. \[ -\frac{b^2 \log \left (a+b x^2\right )}{2 a^3}+\frac{b^2 \log (x)}{a^3}+\frac{b}{2 a^2 x^2}-\frac{1}{4 a x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,a{x}^{4}}}+{\frac{b}{2\,{a}^{2}{x}^{2}}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07039, size = 59, normalized size = 1.2 \begin{align*} -\frac{b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{3}} + \frac{b^{2} \log \left (x\right )}{a^{3}} + \frac{2 \, b x^{2} - a}{4 \, a^{2} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43787, size = 108, normalized size = 2.2 \begin{align*} -\frac{2 \, b^{2} x^{4} \log \left (b x^{2} + a\right ) - 4 \, b^{2} x^{4} \log \left (x\right ) - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.453392, size = 42, normalized size = 0.86 \begin{align*} \frac{- a + 2 b x^{2}}{4 a^{2} x^{4}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{b^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21454, size = 77, normalized size = 1.57 \begin{align*} \frac{b^{2} \log \left (x^{2}\right )}{2 \, a^{3}} - \frac{b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3}} - \frac{3 \, b^{2} x^{4} - 2 \, a b x^{2} + a^{2}}{4 \, a^{3} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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