Optimal. Leaf size=20 \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]
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Rubi [A] time = 0.0230683, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1593, 444, 43} \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{-x+4 x^3}{\left (5+x^2\right )^2} \, dx &=\int \frac{x \left (-1+4 x^2\right )}{\left (5+x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+4 x}{(5+x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{21}{(5+x)^2}+\frac{4}{5+x}\right ) \, dx,x,x^2\right )\\ &=\frac{21}{2 \left (5+x^2\right )}+2 \log \left (5+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0071059, size = 20, normalized size = 1. \[ \frac{21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 19, normalized size = 1. \begin{align*}{\frac{21}{2\,{x}^{2}+10}}+2\,\ln \left ({x}^{2}+5 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987198, size = 24, normalized size = 1.2 \begin{align*} \frac{21}{2 \,{\left (x^{2} + 5\right )}} + 2 \, \log \left (x^{2} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39457, size = 63, normalized size = 3.15 \begin{align*} \frac{4 \,{\left (x^{2} + 5\right )} \log \left (x^{2} + 5\right ) + 21}{2 \,{\left (x^{2} + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.092506, size = 15, normalized size = 0.75 \begin{align*} 2 \log{\left (x^{2} + 5 \right )} + \frac{21}{2 x^{2} + 10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15171, size = 34, normalized size = 1.7 \begin{align*} -\frac{4 \, x^{2} - 1}{2 \,{\left (x^{2} + 5\right )}} + 2 \, \log \left (x^{2} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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