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 = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1593
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*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {21}{2 \left (x^2+5\right )}+2 \log \left (x^2+5\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+4 x^3}{\left (5+x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.80, size = 24, normalized size = 1.20 \begin {gather*} \frac {4 \, {\left (x^{2} + 5\right )} \log \left (x^{2} + 5\right ) + 21}{2 \, {\left (x^{2} + 5\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 25, normalized size = 1.25 \begin {gather*} -\frac {4 \, x^{2} - 1}{2 \, {\left (x^{2} + 5\right )}} + 2 \, \log \left (x^{2} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 19, normalized size = 0.95 \begin {gather*} 2 \ln \left (x^{2}+5\right )+\frac {21}{2 \left (x^{2}+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 18, normalized size = 0.90 \begin {gather*} \frac {21}{2 \, {\left (x^{2} + 5\right )}} + 2 \, \log \left (x^{2} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 20, normalized size = 1.00 \begin {gather*} 2\,\ln \left (x^2+5\right )+\frac {21}{2\,\left (x^2+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 15, normalized size = 0.75 \begin {gather*} 2 \log {\left (x^{2} + 5 \right )} + \frac {21}{2 x^{2} + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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