Optimal. Leaf size=15 \[ \frac {\log \left (a+b x^2\right )}{2 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1598, 266}
\begin {gather*} \frac {\log \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^2}{a x+b x^3} \, dx &=\int \frac {x}{a+b x^2} \, dx\\ &=\frac {\log \left (a+b x^2\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (a+b x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 14, normalized size = 0.93
method | result | size |
default | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
norman | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
risch | \(\frac {\ln \left (b \,x^{2}+a \right )}{2 b}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{2} + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.36, size = 13, normalized size = 0.87 \begin {gather*} \frac {\log \left (b x^{2} + a\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.67 \begin {gather*} \frac {\log {\left (a + b x^{2} \right )}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.15, size = 14, normalized size = 0.93 \begin {gather*} \frac {\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.92, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (b\,x^2+a\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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