Optimal. Leaf size=27 \[ \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {272, 45}
\begin {gather*} \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^3}{a+b x^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x}{a+b x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 27, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 b}-\frac {a \log \left (a+b x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 24, normalized size = 0.89
method | result | size |
default | \(\frac {x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(24\) |
norman | \(\frac {x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(24\) |
risch | \(\frac {x^{2}}{2 b}-\frac {a \ln \left (b \,x^{2}+a \right )}{2 b^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 23, normalized size = 0.85 \begin {gather*} \frac {x^{2}}{2 \, b} - \frac {a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.28, size = 22, normalized size = 0.81 \begin {gather*} \frac {b x^{2} - a \log \left (b x^{2} + a\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 20, normalized size = 0.74 \begin {gather*} - \frac {a \log {\left (a + b x^{2} \right )}}{2 b^{2}} + \frac {x^{2}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 24, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 \, b} - \frac {a \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 22, normalized size = 0.81 \begin {gather*} -\frac {a\,\ln \left (b\,x^2+a\right )-b\,x^2}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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