Optimal. Leaf size=35 \[ -\frac {1}{2 b x^2}-\frac {a \log (x)}{b^2}+\frac {a \log \left (b+a x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 46}
\begin {gather*} \frac {a \log \left (a x^2+b\right )}{2 b^2}-\frac {a \log (x)}{b^2}-\frac {1}{2 b x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^5} \, dx &=\int \frac {1}{x^3 \left (b+a x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 (b+a x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b x^2}-\frac {a}{b^2 x}+\frac {a^2}{b^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{2 b x^2}-\frac {a \log (x)}{b^2}+\frac {a \log \left (b+a x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{2 b x^2}-\frac {a \log (x)}{b^2}+\frac {a \log \left (b+a x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 32, normalized size = 0.91
method | result | size |
default | \(-\frac {1}{2 x^{2} b}-\frac {a \ln \left (x \right )}{b^{2}}+\frac {a \ln \left (a \,x^{2}+b \right )}{2 b^{2}}\) | \(32\) |
norman | \(-\frac {1}{2 x^{2} b}-\frac {a \ln \left (x \right )}{b^{2}}+\frac {a \ln \left (a \,x^{2}+b \right )}{2 b^{2}}\) | \(32\) |
risch | \(-\frac {1}{2 x^{2} b}-\frac {a \ln \left (x \right )}{b^{2}}+\frac {a \ln \left (-a \,x^{2}-b \right )}{2 b^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 0.94 \begin {gather*} \frac {a \log \left (a x^{2} + b\right )}{2 \, b^{2}} - \frac {a \log \left (x^{2}\right )}{2 \, b^{2}} - \frac {1}{2 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 33, normalized size = 0.94 \begin {gather*} \frac {a x^{2} \log \left (a x^{2} + b\right ) - 2 \, a x^{2} \log \left (x\right ) - b}{2 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 31, normalized size = 0.89 \begin {gather*} - \frac {a \log {\left (x \right )}}{b^{2}} + \frac {a \log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{2}} - \frac {1}{2 b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.95, size = 43, normalized size = 1.23 \begin {gather*} -\frac {a \log \left (x^{2}\right )}{2 \, b^{2}} + \frac {a \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac {a x^{2} - b}{2 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 31, normalized size = 0.89 \begin {gather*} \frac {a\,\ln \left (a\,x^2+b\right )}{2\,b^2}-\frac {1}{2\,b\,x^2}-\frac {a\,\ln \left (x\right )}{b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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