Optimal. Leaf size=38 \[ \frac {1}{2 b \left (b+a x^2\right )}+\frac {\log (x)}{b^2}-\frac {\log \left (b+a x^2\right )}{2 b^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 38, 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 {\log \left (a x^2+b\right )}{2 b^2}+\frac {1}{2 b \left (a x^2+b\right )}+\frac {\log (x)}{b^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 )^2 x^5} \, dx &=\int \frac {1}{x \left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2 x}-\frac {a}{b (b+a x)^2}-\frac {a}{b^2 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 b \left (b+a x^2\right )}+\frac {\log (x)}{b^2}-\frac {\log \left (b+a x^2\right )}{2 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 33, normalized size = 0.87 \begin {gather*} \frac {\frac {b}{b+a x^2}+2 \log (x)-\log \left (b+a x^2\right )}{2 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 42, normalized size = 1.11
method | result | size |
risch | \(\frac {1}{2 b \left (a \,x^{2}+b \right )}+\frac {\ln \left (x \right )}{b^{2}}-\frac {\ln \left (a \,x^{2}+b \right )}{2 b^{2}}\) | \(35\) |
norman | \(-\frac {a \,x^{2}}{2 b^{2} \left (a \,x^{2}+b \right )}+\frac {\ln \left (x \right )}{b^{2}}-\frac {\ln \left (a \,x^{2}+b \right )}{2 b^{2}}\) | \(39\) |
default | \(-\frac {a \left (-\frac {b}{a \left (a \,x^{2}+b \right )}+\frac {\ln \left (a \,x^{2}+b \right )}{a}\right )}{2 b^{2}}+\frac {\ln \left (x \right )}{b^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 37, normalized size = 0.97 \begin {gather*} \frac {1}{2 \, {\left (a b x^{2} + b^{2}\right )}} - \frac {\log \left (a x^{2} + b\right )}{2 \, b^{2}} + \frac {\log \left (x^{2}\right )}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 47, normalized size = 1.24 \begin {gather*} -\frac {{\left (a x^{2} + b\right )} \log \left (a x^{2} + b\right ) - 2 \, {\left (a x^{2} + b\right )} \log \left (x\right ) - b}{2 \, {\left (a b^{2} x^{2} + b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 34, normalized size = 0.89 \begin {gather*} \frac {1}{2 a b x^{2} + 2 b^{2}} + \frac {\log {\left (x \right )}}{b^{2}} - \frac {\log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 47, normalized size = 1.24 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, b^{2}} - \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac {a x^{2} + 2 \, b}{2 \, {\left (a x^{2} + b\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 34, normalized size = 0.89 \begin {gather*} \frac {\ln \left (x\right )}{b^2}+\frac {1}{2\,b\,\left (a\,x^2+b\right )}-\frac {\ln \left (a\,x^2+b\right )}{2\,b^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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