Integrand size = 13, antiderivative size = 49 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=-\frac {b^2}{4 a^3 \left (b+a x^2\right )^2}+\frac {b}{a^3 \left (b+a x^2\right )}+\frac {\log \left (b+a x^2\right )}{2 a^3} \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 45} \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=-\frac {b^2}{4 a^3 \left (a x^2+b\right )^2}+\frac {b}{a^3 \left (a x^2+b\right )}+\frac {\log \left (a x^2+b\right )}{2 a^3} \]
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Rule 45
Rule 269
Rule 272
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^5}{\left (b+a x^2\right )^3} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {x^2}{(b+a x)^3} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {b^2}{a^2 (b+a x)^3}-\frac {2 b}{a^2 (b+a x)^2}+\frac {1}{a^2 (b+a x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b^2}{4 a^3 \left (b+a x^2\right )^2}+\frac {b}{a^3 \left (b+a x^2\right )}+\frac {\log \left (b+a x^2\right )}{2 a^3} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {\frac {b \left (3 b+4 a x^2\right )}{\left (b+a x^2\right )^2}+2 \log \left (b+a x^2\right )}{4 a^3} \]
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Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
norman | \(\frac {\frac {b \,x^{2}}{a^{2}}+\frac {3 b^{2}}{4 a^{3}}}{\left (a \,x^{2}+b \right )^{2}}+\frac {\ln \left (a \,x^{2}+b \right )}{2 a^{3}}\) | \(42\) |
risch | \(\frac {\frac {b \,x^{2}}{a^{2}}+\frac {3 b^{2}}{4 a^{3}}}{\left (a \,x^{2}+b \right )^{2}}+\frac {\ln \left (a \,x^{2}+b \right )}{2 a^{3}}\) | \(42\) |
default | \(-\frac {b^{2}}{4 a^{3} \left (a \,x^{2}+b \right )^{2}}+\frac {b}{a^{3} \left (a \,x^{2}+b \right )}+\frac {\ln \left (a \,x^{2}+b \right )}{2 a^{3}}\) | \(46\) |
parallelrisch | \(\frac {2 a^{2} \ln \left (a \,x^{2}+b \right ) x^{4}+4 \ln \left (a \,x^{2}+b \right ) x^{2} a b +4 a b \,x^{2}+2 b^{2} \ln \left (a \,x^{2}+b \right )+3 b^{2}}{4 a^{3} \left (a \,x^{2}+b \right )^{2}}\) | \(72\) |
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Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {4 \, a b x^{2} + 3 \, b^{2} + 2 \, {\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \log \left (a x^{2} + b\right )}{4 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {4 a b x^{2} + 3 b^{2}}{4 a^{5} x^{4} + 8 a^{4} b x^{2} + 4 a^{3} b^{2}} + \frac {\log {\left (a x^{2} + b \right )}}{2 a^{3}} \]
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Time = 0.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {4 \, a b x^{2} + 3 \, b^{2}}{4 \, {\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}} + \frac {\log \left (a x^{2} + b\right )}{2 \, a^{3}} \]
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Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, a^{3}} - \frac {3 \, a x^{4} + 2 \, b x^{2}}{4 \, {\left (a x^{2} + b\right )}^{2} a^{2}} \]
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Time = 6.07 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+\frac {b}{x^2}\right )^3 x} \, dx=\frac {\frac {3\,b^2}{4\,a^3}+\frac {b\,x^2}{a^2}}{a^2\,x^4+2\,a\,b\,x^2+b^2}+\frac {\ln \left (a\,x^2+b\right )}{2\,a^3} \]
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