Integrand size = 14, antiderivative size = 37 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (b+a x^2\right )}{2 a} \]
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Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2505, 269, 266} \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (a x^2+b\right )}{2 a} \]
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Rule 266
Rule 269
Rule 2505
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(b p) \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x} \, dx \\ & = \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+(b p) \int \frac {x}{b+a x^2} \, dx \\ & = \frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log \left (b+a x^2\right )}{2 a} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {b p \log \left (a+\frac {b}{x^2}\right )}{2 a}+\frac {1}{2} x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b p \log (x)}{a} \]
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Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.92
method | result | size |
parts | \(\frac {x^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2}+\frac {b p \ln \left (x^{2} a +b \right )}{2 a}\) | \(34\) |
parallelrisch | \(-\frac {-x^{2} \ln \left (c \left (\frac {x^{2} a +b}{x^{2}}\right )^{p}\right ) a b p -2 \ln \left (x \right ) b^{2} p^{2}-\ln \left (c \left (\frac {x^{2} a +b}{x^{2}}\right )^{p}\right ) b^{2} p}{2 a b p}\) | \(69\) |
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Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {a p x^{2} \log \left (\frac {a x^{2} + b}{x^{2}}\right ) + a x^{2} \log \left (c\right ) + b p \log \left (a x^{2} + b\right )}{2 \, a} \]
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Time = 0.59 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.43 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\begin {cases} \frac {x^{2} \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{2} + \frac {b p \log {\left (a x^{2} + b \right )}}{2 a} & \text {for}\: a \neq 0 \\\frac {p x^{2}}{2} + \frac {x^{2} \log {\left (c \left (\frac {b}{x^{2}}\right )^{p} \right )}}{2} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {1}{2} \, x^{2} \log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right ) + \frac {b p \log \left (a x^{2} + b\right )}{2 \, a} \]
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Time = 0.29 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.27 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {1}{2} \, p x^{2} \log \left (a x^{2} + b\right ) - \frac {1}{2} \, p x^{2} \log \left (x^{2}\right ) + \frac {1}{2} \, x^{2} \log \left (c\right ) + \frac {b p \log \left (a x^{2} + b\right )}{2 \, a} \]
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Time = 1.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.89 \[ \int x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \, dx=\frac {x^2\,\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{2}+\frac {b\,p\,\ln \left (a\,x^2+b\right )}{2\,a} \]
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