Integrand size = 23, antiderivative size = 83 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}-\frac {e p \log \left (f+g x^2\right )}{2 g (e f-d g)} \]
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Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2525, 2442, 36, 31} \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}+\frac {e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac {e p \log \left (f+g x^2\right )}{2 g (e f-d g)} \]
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Rule 31
Rule 36
Rule 2442
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right ) \\ & = -\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}+\frac {(e p) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^2\right )}{2 g} \\ & = -\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}-\frac {(e p) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^2\right )}{2 (e f-d g)}+\frac {\left (e^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 g (e f-d g)} \\ & = \frac {e p \log \left (d+e x^2\right )}{2 g (e f-d g)}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{2 g \left (f+g x^2\right )}-\frac {e p \log \left (f+g x^2\right )}{2 g (e f-d g)} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.88 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {e p \log \left (d+e x^2\right )}{e f-d g}-\frac {\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2}+\frac {e p \log \left (f+g x^2\right )}{-e f+d g}}{2 g} \]
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Time = 1.53 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90
method | result | size |
parts | \(-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 g \left (g \,x^{2}+f \right )}+\frac {p e \left (-\frac {\ln \left (e \,x^{2}+d \right )}{2 \left (d g -e f \right )}+\frac {\ln \left (g \,x^{2}+f \right )}{2 d g -2 e f}\right )}{g}\) | \(75\) |
parallelrisch | \(-\frac {\ln \left (e \,x^{2}+d \right ) x^{2} e^{2} g p -\ln \left (g \,x^{2}+f \right ) x^{2} e^{2} g p +\ln \left (e \,x^{2}+d \right ) e^{2} f p -\ln \left (g \,x^{2}+f \right ) e^{2} f p +\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) d e g -\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) e^{2} f}{2 \left (d g -e f \right ) \left (g \,x^{2}+f \right ) e g}\) | \(127\) |
risch | \(-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right )}{2 g \left (g \,x^{2}+f \right )}-\frac {i \pi d g \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}-i \pi d g \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3} d g +i \pi d g {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-i \pi e f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}+i \pi e f \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+i \pi e f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}-i \pi e f {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-2 \ln \left (g \,x^{2}+f \right ) e g p \,x^{2}+2 \ln \left (-e \,x^{2}-d \right ) e g p \,x^{2}-2 \ln \left (g \,x^{2}+f \right ) e f p +2 \ln \left (-e \,x^{2}-d \right ) e f p +2 \ln \left (c \right ) d g -2 \ln \left (c \right ) e f}{4 g \left (g \,x^{2}+f \right ) \left (d g -e f \right )}\) | \(371\) |
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Time = 0.34 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.10 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {{\left (e g p x^{2} + d g p\right )} \log \left (e x^{2} + d\right ) - {\left (e g p x^{2} + e f p\right )} \log \left (g x^{2} + f\right ) - {\left (e f - d g\right )} \log \left (c\right )}{2 \, {\left (e f^{2} g - d f g^{2} + {\left (e f g^{2} - d g^{3}\right )} x^{2}\right )}} \]
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Timed out. \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.89 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\frac {e p {\left (\frac {\log \left (e x^{2} + d\right )}{e f - d g} - \frac {\log \left (g x^{2} + f\right )}{e f - d g}\right )}}{2 \, g} - \frac {\log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{2 \, {\left (g x^{2} + f\right )} g} \]
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Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.52 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {e p \log \left (e x^{2} + d\right )}{2 \, {\left (e f g + {\left (e x^{2} + d\right )} g^{2} - d g^{2}\right )}} + \frac {e p \log \left (e x^{2} + d\right )}{2 \, {\left (e f g - d g^{2}\right )}} - \frac {e p \log \left (e f + {\left (e x^{2} + d\right )} g - d g\right )}{2 \, {\left (e f g - d g^{2}\right )}} - \frac {e \log \left (c\right )}{2 \, {\left (e f g + {\left (e x^{2} + d\right )} g^{2} - d g^{2}\right )}} \]
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Time = 2.63 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{2\,g\,\left (g\,x^2+f\right )}-\frac {e\,p\,\mathrm {atan}\left (\frac {x^2\,\left (d\,g\,1{}\mathrm {i}-e\,f\,1{}\mathrm {i}\right )}{2\,d\,f+d\,g\,x^2+e\,f\,x^2}\right )\,1{}\mathrm {i}}{d\,g^2-e\,f\,g} \]
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