Integrand size = 25, antiderivative size = 199 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {p x^2}{2 g^2}+\frac {e f^2 p \log \left (d+e x^2\right )}{2 g^3 (e f-d g)}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {e f^2 p \log \left (f+g x^2\right )}{2 g^3 (e f-d g)}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}-\frac {f p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{g^3} \]
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Time = 0.19 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2525, 45, 2463, 2436, 2332, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {e f^2 p \log \left (d+e x^2\right )}{2 g^3 (e f-d g)}-\frac {e f^2 p \log \left (f+g x^2\right )}{2 g^3 (e f-d g)}-\frac {f p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{g^3}-\frac {p x^2}{2 g^2} \]
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Rule 31
Rule 36
Rule 45
Rule 2332
Rule 2436
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2463
Rule 2525
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^2 \log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {\log \left (c (d+e x)^p\right )}{g^2}+\frac {f^2 \log \left (c (d+e x)^p\right )}{g^2 (f+g x)^2}-\frac {2 f \log \left (c (d+e x)^p\right )}{g^2 (f+g x)}\right ) \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g^2}-\frac {f \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{g^2}+\frac {f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{(f+g x)^2} \, dx,x,x^2\right )}{2 g^2} \\ & = -\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}+\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e g^2}+\frac {(e f p) \text {Subst}\left (\int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^2\right )}{g^3}+\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {1}{(d+e x) (f+g x)} \, dx,x,x^2\right )}{2 g^3} \\ & = -\frac {p x^2}{2 g^2}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}+\frac {(f p) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{g^3}+\frac {\left (e^2 f^2 p\right ) \text {Subst}\left (\int \frac {1}{d+e x} \, dx,x,x^2\right )}{2 g^3 (e f-d g)}-\frac {\left (e f^2 p\right ) \text {Subst}\left (\int \frac {1}{f+g x} \, dx,x,x^2\right )}{2 g^2 (e f-d g)} \\ & = -\frac {p x^2}{2 g^2}+\frac {e f^2 p \log \left (d+e x^2\right )}{2 g^3 (e f-d g)}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {e f^2 p \log \left (f+g x^2\right )}{2 g^3 (e f-d g)}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}-\frac {f p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{g^3} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {p x^2-\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}+\frac {e f^2 p \left (\log \left (d+e x^2\right )-\log \left (f+g x^2\right )\right )}{g (-e f+d g)}+\frac {2 f \left (\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )\right )}{g}}{2 g^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.09 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.22
method | result | size |
parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g^{2}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 g^{3} \left (g \,x^{2}+f \right )}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{g^{3}}-p e \left (-\frac {f \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{g^{3} e}-\frac {-\frac {g \,x^{2}}{2 e}+\frac {\left (g^{2} d^{2}-d e f g -e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 \left (d g -e f \right ) e^{2}}+\frac {f^{2} \ln \left (g \,x^{2}+f \right )}{2 d g -2 e f}}{g^{3}}\right )\) | \(442\) |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g^{2}}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{2 g^{3} \left (g \,x^{2}+f \right )}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{g^{3}}-\frac {p \,x^{2}}{2 g^{2}}+\frac {p \ln \left (e \,x^{2}+d \right ) d^{2}}{2 e g \left (d g -e f \right )}-\frac {p \ln \left (e \,x^{2}+d \right ) d f}{2 g^{2} \left (d g -e f \right )}-\frac {p e \ln \left (e \,x^{2}+d \right ) f^{2}}{2 g^{3} \left (d g -e f \right )}+\frac {p e \,f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3} \left (d g -e f \right )}+\frac {p f \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{g^{3}}+\left (\frac {i \pi \,\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}}{2}-\frac {i \pi \,\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 )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {x^{2}}{2 g^{2}}-\frac {f \left (\frac {f}{g \left (g \,x^{2}+f \right )}+\frac {2 \ln \left (g \,x^{2}+f \right )}{g}\right )}{2 g^{2}}\right )\) | \(627\) |
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^5\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]
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