Integrand size = 25, antiderivative size = 188 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {f p x^2}{2 g^2}+\frac {d p x^2}{4 e g}-\frac {p x^4}{8 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}+\frac {f^2 p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3} \]
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Time = 0.19 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2525, 45, 2463, 2436, 2332, 2442, 2441, 2440, 2438} \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {f^2 p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{2 g^3}+\frac {d p x^2}{4 e g}+\frac {f p x^2}{2 g^2}-\frac {p x^4}{8 g} \]
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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} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (-\frac {f \log \left (c (d+e x)^p\right )}{g^2}+\frac {x \log \left (c (d+e x)^p\right )}{g}+\frac {f^2 \log \left (c (d+e x)^p\right )}{g^2 (f+g x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {f \text {Subst}\left (\int \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g^2}+\frac {f^2 \text {Subst}\left (\int \frac {\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^2\right )}{2 g^2}+\frac {\text {Subst}\left (\int x \log \left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 g} \\ & = \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}+\frac {f^2 \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 )}{2 g^3}-\frac {f \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e g^2}-\frac {\left (e f^2 p\right ) \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 )}{2 g^3}-\frac {(e p) \text {Subst}\left (\int \frac {x^2}{d+e x} \, dx,x,x^2\right )}{4 g} \\ & = \frac {f p x^2}{2 g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \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 ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}-\frac {\left (f^2 p\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x^2\right )}{2 g^3}-\frac {(e p) \text {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x}{e}+\frac {d^2}{e^2 (d+e x)}\right ) \, dx,x,x^2\right )}{4 g} \\ & = \frac {f p x^2}{2 g^2}+\frac {d p x^2}{4 e g}-\frac {p x^4}{8 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \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 ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}+\frac {f^2 p \text {Li}_2\left (-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {e g p x^2 \left (4 e f+2 d g-e g x^2\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+e \log \left (c \left (d+e x^2\right )^p\right ) \left (2 g \left (-2 d f-2 e f x^2+e g x^4\right )+4 e f^2 \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )\right )+4 e^2 f^2 p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{8 e^2 g^3} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.48 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.18
method | result | size |
parts | \(\frac {x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 g}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \,x^{2}}{2 g^{2}}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}-p e \left (\frac {f^{2} \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 )}{2 g^{3} e}-\frac {-\frac {\frac {1}{2} e g \,x^{4}-d g \,x^{2}-2 f e \,x^{2}}{2 e^{2}}-\frac {d \left (d g +2 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}}{2 g^{2}}\right )\) | \(410\) |
risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{4}}{4 g}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \,x^{2}}{2 g^{2}}+\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}-\frac {p \,f^{2} \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 )}{2 g^{3}}-\frac {p \,x^{4}}{8 g}+\frac {d p \,x^{2}}{4 e g}+\frac {f p \,x^{2}}{2 g^{2}}-\frac {d^{2} p \ln \left (e \,x^{2}+d \right )}{4 e^{2} g}-\frac {p d \ln \left (e \,x^{2}+d \right ) f}{2 e \,g^{2}}+\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 {\frac {1}{2} g \,x^{4}-f \,x^{2}}{2 g^{2}}+\frac {f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}\right )\) | \(566\) |
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \]
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
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Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^5\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]
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