Integrand size = 22, antiderivative size = 22 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=-48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {208 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {48 \sqrt {d} f p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {64 d^{3/2} g p^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {24 i \sqrt {d} f p^3 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d}-i \sqrt {e} x}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \operatorname {PolyLog}\left (2,-\frac {\sqrt {d}-i \sqrt {e} x}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}-\frac {2 d (-3 e f+d g) p \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e} \]
[Out]
Not integrable
Time = 0.92 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (f \log ^3\left (c \left (d+e x^2\right )^p\right )+g x^2 \log ^3\left (c \left (d+e x^2\right )^p\right )\right ) \, dx \\ & = f \int \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+g \int x^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \\ & = f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 e f p) \int \frac {x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-(2 e g p) \int \frac {x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 e f p) \int \left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-(2 e g p) \int \left (-\frac {d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 f p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-(2 g p) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {(2 d g p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{e}-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e} \\ & = -6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 e f p^2\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\left (8 d g p^2\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac {1}{3} \left (8 e g p^2\right ) \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx \\ & = -6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 e f p^2\right ) \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-\left (8 d g p^2\right ) \int \left (\frac {\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx+\frac {1}{3} \left (8 e g p^2\right ) \int \left (-\frac {d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = -6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 f p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\left (24 d f p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac {1}{3} \left (8 g p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac {\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac {\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{e}+\frac {\left (8 d^2 g p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}+\frac {\left (8 d^2 g p^2\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e} \\ & = 24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\left (48 e f p^3\right ) \int \frac {x^2}{d+e x^2} \, dx+\left (48 d e f p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx+\frac {1}{3} \left (16 d g p^3\right ) \int \frac {x^2}{d+e x^2} \, dx+\left (16 d g p^3\right ) \int \frac {x^2}{d+e x^2} \, dx-\frac {1}{3} \left (16 d^2 g p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx-\left (16 d^2 g p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx-\frac {1}{9} \left (16 e g p^3\right ) \int \frac {x^4}{d+e x^2} \, dx \\ & = -48 f p^3 x+\frac {64 d g p^3 x}{3 e}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (48 d f p^3\right ) \int \frac {1}{d+e x^2} \, dx+\left (48 \sqrt {d} \sqrt {e} f p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx-\frac {\left (16 d^2 g p^3\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}-\frac {\left (16 d^2 g p^3\right ) \int \frac {1}{d+e x^2} \, dx}{e}-\frac {\left (16 d^{3/2} g p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{3 \sqrt {e}}-\frac {\left (16 d^{3/2} g p^3\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{\sqrt {e}}-\frac {1}{9} \left (16 e g p^3\right ) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx \\ & = -48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {64 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\left (48 f p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx+\frac {\left (16 d g p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{3 e}+\frac {\left (16 d g p^3\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{e}-\frac {\left (16 d^2 g p^3\right ) \int \frac {1}{d+e x^2} \, dx}{9 e} \\ & = -48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {208 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {64 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (48 f p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx-\frac {\left (16 d g p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx}{3 e}-\frac {\left (16 d g p^3\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx}{e} \\ & = -48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {208 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {64 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\frac {\left (48 i \sqrt {d} f p^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{\sqrt {e}}+\frac {\left (16 i d^{3/2} g p^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{3 e^{3/2}}+\frac {\left (16 i d^{3/2} g p^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{e^{3/2}} \\ & = -48 f p^3 x+\frac {208 d g p^3 x}{9 e}-\frac {16}{27} g p^3 x^3+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {208 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {64 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {24 i \sqrt {d} f p^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}+\frac {32 i d^{3/2} g p^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+(6 d f p) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {\left (2 d^2 g p\right ) \int \frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1772\) vs. \(2(683)=1366\).
Time = 9.07 (sec) , antiderivative size = 1772, normalized size of antiderivative = 80.55 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\frac {2 d g p x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{e}+\frac {6 \sqrt {d} f p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{\sqrt {e}}-\frac {2 d^{3/2} g p \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2}{e^{3/2}}+3 f p x \log \left (d+e x^2\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+g p x^3 \log \left (d+e x^2\right ) \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2+f x \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2 \left (-6 p-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )+\frac {1}{3} g x^3 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )^2 \left (-2 p-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right )+3 f p^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (x \log ^2\left (d+e x^2\right )-\frac {4 \left (-i \sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+\sqrt {e} x \left (-2+\log \left (d+e x^2\right )\right )-\sqrt {d} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2+2 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+\log \left (d+e x^2\right )\right )-i \sqrt {d} \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{\sqrt {e}}\right )+3 g p^2 \left (-p \log \left (d+e x^2\right )+\log \left (c \left (d+e x^2\right )^p\right )\right ) \left (\frac {1}{3} x^3 \log ^2\left (d+e x^2\right )-\frac {4 \left (9 i d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+3 d^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-8+6 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+3 \log \left (d+e x^2\right )\right )+\sqrt {e} x \left (24 d-2 e x^2+\left (-9 d+3 e x^2\right ) \log \left (d+e x^2\right )\right )+9 i d^{3/2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )\right )}{27 e^{3/2}}\right )+\frac {g p^3 \left (416 \sqrt {-d} d^{3/2} \sqrt {d+e x^2} \sqrt {1-\frac {d}{d+e x^2}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right )+36 \sqrt {-d} d^{3/2} \sqrt {1-\frac {d}{d+e x^2}} \left (8 \sqrt {d} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right )+4 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )+\sqrt {d+e x^2} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right ) \log ^2\left (d+e x^2\right )\right )-\frac {2}{3} \sqrt {-d} e x^2 \left (\left (d+e x^2\right ) \left (16-24 \log \left (d+e x^2\right )+18 \log ^2\left (d+e x^2\right )-9 \log ^3\left (d+e x^2\right )\right )+d \left (-640+312 \log \left (d+e x^2\right )-72 \log ^2\left (d+e x^2\right )+9 \log ^3\left (d+e x^2\right )\right )\right )-48 d^2 \left (4 \sqrt {e x^2} \text {arctanh}\left (\frac {\sqrt {e x^2}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^2\right )-\log \left (\frac {d+e x^2}{d}\right )\right )-\sqrt {-d} \sqrt {1-\frac {d+e x^2}{d}} \left (\log ^2\left (\frac {d+e x^2}{d}\right )-4 \log \left (\frac {d+e x^2}{d}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {d+e x^2}{d}}\right )\right )\right )\right )}{18 \sqrt {-d} e^2 x}+\frac {f p^3 \left (-48 \sqrt {-d^2} \sqrt {d+e x^2} \sqrt {1-\frac {d}{d+e x^2}} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right )-6 \sqrt {-d^2} \sqrt {1-\frac {d}{d+e x^2}} \left (8 \sqrt {d} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right )+4 \sqrt {d} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {d}{d+e x^2}\right ) \log \left (d+e x^2\right )+\sqrt {d+e x^2} \arcsin \left (\frac {\sqrt {d}}{\sqrt {d+e x^2}}\right ) \log ^2\left (d+e x^2\right )\right )+\sqrt {-d} e x^2 \left (-48+24 \log \left (d+e x^2\right )-6 \log ^2\left (d+e x^2\right )+\log ^3\left (d+e x^2\right )\right )+24 d \sqrt {e x^2} \text {arctanh}\left (\frac {\sqrt {e x^2}}{\sqrt {-d}}\right ) \left (\log \left (d+e x^2\right )-\log \left (\frac {d+e x^2}{d}\right )\right )+6 (-d)^{3/2} \sqrt {1-\frac {d+e x^2}{d}} \left (\log ^2\left (\frac {d+e x^2}{d}\right )-4 \log \left (\frac {d+e x^2}{d}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {1-\frac {d+e x^2}{d}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {d+e x^2}{d}}\right )\right )\right )}{\sqrt {-d} e x} \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
\[\int \left (g \,x^{2}+f \right ) {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}^{3}d x\]
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Not integrable
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 10.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int \left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]
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Exception generated. \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int { {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3} \,d x } \]
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Not integrable
Time = 1.40 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx=\int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,\left (g\,x^2+f\right ) \,d x \]
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