Optimal. Leaf size=683 \[ -\frac {2 d p (d g-3 e f) \text {Int}\left (\frac {\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e}+\frac {32 d^{3/2} g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac {24 \sqrt {d} f p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}+f x \log ^3\left (c \left (d+e x^2\right )^p\right )-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac {8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac {1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac {2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {32 i d^{3/2} g p^3 \text {Li}_2\left (-\frac {\sqrt {d}-i \sqrt {e} x}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}}+\frac {32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {208 d^{3/2} g p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {64 d^{3/2} g p^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {24 i \sqrt {d} f p^3 \text {Li}_2\left (-\frac {\sqrt {d}-i \sqrt {e} x}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {24 i \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {48 \sqrt {d} f p^3 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {48 \sqrt {d} f p^3 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {208 d g p^3 x}{9 e}-48 f p^3 x-\frac {16}{27} g p^3 x^3 \]
[Out]
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Rubi [A] time = 1.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx &=\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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [A] time = 4.57, size = 1460, normalized size = 2.14 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 76.68, size = 0, normalized size = 0.00 \[ \int \left (g \,x^{2}+f \right ) \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, {\left (g p^{3} x^{3} + 3 \, f p^{3} x\right )} \log \left (e x^{2} + d\right )^{3} + \int \frac {e g x^{4} \log \relax (c)^{3} + {\left (e f + d g\right )} x^{2} \log \relax (c)^{3} + d f \log \relax (c)^{3} - {\left ({\left (2 \, e g p^{3} - 3 \, e g p^{2} \log \relax (c)\right )} x^{4} - 3 \, d f p^{2} \log \relax (c) + 3 \, {\left (2 \, e f p^{3} - {\left (e f p^{2} + d g p^{2}\right )} \log \relax (c)\right )} x^{2}\right )} \log \left (e x^{2} + d\right )^{2} + 3 \, {\left (e g p x^{4} \log \relax (c)^{2} + d f p \log \relax (c)^{2} + {\left (e f p + d g p\right )} x^{2} \log \relax (c)^{2}\right )} \log \left (e x^{2} + d\right )}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^3\,\left (g\,x^2+f\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (f + g x^{2}\right ) \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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