Optimal. Leaf size=945 \[ 8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^2 p^2 \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 {16 d^{3/2} f g p^2 \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}}+\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.79, antiderivative size = 945, normalized size of antiderivative = 1.00, number of steps
used = 50, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used =
{2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2507, 2505, 308}
\begin {gather*} \frac {8}{125} g^2 p^2 x^5+\frac {1}{5} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^5-\frac {4}{25} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {16}{27} f g p^2 x^3+\frac {2}{3} f g \log ^2\left (c \left (e x^2+d\right )^p\right ) x^3+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{15 e}-\frac {8}{9} f g p \log \left (c \left (e x^2+d\right )^p\right ) x^3+8 f^2 p^2 x+\frac {184 d^2 g^2 p^2 x}{75 e^2}-\frac {64 d f g p^2 x}{9 e}+f^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^2 p \log \left (c \left (e x^2+d\right )^p\right ) x-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{5 e^2}+\frac {8 d f g p \log \left (c \left (e x^2+d\right )^p\right ) x}{3 e}+\frac {4 i \sqrt {d} f^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}+\frac {4 i d^{5/2} g^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}-\frac {8 i d^{3/2} f g p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}-\frac {8 \sqrt {d} f^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}-\frac {184 d^{5/2} g^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {64 d^{3/2} f g p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}+\frac {8 \sqrt {d} f^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 d^{5/2} g^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{5 e^{5/2}}-\frac {16 d^{3/2} f g p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}}+\frac {4 \sqrt {d} f^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{\sqrt {e}}+\frac {4 d^{5/2} g^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{5 e^{5/2}}-\frac {8 d^{3/2} f g p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{3 e^{3/2}}+\frac {4 i \sqrt {d} f^2 p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}+\frac {4 i d^{5/2} g^2 p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{5 e^{5/2}}-\frac {8 i d^{3/2} f g p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{3 e^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 211
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 2498
Rule 2500
Rule 2505
Rule 2507
Rule 2520
Rule 2521
Rule 2526
Rule 4964
Rule 5040
Rubi steps
\begin {align*} \int \left (f+g x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^2 \log ^2\left (c \left (d+e x^2\right )^p\right )+2 f g x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^2 x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^2 \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+(2 f g) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^4 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 e f^2 p\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (8 e f g p) \int \frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{5} \left (4 e g^2 p\right ) \int \frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 e f^2 p\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} (8 e f g p) \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-\frac {1}{5} \left (4 e g^2 p\right ) \int \left (\frac {d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx\\ &=f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (4 f^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (4 d f^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{3} (8 f g p) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {(8 d f g p) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac {\left (8 d^2 f g p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}-\frac {1}{5} \left (4 g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac {\left (4 d^2 g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{5 e^2}+\frac {\left (4 d^3 g^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{5 e^2}+\frac {\left (4 d g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{5 e}\\ &=-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 e f^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx-\left (8 d e f^2 p^2\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 f g p^2\right ) \int \frac {x^2}{d+e x^2} \, dx+\frac {1}{3} \left (16 d^2 f g p^2\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 f g p^2\right ) \int \frac {x^4}{d+e x^2} \, dx-\frac {1}{15} \left (8 d g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx+\frac {\left (8 d^2 g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{5 e}-\frac {\left (8 d^3 g^2 p^2\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}{5 e}+\frac {1}{25} \left (8 e g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx\\ &=8 f^2 p^2 x-\frac {16 d f g p^2 x}{3 e}+\frac {8 d^2 g^2 p^2 x}{5 e^2}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 d f^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx-\left (8 \sqrt {d} \sqrt {e} f^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx+\frac {\left (16 d^2 f g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{3 e}+\frac {\left (16 d^{3/2} f g p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{3 \sqrt {e}}+\frac {1}{9} \left (16 e f g p^2\right ) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac {1}{15} \left (8 d g^2 p^2\right ) \int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{5 e^2}-\frac {\left (8 d^{5/2} g^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{5 e^{3/2}}+\frac {1}{25} \left (8 e g^2 p^2\right ) \int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx\\ &=8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {16 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{3 e^{3/2}}-\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{5 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\left (8 f^2 p^2\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 f g p^2\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^2 f g p^2\right ) \int \frac {1}{d+e x^2} \, dx}{9 e}+\frac {\left (8 d^2 g^2 p^2\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx}{5 e^2}-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{25 e^2}-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{15 e^2}\\ &=8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^2 p^2 \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 {16 d^{3/2} f g p^2 \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}}+\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )-\left (8 f^2 p^2\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 f g p^2\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 (8 d^2 g^2 p^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx}{5 e^2}\\ &=8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^2 p^2 \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 {16 d^{3/2} f g p^2 \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}}+\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (8 i \sqrt {d} f^2 p^2\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} f g p^2\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 (8 i d^{5/2} g^2 p^2\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 )}{5 e^{5/2}}\\ &=8 f^2 p^2 x-\frac {64 d f g p^2 x}{9 e}+\frac {184 d^2 g^2 p^2 x}{75 e^2}+\frac {16}{27} f g p^2 x^3-\frac {64 d g^2 p^2 x^3}{225 e}+\frac {8}{125} g^2 p^2 x^5-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {64 d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{9 e^{3/2}}-\frac {184 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{75 e^{5/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{5 e^{5/2}}+\frac {8 \sqrt {d} f^2 p^2 \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 {16 d^{3/2} f g p^2 \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}}+\frac {8 d^{5/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {8 d f g p x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac {4 d^2 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{5 e^2}-\frac {8}{9} f g p x^3 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{15 e}-\frac {4}{25} g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt {e}}-\frac {8 d^{3/2} f g p \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}}+\frac {4 d^{5/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^{5/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {2}{3} f g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{5} g^2 x^5 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {4 i \sqrt {d} f^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {8 i d^{3/2} f g p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{3 e^{3/2}}+\frac {4 i d^{5/2} g^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{5 e^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.35, size = 435, normalized size = 0.46 \begin {gather*} \frac {900 i \sqrt {d} \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2+60 \sqrt {d} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-2 \left (225 e^2 f^2-200 d e f g+69 d^2 g^2\right ) p+30 \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+15 \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )+\sqrt {e} x \left (8 p^2 \left (1035 d^2 g^2-120 d e g \left (25 f+g x^2\right )+e^2 \left (3375 f^2+250 f g x^2+27 g^2 x^4\right )\right )-60 p \left (45 d^2 g^2-15 d e g \left (10 f+g x^2\right )+e^2 \left (225 f^2+50 f g x^2+9 g^2 x^4\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+225 e^2 \left (15 f^2+10 f g x^2+3 g^2 x^4\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )\right )+900 i \sqrt {d} \left (15 e^2 f^2-10 d e f g+3 d^2 g^2\right ) p^2 \text {Li}_2\left (\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{3375 e^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \left (g \,x^{2}+f \right )^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f + g x^{2}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^2+f\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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