Optimal. Leaf size=1221 \[ 8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {d^4 g^3 p^2 x^2}{e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac {2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac {d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac {g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 )}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {d^4 g^3 p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {d^3 g^3 p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^5}-\frac {2 d^2 g^3 p \left (d+e x^2\right )^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 e^5}+\frac {d g^3 p \left (d+e x^2\right )^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^5}-\frac {g^3 p \left (d+e x^2\right )^5 \log \left (c \left (d+e x^2\right )^p\right )}{25 e^5}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.02, antiderivative size = 1221, normalized size of antiderivative = 1.00, number of
steps used = 55, number of rules used = 29, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.208, Rules used
= {2521, 2500, 2526, 2498, 327, 211, 2520, 12, 5040, 4964, 2449, 2352, 2504, 2448,
2436, 2333, 2332, 2437, 2342, 2341, 2507, 2505, 308, 2445, 2458, 45, 2372, 14, 2338}
\begin {gather*} \frac {1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}+\frac {24}{343} f g^2 p^2 x^7+\frac {3}{7} f g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {12}{49} f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {12 d f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {4 d^2 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{7 e^2}+\frac {d^4 g^3 p^2 x^2}{e^4}-\frac {3 d f^2 g p^2 x^2}{e}+8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}+f^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x-4 f^3 p \log \left (c \left (e x^2+d\right )^p\right ) x+\frac {12 d^3 f g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {g^3 p^2 \left (e x^2+d\right )^5}{125 e^5}-\frac {d g^3 p^2 \left (e x^2+d\right )^4}{16 e^5}+\frac {2 d^2 g^3 p^2 \left (e x^2+d\right )^3}{9 e^5}-\frac {d^3 g^3 p^2 \left (e x^2+d\right )^2}{2 e^5}+\frac {3 f^2 g p^2 \left (e x^2+d\right )^2}{8 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (e x^2+d\right )}{10 e^5}+\frac {3 f^2 g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {3 d f^2 g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {8 \sqrt {d} f^3 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 )}{7 e^{7/2}}-\frac {g^3 p \left (e x^2+d\right )^5 \log \left (c \left (e x^2+d\right )^p\right )}{25 e^5}+\frac {d g^3 p \left (e x^2+d\right )^4 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^5}-\frac {2 d^2 g^3 p \left (e x^2+d\right )^3 \log \left (c \left (e x^2+d\right )^p\right )}{3 e^5}+\frac {d^3 g^3 p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{e^5}-\frac {3 f^2 g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {d^4 g^3 p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^5}+\frac {3 d f^2 g p \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{e^2}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}+\frac {d^5 g^3 p \log \left (e x^2+d\right ) \log \left (c \left (e x^2+d\right )^p\right )}{5 e^5}+\frac {4 i \sqrt {d} f^3 p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 45
Rule 211
Rule 308
Rule 327
Rule 2332
Rule 2333
Rule 2338
Rule 2341
Rule 2342
Rule 2352
Rule 2372
Rule 2436
Rule 2437
Rule 2445
Rule 2448
Rule 2449
Rule 2458
Rule 2498
Rule 2500
Rule 2504
Rule 2505
Rule 2507
Rule 2520
Rule 2521
Rule 2526
Rule 4964
Rule 5040
Rubi steps
\begin {align*} \int \left (f+g x^3\right )^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f^2 g x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+3 f g^2 x^6 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^3 x^9 \log ^2\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f^3 \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f^2 g\right ) \int x^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\left (3 f g^2\right ) \int x^6 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^3 \int x^9 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} \left (3 f^2 g\right ) \text {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^3 \text {Subst}\left (\int x^4 \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )-\left (4 e f^3 p\right ) \int \frac {x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (12 e f g^2 p\right ) \int \frac {x^8 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{2} \left (3 f^2 g\right ) \text {Subst}\left (\int \left (-\frac {d \log ^2\left (c (d+e x)^p\right )}{e}+\frac {(d+e x) \log ^2\left (c (d+e x)^p\right )}{e}\right ) \, dx,x,x^2\right )-\left (4 e f^3 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}{7} \left (12 e f g^2 p\right ) \int \left (-\frac {d^3 \log \left (c \left (d+e x^2\right )^p\right )}{e^4}+\frac {d^2 x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^3}-\frac {d x^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {x^6 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {d^4 \log \left (c \left (d+e x^2\right )^p\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx-\frac {1}{5} \left (e g^3 p\right ) \text {Subst}\left (\int \frac {x^5 \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^2\right )\\ &=f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (3 f^2 g\right ) \text {Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\frac {\left (3 d f^2 g\right ) \text {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{2 e}-\left (4 f^3 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\left (4 d f^3 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac {1}{7} \left (12 f g^2 p\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {\left (12 d^3 f g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (12 d^4 f g^2 p\right ) \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{7 e^3}-\frac {\left (12 d^2 f g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac {\left (12 d f g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e}-\frac {1}{5} \left (g^3 p\right ) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^5 \log \left (c x^p\right )}{x} \, dx,x,d+e x^2\right )\\ &=-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {\left (3 f^2 g\right ) \text {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}-\frac {\left (3 d f^2 g\right ) \text {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\left (8 e f^3 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx-\left (8 d e f^3 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}{35} \left (24 d f g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx-\frac {\left (24 d^3 f g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{7 e^2}+\frac {\left (24 d^4 f 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}{7 e^2}+\frac {\left (8 d^2 f g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx}{7 e}+\frac {1}{49} \left (24 e f g^2 p^2\right ) \int \frac {x^8}{d+e x^2} \, dx+\frac {1}{5} \left (g^3 p^2\right ) \text {Subst}\left (\int \frac {300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{60 e^5 x} \, dx,x,d+e x^2\right )\\ &=8 f^3 p^2 x-\frac {24 d^3 f g^2 p^2 x}{7 e^3}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\frac {\left (3 f^2 g p\right ) \text {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{2 e^2}+\frac {\left (3 d f^2 g p\right ) \text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\left (8 d f^3 p^2\right ) \int \frac {1}{d+e x^2} \, dx-\left (8 \sqrt {d} \sqrt {e} f^3 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx-\frac {1}{35} \left (24 d f 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+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (24 d^{7/2} f g^2 p^2\right ) \int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx}{7 e^{5/2}}+\frac {\left (8 d^2 f 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}{7 e}+\frac {1}{49} \left (24 e f g^2 p^2\right ) \int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx+\frac {\left (g^3 p^2\right ) \text {Subst}\left (\int \frac {300 d^4 x-300 d^3 x^2+200 d^2 x^3-75 d x^4+12 x^5-60 d^5 \log (x)}{x} \, dx,x,d+e x^2\right )}{300 e^5}\\ &=8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {24 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\left (8 f^3 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 (24 d^3 f 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}{7 e^3}+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{49 e^3}+\frac {\left (24 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{35 e^3}+\frac {\left (8 d^4 f g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (g^3 p^2\right ) \text {Subst}\left (\int \left (300 d^4-300 d^3 x+200 d^2 x^2-75 d x^3+12 x^4-\frac {60 d^5 \log (x)}{x}\right ) \, dx,x,d+e x^2\right )}{300 e^5}\\ &=8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {d^4 g^3 p^2 x^2}{e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac {2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac {d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac {g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 )}{7 e^{7/2}}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}-\left (8 f^3 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 (24 d^3 f 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}{7 e^3}-\frac {\left (d^5 g^3 p^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x^2\right )}{5 e^5}\\ &=8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {d^4 g^3 p^2 x^2}{e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac {2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac {d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac {g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 )}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {\left (8 i \sqrt {d} f^3 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 (24 i d^{7/2} f 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 )}{7 e^{7/2}}\\ &=8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {3 d f^2 g p^2 x^2}{e}+\frac {d^4 g^3 p^2 x^2}{e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {24}{343} f g^2 p^2 x^7+\frac {3 f^2 g p^2 \left (d+e x^2\right )^2}{8 e^2}-\frac {d^3 g^3 p^2 \left (d+e x^2\right )^2}{2 e^5}+\frac {2 d^2 g^3 p^2 \left (d+e x^2\right )^3}{9 e^5}-\frac {d g^3 p^2 \left (d+e x^2\right )^4}{16 e^5}+\frac {g^3 p^2 \left (d+e x^2\right )^5}{125 e^5}-\frac {8 \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{245 e^{7/2}}+\frac {4 i \sqrt {d} f^3 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {8 \sqrt {d} f^3 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 {24 d^{7/2} f 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 )}{7 e^{7/2}}-\frac {d^5 g^3 p^2 \log ^2\left (d+e x^2\right )}{10 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {3 d f^2 g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {3 f^2 g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 \sqrt {d} f^3 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 {12 d^{7/2} f g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (\frac {300 d^4 \left (d+e x^2\right )}{e^5}-\frac {300 d^3 \left (d+e x^2\right )^2}{e^5}+\frac {200 d^2 \left (d+e x^2\right )^3}{e^5}-\frac {75 d \left (d+e x^2\right )^4}{e^5}+\frac {12 \left (d+e x^2\right )^5}{e^5}-\frac {60 d^5 \log \left (d+e x^2\right )}{e^5}\right ) \log \left (c \left (d+e x^2\right )^p\right )+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {3 d f^2 g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {3 f^2 g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {4 i \sqrt {d} f^3 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{\sqrt {e}}-\frac {12 i d^{7/2} f g^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )}{7 e^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 1020, normalized size = 0.84 \begin {gather*} 8 f^3 p^2 x-\frac {1408 d^3 f g^2 p^2 x}{245 e^3}-\frac {9 d f^2 g p^2 x^2}{4 e}+\frac {137 d^4 g^3 p^2 x^2}{300 e^4}+\frac {568 d^2 f g^2 p^2 x^3}{735 e^2}+\frac {3}{8} f^2 g p^2 x^4-\frac {77 d^3 g^3 p^2 x^4}{600 e^3}-\frac {288 d f g^2 p^2 x^5}{1225 e}+\frac {47 d^2 g^3 p^2 x^6}{900 e^2}+\frac {24}{343} f g^2 p^2 x^7-\frac {9 d g^3 p^2 x^8}{400 e}+\frac {1}{125} g^3 p^2 x^{10}-\frac {4 i \sqrt {d} f \left (-7 e^3 f^2+3 d^3 g^2\right ) p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {3 d^2 f^2 g p^2 \log \left (d+e x^2\right )}{4 e^2}-\frac {77 d^5 g^3 p^2 \log \left (d+e x^2\right )}{300 e^5}+\frac {3 d^2 f^2 g p \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {d^5 g^3 p \log \left (c \left (d+e x^2\right )^p\right )}{5 e^5}-4 f^3 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {12 d^3 f g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}+\frac {3 d f^2 g p x^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e}-\frac {d^4 g^3 p x^2 \log \left (c \left (d+e x^2\right )^p\right )}{5 e^4}-\frac {4 d^2 f g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{7 e^2}-\frac {3}{4} f^2 g p x^4 \log \left (c \left (d+e x^2\right )^p\right )+\frac {d^3 g^3 p x^4 \log \left (c \left (d+e x^2\right )^p\right )}{10 e^3}+\frac {12 d f g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {d^2 g^3 p x^6 \log \left (c \left (d+e x^2\right )^p\right )}{15 e^2}-\frac {12}{49} f g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {d g^3 p x^8 \log \left (c \left (d+e x^2\right )^p\right )}{20 e}-\frac {1}{25} g^3 p x^{10} \log \left (c \left (d+e x^2\right )^p\right )-\frac {3 d^2 f^2 g \log ^2\left (c \left (d+e x^2\right )^p\right )}{4 e^2}+\frac {d^5 g^3 \log ^2\left (c \left (d+e x^2\right )^p\right )}{10 e^5}+f^3 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{4} f^2 g x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {3}{7} f g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{10} g^3 x^{10} \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {4 \sqrt {d} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (490 e^3 f^2 p-352 d^3 g^2 p-70 \left (7 e^3 f^2-3 d^3 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )-35 \left (7 e^3 f^2-3 d^3 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )\right )}{245 e^{7/2}}-\frac {4 i \sqrt {d} f \left (-7 e^3 f^2+3 d^3 g^2\right ) p^2 \text {Li}_2\left (\frac {i \sqrt {d}+\sqrt {e} x}{-i \sqrt {d}+\sqrt {e} x}\right )}{7 e^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.23, size = 0, normalized size = 0.00 \[\int \left (g \,x^{3}+f \right )^{3} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}\, dx\]
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^{3}\right )^{3} \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^3+f\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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