Optimal. Leaf size=1221 \[ \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 \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 {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 \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 {8 \sqrt {d} f^3 p^2 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 \tan ^{-1}\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 {Li}_2\left (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 {Li}_2\left (1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{7 e^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.64, antiderivative size = 1139, normalized size of antiderivative = 0.93, 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 = {2471, 2450, 2476, 2448, 321, 205, 2470, 12, 4920, 4854, 2402, 2315, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2457, 2455, 302, 2398, 2411, 43, 2334, 14, 2301} \[ \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 \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 {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 \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 {8 \sqrt {d} f^3 p^2 \tan ^{-1}\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 \tan ^{-1}\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 {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 {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 \tan ^{-1}\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 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (e x^2+d\right )^p\right )}{7 e^{7/2}}-\frac {1}{300} g^3 p \left (-\frac {60 \log \left (e x^2+d\right ) d^5}{e^5}+\frac {300 \left (e x^2+d\right ) d^4}{e^5}-\frac {300 \left (e x^2+d\right )^2 d^3}{e^5}+\frac {200 \left (e x^2+d\right )^3 d^2}{e^5}-\frac {75 \left (e x^2+d\right )^4 d}{e^5}+\frac {12 \left (e x^2+d\right )^5}{e^5}\right ) \log \left (c \left (e x^2+d\right )^p\right )+\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}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 43
Rule 205
Rule 302
Rule 321
Rule 2295
Rule 2296
Rule 2301
Rule 2304
Rule 2305
Rule 2315
Rule 2334
Rule 2389
Rule 2390
Rule 2398
Rule 2401
Rule 2402
Rule 2411
Rule 2448
Rule 2450
Rule 2454
Rule 2455
Rule 2457
Rule 2470
Rule 2471
Rule 2476
Rule 4854
Rule 4920
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 ) \operatorname {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )+\frac {1}{2} g^3 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \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 (24 i d^{7/2} f g^2 p^2\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 )}{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 = 0.99, size = 1020, normalized size = 0.84 \[ \frac {1}{125} g^3 p^2 x^{10}+\frac {1}{10} g^3 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^{10}-\frac {1}{25} g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^{10}-\frac {9 d g^3 p^2 x^8}{400 e}+\frac {d g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^8}{20 e}+\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 {47 d^2 g^3 p^2 x^6}{900 e^2}-\frac {d^2 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^6}{15 e^2}-\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 {77 d^3 g^3 p^2 x^4}{600 e^3}+\frac {3}{8} f^2 g p^2 x^4+\frac {3}{4} f^2 g \log ^2\left (c \left (e x^2+d\right )^p\right ) x^4+\frac {d^3 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^4}{10 e^3}-\frac {3}{4} f^2 g p \log \left (c \left (e x^2+d\right )^p\right ) x^4+\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 {137 d^4 g^3 p^2 x^2}{300 e^4}-\frac {9 d f^2 g p^2 x^2}{4 e}-\frac {d^4 g^3 p \log \left (c \left (e x^2+d\right )^p\right ) x^2}{5 e^4}+\frac {3 d f^2 g p \log \left (c \left (e x^2+d\right )^p\right ) x^2}{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 {4 i \sqrt {d} f \left (3 d^3 g^2-7 e^3 f^2\right ) p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {d^5 g^3 \log ^2\left (c \left (e x^2+d\right )^p\right )}{10 e^5}-\frac {3 d^2 f^2 g \log ^2\left (c \left (e x^2+d\right )^p\right )}{4 e^2}-\frac {77 d^5 g^3 p^2 \log \left (e x^2+d\right )}{300 e^5}+\frac {3 d^2 f^2 g p^2 \log \left (e x^2+d\right )}{4 e^2}-\frac {d^5 g^3 p \log \left (c \left (e x^2+d\right )^p\right )}{5 e^5}+\frac {3 d^2 f^2 g p \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {4 \sqrt {d} f p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-352 g^2 p d^3+490 e^3 f^2 p-70 \left (7 e^3 f^2-3 d^3 g^2\right ) p \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )-35 \left (7 e^3 f^2-3 d^3 g^2\right ) \log \left (c \left (e x^2+d\right )^p\right )\right )}{245 e^{7/2}}-\frac {4 i \sqrt {d} f \left (3 d^3 g^2-7 e^3 f^2\right ) p^2 \text {Li}_2\left (\frac {\sqrt {e} x+i \sqrt {d}}{\sqrt {e} x-i \sqrt {d}}\right )}{7 e^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g^{3} x^{9} + 3 \, f g^{2} x^{6} + 3 \, f^{2} g x^{3} + f^{3}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}, x\right ) \]
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 \[ \int {\left (g x^{3} + f\right )}^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.63, 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 \[ \frac {1}{140} \, {\left (14 \, g^{3} p^{2} x^{10} + 60 \, f g^{2} p^{2} x^{7} + 105 \, f^{2} g p^{2} x^{4} + 140 \, f^{3} p^{2} x\right )} \log \left (e x^{2} + d\right )^{2} + \int \frac {35 \, e g^{3} x^{11} \log \relax (c)^{2} + 35 \, d g^{3} x^{9} \log \relax (c)^{2} + 105 \, e f g^{2} x^{8} \log \relax (c)^{2} + 105 \, d f g^{2} x^{6} \log \relax (c)^{2} + 105 \, e f^{2} g x^{5} \log \relax (c)^{2} + 105 \, d f^{2} g x^{3} \log \relax (c)^{2} + 35 \, e f^{3} x^{2} \log \relax (c)^{2} + 35 \, d f^{3} \log \relax (c)^{2} + {\left (70 \, d g^{3} p x^{9} \log \relax (c) - 14 \, {\left (e g^{3} p^{2} - 5 \, e g^{3} p \log \relax (c)\right )} x^{11} + 210 \, d f g^{2} p x^{6} \log \relax (c) - 30 \, {\left (2 \, e f g^{2} p^{2} - 7 \, e f g^{2} p \log \relax (c)\right )} x^{8} + 210 \, d f^{2} g p x^{3} \log \relax (c) - 105 \, {\left (e f^{2} g p^{2} - 2 \, e f^{2} g p \log \relax (c)\right )} x^{5} + 70 \, d f^{3} p \log \relax (c) - 70 \, {\left (2 \, e f^{3} p^{2} - e f^{3} p \log \relax (c)\right )} x^{2}\right )} \log \left (e x^{2} + d\right )}{35 \, {\left (e x^{2} + d\right )}}\,{d x} \]
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 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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