Optimal. Leaf size=835 \[ \frac {8}{343} g^2 p^2 x^7+\frac {1}{7} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {4}{49} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{21 e^2}-\frac {2 d f g p^2 x^2}{e}+8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}+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^3 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {f g p^2 \left (e x^2+d\right )^2}{4 e^2}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {f g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {d f g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {8 d^{7/2} 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 {f g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}+\frac {2 d f 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^2 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 {4 d^{7/2} 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 {4 i \sqrt {d} f^2 p^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {4 i d^{7/2} 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 = 1.08, antiderivative size = 835, normalized size of antiderivative = 1.00, number of steps used = 47, number of rules used = 23, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.958, 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} \[ \frac {8}{343} g^2 p^2 x^7+\frac {1}{7} g^2 \log ^2\left (c \left (e x^2+d\right )^p\right ) x^7-\frac {4}{49} g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^7-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {4 d g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^5}{35 e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {4 d^2 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x^3}{21 e^2}-\frac {2 d f g p^2 x^2}{e}+8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}+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^3 g^2 p \log \left (c \left (e x^2+d\right )^p\right ) x}{7 e^3}+\frac {f g p^2 \left (e x^2+d\right )^2}{4 e^2}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\frac {f g \left (e x^2+d\right )^2 \log ^2\left (c \left (e x^2+d\right )^p\right )}{2 e^2}-\frac {d f g \left (e x^2+d\right ) \log ^2\left (c \left (e x^2+d\right )^p\right )}{e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{i \sqrt {e} x+\sqrt {d}}\right )}{\sqrt {e}}-\frac {8 d^{7/2} 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 {f g p \left (e x^2+d\right )^2 \log \left (c \left (e x^2+d\right )^p\right )}{2 e^2}+\frac {2 d f 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^2 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 {4 d^{7/2} 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 {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^{7/2} 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 205
Rule 302
Rule 321
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2315
Rule 2389
Rule 2390
Rule 2401
Rule 2402
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 )^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^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+g^2 x^6 \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^3 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+g^2 \int x^6 \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 {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+(f g) \operatorname {Subst}\left (\int x \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\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}{7} \left (4 e 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^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+(f g) \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^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}{7} \left (4 e 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\\ &=f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \operatorname {Subst}\left (\int (d+e x) \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\frac {(d f g) \operatorname {Subst}\left (\int \log ^2\left (c (d+e x)^p\right ) \, dx,x,x^2\right )}{e}-\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}{7} \left (4 g^2 p\right ) \int x^6 \log \left (c \left (d+e x^2\right )^p\right ) \, dx+\frac {\left (4 d^3 g^2 p\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^3}-\frac {\left (4 d^4 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 (4 d^2 g^2 p\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e^2}+\frac {\left (4 d g^2 p\right ) \int x^4 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{7 e}\\ &=-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {(f g) \operatorname {Subst}\left (\int x \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\frac {(d f g) \operatorname {Subst}\left (\int \log ^2\left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\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}{35} \left (8 d g^2 p^2\right ) \int \frac {x^6}{d+e x^2} \, dx-\frac {\left (8 d^3 g^2 p^2\right ) \int \frac {x^2}{d+e x^2} \, dx}{7 e^2}+\frac {\left (8 d^4 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 g^2 p^2\right ) \int \frac {x^4}{d+e x^2} \, dx}{21 e}+\frac {1}{49} \left (8 e g^2 p^2\right ) \int \frac {x^8}{d+e x^2} \, dx\\ &=8 f^2 p^2 x-\frac {8 d^3 g^2 p^2 x}{7 e^3}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\frac {(f g p) \operatorname {Subst}\left (\int x \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}+\frac {(2 d f g p) \operatorname {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,d+e x^2\right )}{e^2}-\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 {1}{35} \left (8 d 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 (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{7 e^3}+\frac {\left (8 d^{7/2} 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 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}{21 e}+\frac {1}{49} \left (8 e 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\\ &=8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {8 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{7 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 (8 d^3 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 (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{49 e^3}+\frac {\left (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{35 e^3}+\frac {\left (8 d^4 g^2 p^2\right ) \int \frac {1}{d+e x^2} \, dx}{21 e^3}\\ &=8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}-\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 (8 d^3 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}\\ &=8 f^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\frac {\left (8 i \sqrt {d} f^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 )}{\sqrt {e}}-\frac {\left (8 i d^{7/2} 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^2 p^2 x-\frac {1408 d^3 g^2 p^2 x}{735 e^3}-\frac {2 d f g p^2 x^2}{e}+\frac {568 d^2 g^2 p^2 x^3}{2205 e^2}-\frac {96 d g^2 p^2 x^5}{1225 e}+\frac {8}{343} g^2 p^2 x^7+\frac {f g p^2 \left (d+e x^2\right )^2}{4 e^2}-\frac {8 \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {e}}+\frac {1408 d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{735 e^{7/2}}+\frac {4 i \sqrt {d} f^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{\sqrt {e}}-\frac {4 i d^{7/2} g^2 p^2 \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{7 e^{7/2}}+\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 {8 d^{7/2} 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 )}{7 e^{7/2}}-4 f^2 p x \log \left (c \left (d+e x^2\right )^p\right )+\frac {4 d^3 g^2 p x \log \left (c \left (d+e x^2\right )^p\right )}{7 e^3}-\frac {4 d^2 g^2 p x^3 \log \left (c \left (d+e x^2\right )^p\right )}{21 e^2}+\frac {4 d g^2 p x^5 \log \left (c \left (d+e x^2\right )^p\right )}{35 e}-\frac {4}{49} g^2 p x^7 \log \left (c \left (d+e x^2\right )^p\right )+\frac {2 d f g p \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e^2}-\frac {f g p \left (d+e x^2\right )^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 d^{7/2} g^2 p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{7 e^{7/2}}+f^2 x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac {1}{7} g^2 x^7 \log ^2\left (c \left (d+e x^2\right )^p\right )-\frac {d f g \left (d+e x^2\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac {f g \left (d+e x^2\right )^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{2 e^2}+\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 {4 i d^{7/2} 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 = 0.57, size = 475, normalized size = 0.57 \[ \frac {-1680 \sqrt {d} p \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (-105 \left (7 e^3 f^2-d^3 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )-210 p \left (7 e^3 f^2-d^3 g^2\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} x}\right )+2 p \left (735 e^3 f^2-176 d^3 g^2\right )\right )+\sqrt {e} \left (22050 \left (e^3 x \left (14 f^2+7 f g x^3+2 g^2 x^6\right )-7 d^2 e f g\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )-210 p \left (-840 d^3 g^2 x+70 d^2 e g \left (4 g x^3-21 f\right )-42 d e^2 g x^2 \left (35 f+4 g x^3\right )+15 e^3 x \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+154350 d^2 e f g p^2 \log \left (d+e x^2\right )+p^2 x \left (-591360 d^3 g^2+79520 d^2 e g^2 x^2-378 d e^2 g x \left (1225 f+64 g x^3\right )+225 e^3 \left (10976 f^2+343 f g x^3+32 g^2 x^6\right )\right )\right )-176400 i \sqrt {d} p^2 \left (d^3 g^2-7 e^3 f^2\right ) \text {Li}_2\left (\frac {\sqrt {e} x+i \sqrt {d}}{\sqrt {e} x-i \sqrt {d}}\right )-176400 i \sqrt {d} p^2 \left (d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )^2}{308700 e^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (g^{2} x^{6} + 2 \, f g x^{3} + f^{2}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}, x\right ) \]
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 \[ \int {\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.49, size = 0, normalized size = 0.00 \[ \int \left (g \,x^{3}+f \right )^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )^{2}\, dx \]
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 \[ \frac {1}{14} \, {\left (2 \, g^{2} p^{2} x^{7} + 7 \, f g p^{2} x^{4} + 14 \, f^{2} p^{2} x\right )} \log \left (e x^{2} + d\right )^{2} + \int \frac {7 \, e g^{2} x^{8} \log \relax (c)^{2} + 7 \, d g^{2} x^{6} \log \relax (c)^{2} + 14 \, e f g x^{5} \log \relax (c)^{2} + 14 \, d f g x^{3} \log \relax (c)^{2} + 7 \, e f^{2} x^{2} \log \relax (c)^{2} + 7 \, d f^{2} \log \relax (c)^{2} + 2 \, {\left (7 \, d g^{2} p x^{6} \log \relax (c) - {\left (2 \, e g^{2} p^{2} - 7 \, e g^{2} p \log \relax (c)\right )} x^{8} + 14 \, d f g p x^{3} \log \relax (c) - 7 \, {\left (e f g p^{2} - 2 \, e f g p \log \relax (c)\right )} x^{5} + 7 \, d f^{2} p \log \relax (c) - 7 \, {\left (2 \, e f^{2} p^{2} - e f^{2} p \log \relax (c)\right )} x^{2}\right )} \log \left (e x^{2} + d\right )}{7 \, {\left (e x^{2} + d\right )}}\,{d x} \]
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 \[ \int {\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}^2\,{\left (g\,x^3+f\right )}^2 \,d x \]
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 \[ \int \left (f + g x^{3}\right )^{2} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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