Optimal. Leaf size=835 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.07923, antiderivative size = 835, normalized size of antiderivative = 1., 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 2471
Rule 2450
Rule 2476
Rule 2448
Rule 321
Rule 205
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 2454
Rule 2401
Rule 2389
Rule 2296
Rule 2295
Rule 2390
Rule 2305
Rule 2304
Rule 2457
Rule 2455
Rule 302
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*}
Mathematica [A] time = 0.561695, size = 475, normalized size = 0.57 \[ \frac{-176400 i \sqrt{d} p^2 \left (d^3 g^2-7 e^3 f^2\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} x+i \sqrt{d}}{\sqrt{e} x-i \sqrt{d}}\right )+\sqrt{e} \left (22050 \left (e^3 x \left (14 f^2+7 f g x^3+2 g^2 x^6\right )-7 d^2 e f g\right ) \log ^2\left (c \left (d+e x^2\right )^p\right )-210 p \left (70 d^2 e g \left (4 g x^3-21 f\right )-840 d^3 g^2 x-42 d e^2 g x^2 \left (35 f+4 g x^3\right )+15 e^3 x \left (392 f^2+49 f g x^3+8 g^2 x^6\right )\right ) \log \left (c \left (d+e x^2\right )^p\right )+p^2 x \left (79520 d^2 e g^2 x^2-591360 d^3 g^2-378 d e^2 g x \left (1225 f+64 g x^3\right )+225 e^3 \left (10976 f^2+343 f g x^3+32 g^2 x^6\right )\right )+154350 d^2 e f g p^2 \log \left (d+e x^2\right )\right )-1680 \sqrt{d} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (-105 \left (7 e^3 f^2-d^3 g^2\right ) \log \left (c \left (d+e x^2\right )^p\right )-210 p \left (7 e^3 f^2-d^3 g^2\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )+2 p \left (735 e^3 f^2-176 d^3 g^2\right )\right )-176400 i \sqrt{d} p^2 \left (d^3 g^2-7 e^3 f^2\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{308700 e^{7/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.776, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{3}+f \right ) ^{2} \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{3} + f\right )}^{2} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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