Optimal. Leaf size=682 \[ -\frac{2 d p (d g-3 e f) \text{Unintegrable}\left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2},x\right )}{e}+\frac{32 i d^{3/2} g p^3 \text{PolyLog}\left (2,-\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \text{PolyLog}\left (2,-\frac{\sqrt{d}-i \sqrt{e} x}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}+\frac{64 d^{3/2} g p^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{48 \sqrt{d} f p^3 \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right ) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}+\frac{208 d g p^3 x}{9 e}-48 f p^3 x-\frac{16}{27} g p^3 x^3 \]
[Out]
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Rubi [A] time = 1.38604, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \left (f+g x^2\right ) \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx &=\int \left (f \log ^3\left (c \left (d+e x^2\right )^p\right )+g x^2 \log ^3\left (c \left (d+e x^2\right )^p\right )\right ) \, dx\\ &=f \int \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx+g \int x^2 \log ^3\left (c \left (d+e x^2\right )^p\right ) \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 e f p) \int \frac{x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-(2 e g p) \int \frac{x^4 \log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 e f p) \int \left (\frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx-(2 e g p) \int \left (-\frac{d \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^2 \log ^2\left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-(6 f p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-(2 g p) \int x^2 \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx+\frac{(2 d g p) \int \log ^2\left (c \left (d+e x^2\right )^p\right ) \, dx}{e}-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 e f p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\left (8 d g p^2\right ) \int \frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac{1}{3} \left (8 e g p^2\right ) \int \frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 e f p^2\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-\left (8 d g p^2\right ) \int \left (\frac{\log \left (c \left (d+e x^2\right )^p\right )}{e}-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e \left (d+e x^2\right )}\right ) \, dx+\frac{1}{3} \left (8 e g p^2\right ) \int \left (-\frac{d \log \left (c \left (d+e x^2\right )^p\right )}{e^2}+\frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac{d^2 \log \left (c \left (d+e x^2\right )^p\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (24 f p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\left (24 d f p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx+\frac{1}{3} \left (8 g p^2\right ) \int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx-\frac{\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{3 e}-\frac{\left (8 d g p^2\right ) \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{e}+\frac{\left (8 d^2 g p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{3 e}+\frac{\left (8 d^2 g p^2\right ) \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ &=24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\left (48 e f p^3\right ) \int \frac{x^2}{d+e x^2} \, dx+\left (48 d e f p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx+\frac{1}{3} \left (16 d g p^3\right ) \int \frac{x^2}{d+e x^2} \, dx+\left (16 d g p^3\right ) \int \frac{x^2}{d+e x^2} \, dx-\frac{1}{3} \left (16 d^2 g p^3\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-\left (16 d^2 g p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} \left (d+e x^2\right )} \, dx-\frac{1}{9} \left (16 e g p^3\right ) \int \frac{x^4}{d+e x^2} \, dx\\ &=-48 f p^3 x+\frac{64 d g p^3 x}{3 e}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (48 d f p^3\right ) \int \frac{1}{d+e x^2} \, dx+\left (48 \sqrt{d} \sqrt{e} f p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{3 e}-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{e}-\frac{\left (16 d^{3/2} g p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{3 \sqrt{e}}-\frac{\left (16 d^{3/2} g p^3\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{d+e x^2} \, dx}{\sqrt{e}}-\frac{1}{9} \left (16 e g p^3\right ) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{64 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\left (48 f p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx+\frac{\left (16 d g p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{3 e}+\frac{\left (16 d g p^3\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{i-\frac{\sqrt{e} x}{\sqrt{d}}} \, dx}{e}-\frac{\left (16 d^2 g p^3\right ) \int \frac{1}{d+e x^2} \, dx}{9 e}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{48 \sqrt{d} f p^3 \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{64 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}+\left (48 f p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx-\frac{\left (16 d g p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{3 e}-\frac{\left (16 d g p^3\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{e} x}{\sqrt{d}}}\right )}{1+\frac{e x^2}{d}} \, dx}{e}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{48 \sqrt{d} f p^3 \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{64 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}-\frac{\left (48 i \sqrt{d} f p^3\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 (16 i d^{3/2} g p^3\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 )}{3 e^{3/2}}+\frac{\left (16 i d^{3/2} g p^3\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 )}{e^{3/2}}\\ &=-48 f p^3 x+\frac{208 d g p^3 x}{9 e}-\frac{16}{27} g p^3 x^3+\frac{48 \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-\frac{208 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{9 e^{3/2}}-\frac{24 i \sqrt{d} f p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )^2}{3 e^{3/2}}-\frac{48 \sqrt{d} f p^3 \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{64 d^{3/2} g p^3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}+24 f p^2 x \log \left (c \left (d+e x^2\right )^p\right )-\frac{32 d g p^2 x \log \left (c \left (d+e x^2\right )^p\right )}{3 e}+\frac{8}{9} g p^2 x^3 \log \left (c \left (d+e x^2\right )^p\right )-\frac{24 \sqrt{d} f p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{\sqrt{e}}+\frac{32 d^{3/2} g p^2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{3 e^{3/2}}-6 f p x \log ^2\left (c \left (d+e x^2\right )^p\right )+\frac{2 d g p x \log ^2\left (c \left (d+e x^2\right )^p\right )}{e}-\frac{2}{3} g p x^3 \log ^2\left (c \left (d+e x^2\right )^p\right )+f x \log ^3\left (c \left (d+e x^2\right )^p\right )+\frac{1}{3} g x^3 \log ^3\left (c \left (d+e x^2\right )^p\right )-\frac{24 i \sqrt{d} f p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{\sqrt{e}}+\frac{32 i d^{3/2} g p^3 \text{Li}_2\left (1-\frac{2 \sqrt{d}}{\sqrt{d}+i \sqrt{e} x}\right )}{3 e^{3/2}}+(6 d f p) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx-\frac{\left (2 d^2 g p\right ) \int \frac{\log ^2\left (c \left (d+e x^2\right )^p\right )}{d+e x^2} \, dx}{e}\\ \end{align*}
Mathematica [A] time = 4.48061, size = 1460, normalized size = 2.14 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 19.954, size = 0, normalized size = 0. \begin{align*} \int \left ( g{x}^{2}+f \right ) \left ( \ln \left ( c \left ( e{x}^{2}+d \right ) ^{p} \right ) \right ) ^{3}\, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}, 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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (g x^{2} + f\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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