Optimal. Leaf size=482 \[ -\frac{2 b d^2 n \text{Unintegrable}\left (\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x^{2/3} \left (d x^{2/3}+e\right )},x\right )}{e}-\frac{32 i b^3 d^{3/2} n^3 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}+\frac{64 b^3 d^{3/2} n^3 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}+\frac{16 b^3 n^3}{9 x} \]
[Out]
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Rubi [A] time = 1.38325, 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 = {} \[ \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^3}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-(6 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-(6 b e n) \operatorname{Subst}\left (\int \left (\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e x^4}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-(6 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )+\frac{(6 b d n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\left (24 b^2 d n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^2}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^4}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (8 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (24 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (24 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )+\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )-\frac{1}{3} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\left (48 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (16 b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}+\frac{\left (48 b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}-\frac{1}{3} \left (16 b^3 e n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{16 b^3 n^3}{9 x}-\frac{64 b^3 d n^3}{e \sqrt [3]{x}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{1}{3} \left (16 b^3 d n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (48 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (16 b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}+\frac{\left (48 b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}+\frac{\left (16 i b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{e^{3/2}}+\frac{\left (48 i b^3 d^{3/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{e^{3/2}}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (16 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{e^2}-\frac{\left (48 b^3 d^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{e^2}\\ &=\frac{16 b^3 n^3}{9 x}-\frac{208 b^3 d n^3}{3 e \sqrt [3]{x}}-\frac{208 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{32 i b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{64 b^3 d^{3/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{8 b^2 n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}+\frac{32 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}+\frac{2 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{6 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{e \sqrt [3]{x}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3}{x}-\frac{32 i b^3 d^{3/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{\left (6 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ \end{align*}
Mathematica [A] time = 2.23848, size = 1097, normalized size = 2.28 \[ \frac{b^3 \left (18 \left (x^{2/3} d+e\right ) \, _5F_4\left (-\frac{1}{2},1,1,1,1;2,2,2,2;\frac{e}{d x^{2/3}}+1\right )-\log \left (d+\frac{e}{x^{2/3}}\right ) \left (18 \left (x^{2/3} d+e\right ) \, _4F_3\left (-\frac{1}{2},1,1,1;2,2,2;\frac{e}{d x^{2/3}}+1\right )+\log \left (d+\frac{e}{x^{2/3}}\right ) \left (2 \left (x^{2/3} d+e \sqrt{-\frac{e}{d x^{2/3}}}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )-9 \left (x^{2/3} d+e\right ) \, _3F_2\left (-\frac{1}{2},1,1;2,2;\frac{e}{d x^{2/3}}+1\right )\right )\right )\right ) n^3}{2 e \sqrt{-\frac{e}{d x^{2/3}}} x}+\frac{b^2 \left (-a+b n \log \left (d+\frac{e}{x^{2/3}}\right )-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \left (9 e^{3/2} \log ^2\left (d+\frac{e}{x^{2/3}}\right )-12 e^{3/2} \log \left (d+\frac{e}{x^{2/3}}\right )+18 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+18 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (d+\frac{e}{x^{2/3}}\right )+36 d \sqrt{e} x^{2/3} \log \left (d+\frac{e}{x^{2/3}}\right )+9 (-d)^{3/2} x \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+9 \sqrt{-d} d x \log ^2\left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right )+8 e^{3/2}+96 d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )+18 \sqrt{-d} d x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 (-d)^{3/2} x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \log \left (\sqrt [3]{x} \sqrt{-d}+\sqrt{e}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+36 \sqrt{-d} d x \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+18 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+18 \sqrt{-d} d x \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )\right )+36 (-d)^{3/2} x \text{PolyLog}\left (2,\frac{\sqrt [3]{x} \sqrt{-d}}{\sqrt{e}}+1\right )-96 d \sqrt{e} x^{2/3}\right ) n^2}{3 e^{3/2} x}-\frac{6 b d^{3/2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{e^{3/2}}-\frac{3 b \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{x}-\frac{6 b d \left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 n}{e \sqrt [3]{x}}-\frac{\left (a-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2 \left (a-2 b n-b n \log \left (d+\frac{e}{x^{2/3}}\right )+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \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 (\frac{b^{3} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{3}}{x^{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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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