Optimal. Leaf size=1277 \[ \text{result too large to display} \]
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Rubi [A] time = 3.16869, 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 = {} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx \]
Verification is Not applicable to the result.
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[Out]
Rubi steps
\begin{align*} \int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3 \, dx &=3 \operatorname{Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \operatorname{Subst}\left (\int \frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+(2 b e n) \operatorname{Subst}\left (\int \left (-\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^4}+\frac{e^2 x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^3}-\frac{e x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^2}+\frac{x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d}+\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{d^4 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{(2 b e n) \operatorname{Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d}-\frac{\left (2 b e^2 n\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^2}+\frac{\left (2 b e^3 n\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^3}-\frac{\left (2 b e^4 n\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (2 b e^5 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 )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (8 b^2 e^5 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^2} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^3}-\frac{e x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2}+\frac{x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d}-\frac{e^3 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^3 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \left (-\frac{e \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2}+\frac{x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d}+\frac{e^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^2}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{d}-\frac{e \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{d \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac{\left (8 b^2 e^5 n^2\right ) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{2 \sqrt{e} \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{2 \sqrt{e} \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (8 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^2}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^3}-\frac{\left (8 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^3}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (8 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac{\left (4 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^2 e^{9/2} n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (8 b^2 e^5 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 )}{7 d^4}-\frac{\left (8 b^2 e^5 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 )}{5 d^4}-\frac{\left (8 b^2 e^5 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 )}{3 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (8 b^3 e^4 n^2\right ) \operatorname{Subst}\left (\int \log \left (c \left (d+\frac{e}{x^2}\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{d+\frac{e}{x^2}} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (16 b^3 e^6 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 )}{7 d^4}-\frac{\left (16 b^3 e^6 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 )}{5 d^4}-\frac{\left (16 b^3 e^6 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 )}{3 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{x^4}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^2}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}-\frac{\left (16 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{x^2}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{e x}-\frac{d x \log \left (\sqrt{e}-\sqrt{-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{11/2} n^3\right ) \operatorname{Subst}\left (\int \left (\frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{e x}-\frac{d x \log \left (\sqrt{e}+\sqrt{-d} x\right )}{e \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (16 b^3 e^{11/2} 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 )}{7 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} 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 )}{5 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} 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 )}{3 d^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{64 b^3 e^4 n^3 \sqrt [3]{x}}{35 d^4}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (16 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{e}{d^2}+\frac{x^2}{d}+\frac{e^2}{d^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{35 d^2}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{x} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \log \left (\sqrt{e}-\sqrt{-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{x \log \left (\sqrt{e}+\sqrt{-d} x\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{7 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{5 d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac{\left (16 b^3 e^{11/2} 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 )}{7 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} 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 )}{5 d^{9/2}}-\frac{\left (16 b^3 e^{11/2} 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 )}{3 d^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1328 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \log \left (\sqrt{e}-\sqrt{-d} x\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \log \left (\sqrt{e}-\sqrt{-d} x\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \left (-\frac{\sqrt{-d} \log \left (\sqrt{e}+\sqrt{-d} x\right )}{2 d \left (\sqrt{e}-\sqrt{-d} x\right )}+\frac{\sqrt{-d} \log \left (\sqrt{e}+\sqrt{-d} x\right )}{2 d \left (\sqrt{e}+\sqrt{-d} x\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{(-d)^{7/2}}-\frac{\left (16 i b^3 e^{9/2} 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 )}{7 d^{9/2}}-\frac{\left (16 i b^3 e^{9/2} 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 )}{5 d^{9/2}}-\frac{\left (16 i b^3 e^{9/2} 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 )}{3 d^{9/2}}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{\sqrt{-d} x}{\sqrt{e}}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (8 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{-d} x}{\sqrt{e}}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (16 b^3 e^5 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{35 d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} 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 )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (16 b^3 e^4 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 )}{7 d^4}+\frac{\left (16 b^3 e^4 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 )}{5 d^4}+\frac{\left (16 b^3 e^4 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 )}{3 d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}-\sqrt{-d} x\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{e}+\sqrt{-d} x\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} 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 )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{e}-\sqrt{-d} x}{2 \sqrt{e}}\right )}{\sqrt{e}+\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{\sqrt{e}+\sqrt{-d} x}{2 \sqrt{e}}\right )}{\sqrt{e}-\sqrt{-d} x} \, dx,x,\sqrt [3]{x}\right )}{d^4}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} 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 )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}+\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 \sqrt{e}}\right )}{x} \, dx,x,\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{\left (4 b^3 e^{9/2} n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{2 \sqrt{e}}\right )}{x} \, dx,x,\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}\\ &=\frac{568 a b^2 e^4 n^2 \sqrt [3]{x}}{105 d^4}-\frac{16 b^3 e^4 n^3 \sqrt [3]{x}}{7 d^4}+\frac{16 b^3 e^3 n^3 x}{105 d^3}+\frac{1376 b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{105 d^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{105 d^{9/2}}-\frac{1136 b^3 e^{9/2} 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 )}{105 d^{9/2}}+\frac{568 b^3 e^4 n^2 \sqrt [3]{x} \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )}{105 d^4}-\frac{32 b^2 e^3 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^3}+\frac{8 b^2 e^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{35 d^2}-\frac{568 b^2 e^{9/2} 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 )}{105 d^{9/2}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{5 d^2}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^3+\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}-\frac{4 b^2 e^{9/2} n^2 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right ) \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{2 b^3 e^{9/2} n^3 \log ^2\left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}+\sqrt{-d} \sqrt [3]{x}\right ) \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{568 i b^3 e^{9/2} n^3 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{105 d^{9/2}}+\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}-\frac{4 b^3 e^{9/2} n^3 \text{Li}_2\left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )}{(-d)^{9/2}}+\frac{4 b^3 e^{9/2} n^3 \text{Li}_2\left (\frac{1}{2} \left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )}{(-d)^{9/2}}-\frac{8 b^3 e^{9/2} n^3 \text{Li}_2\left (1+\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )}{(-d)^{9/2}}+\frac{\left (2 b e^5 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 )}{d^4}\\ \end{align*}
Mathematica [A] time = 4.8286, size = 764, normalized size = 0.6 \[ -\frac{b^2 n^2 \left (-a-b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )+b n \log \left (d+\frac{e}{x^{2/3}}\right )\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (9 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-9 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}+\frac{b^3 n^3 \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (\log \left (d+\frac{e}{x^{2/3}}\right ) \left (27 e^5 \left (d x^{2/3}+e\right ) \, _3F_2\left (1,1,\frac{11}{2};2,2;\frac{e}{d x^{2/3}}+1\right )+2 d x^{2/3} \left (d^5 x^{10/3} \sqrt{-\frac{e}{d x^{2/3}}}+e^5\right ) \log \left (d+\frac{e}{x^{2/3}}\right )\right )-54 e^5 \left (d x^{2/3}+e\right ) \, _4F_3\left (1,1,1,\frac{11}{2};2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )+54 e^5 \left (d x^{2/3}+e\right ) \, _5F_4\left (1,1,1,1,\frac{11}{2};2,2,2,2;\frac{e}{d x^{2/3}}+1\right )\right )}{6 d^6 x \sqrt{-\frac{e}{d x^{2/3}}}}-\frac{2 b e^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^4}+\frac{2 b e^3 n x \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{3 d^3}-\frac{2 b e^2 n x^{5/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{5 d^2}+\frac{2 b e^{9/2} n \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 )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{d^{9/2}}+\frac{2 b e n x^{7/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2}{7 d}+\frac{1}{3} x^3 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^3+b n x^3 \log \left (d+\frac{e}{x^{2/3}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )-b n \log \left (d+\frac{e}{x^{2/3}}\right )\right )^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.355, size = 0, normalized size = 0. \begin{align*} \int{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 (b^{3} x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} x^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b x^{2} \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{\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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