Integrand size = 20, antiderivative size = 227 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3} \]
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Time = 0.31 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {2501, 2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}-\frac {2 b e^2 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {2 b^2 e^3 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2} \]
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Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2501
Rule 2504
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right ) \\ & = -\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^3 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {(2 b n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d} \\ & = \frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {(2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^2} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d} \\ & = -\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^2}{d (d-x)^2}+\frac {e^2}{d^2 (d-x)}+\frac {e^2}{d^2 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {\left (2 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{x}\right )}{x} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{d^3} \\ & = \frac {b^2 e^2 n^2 \sqrt [3]{x}}{d^2}-\frac {b^2 e^3 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {2 b e^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+\frac {b e n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d}-\frac {2 b e^3 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^3}+x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^3 n^2 \log (x)}{d^3}+\frac {2 b^2 e^3 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^3} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.06 \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b e n \left (6 a d e \sqrt [3]{x}+6 b e^2 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+6 b d e \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-3 d^2 x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )-6 e^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \log \left (e+d \sqrt [3]{x}\right )+3 b e n \left (-d \sqrt [3]{x}+e \log \left (e+d \sqrt [3]{x}\right )\right )+2 b e^2 n \log (x)+3 b e^2 n \left (\log \left (e+d \sqrt [3]{x}\right ) \left (\log \left (e+d \sqrt [3]{x}\right )-2 \log \left (-\frac {d \sqrt [3]{x}}{e}\right )\right )-2 \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )\right )}{3 d^3} \]
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\[\int {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}d x\]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{2}\, dx \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
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\[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \]
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