Integrand size = 24, antiderivative size = 476 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4 i b^2 e^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}} \]
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Time = 0.40 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2508, 2507, 2526, 2505, 331, 211, 2520, 12, 5040, 4964, 2449, 2352} \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\frac {4 b e^{9/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4 i b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}-\frac {1408 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}} \]
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Rule 12
Rule 211
Rule 331
Rule 2352
Rule 2449
Rule 2505
Rule 2507
Rule 2508
Rule 2520
Rule 2526
Rule 4964
Rule 5040
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{x^{10}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {1}{3} (4 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^8 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {1}{3} (4 b e n) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d x^8}-\frac {e \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^2 x^6}+\frac {e^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^3 x^4}-\frac {e^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^4 x^2}+\frac {e^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {(4 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^8} \, dx,x,\sqrt [3]{x}\right )}{3 d}-\frac {\left (4 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^6} \, dx,x,\sqrt [3]{x}\right )}{3 d^2}+\frac {\left (4 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )}{3 d^3}-\frac {\left (4 b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}+\frac {\left (4 b e^5 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4} \\ & = -\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {\left (8 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x^6 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 d}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{9 d^3}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^4}-\frac {\left (8 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{3 d^4} \\ & = -\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {8 b^2 e^3 n^2}{45 d^3 x}-\frac {8 b^2 e^4 n^2}{9 d^4 \sqrt [3]{x}}-\frac {8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}-\frac {\left (8 b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x^4 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 d^2}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{15 d^3}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 d^4}-\frac {\left (8 b^2 e^{11/2} n^2\right ) \text {Subst}\left (\int \frac {x \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 d^{9/2}} \\ & = -\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {64 b^2 e^4 n^2}{45 d^4 \sqrt [3]{x}}-\frac {32 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{9 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {\left (8 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (d+e x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{21 d^3}+\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{i-\frac {\sqrt {e} x}{\sqrt {d}}} \, dx,x,\sqrt [3]{x}\right )}{3 d^5}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 d^4} \\ & = -\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}-\frac {184 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{45 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1+\frac {i \sqrt {e} x}{\sqrt {d}}}\right )}{1+\frac {e x^2}{d}} \, dx,x,\sqrt [3]{x}\right )}{3 d^5}-\frac {\left (8 b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 d^4} \\ & = -\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {\left (8 i b^2 e^{9/2} n^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{3 d^{9/2}} \\ & = -\frac {8 b^2 e^2 n^2}{105 d^2 x^{5/3}}+\frac {32 b^2 e^3 n^2}{105 d^3 x}-\frac {568 b^2 e^4 n^2}{315 d^4 \sqrt [3]{x}}-\frac {1408 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{315 d^{9/2}}+\frac {4 i b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 d^{9/2}}+\frac {8 b^2 e^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 d^{9/2}}-\frac {4 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 d x^{7/3}}+\frac {4 b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 d^2 x^{5/3}}-\frac {4 b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 d^3 x}+\frac {4 b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^4 \sqrt [3]{x}}+\frac {4 b e^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^{9/2}}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4 i b^2 e^{9/2} n^2 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{3 d^{9/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.39 (sec) , antiderivative size = 473, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 x^3}+\frac {4}{3} b e n \left (-\frac {2 b e^{7/2} n \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{d^{9/2}}-\frac {2 b e n \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},-\frac {e x^{2/3}}{d}\right )}{35 d^2 x^{5/3}}+\frac {2 b e^2 n \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{15 d^3 x}-\frac {2 b e^3 n \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {e x^{2/3}}{d}\right )}{3 d^4 \sqrt [3]{x}}-\frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{7 d x^{7/3}}+\frac {e \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{5 d^2 x^{5/3}}-\frac {e^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 d^3 x}+\frac {e^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^4 \sqrt [3]{x}}+\frac {e^{7/2} \arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{d^{9/2}}+\frac {i b e^{7/2} n \left (\arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (\arctan \left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )-2 i \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{-i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}\right )\right )}{d^{9/2}}\right ) \]
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\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x^{4}}d x\]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^4} \,d x \]
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