Integrand size = 24, antiderivative size = 572 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}-\frac {2 b e^9 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {2 b^2 e^9 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9} \]
[Out]
Time = 1.06 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 36, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2504, 2445, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=-\frac {2 b e^9 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}-\frac {2 b e^8 n \sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {2 b^2 e^9 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2} \]
[In]
[Out]
Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2445
Rule 2458
Rule 2504
Rubi steps \begin{align*} \text {integral}& = -\left (3 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^{10}} \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {1}{3} (2 b e n) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^9 (d+e x)} \, dx,x,\frac {1}{\sqrt [3]{x}}\right ) \\ & = \frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {1}{3} (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 )^9} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right ) \\ & = \frac {1}{3} x^3 \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 )^9} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 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 )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d} \\ & = \frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \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 )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 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 )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 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 )^8} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{12 d} \\ & = -\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {\left (2 b e^2 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^3}-\frac {\left (b^2 e n^2\right ) \text {Subst}\left (\int \left (\frac {e^8}{d (d-x)^8}+\frac {e^8}{d^2 (d-x)^7}+\frac {e^8}{d^3 (d-x)^6}+\frac {e^8}{d^4 (d-x)^5}+\frac {e^8}{d^5 (d-x)^4}+\frac {e^8}{d^6 (d-x)^3}+\frac {e^8}{d^7 (d-x)^2}+\frac {e^8}{d^8 (d-x)}+\frac {e^8}{d^8 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{12 d}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^7} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{21 d^2} \\ & = \frac {b^2 e^8 n^2 \sqrt [3]{x}}{12 d^8}-\frac {b^2 e^7 n^2 x^{2/3}}{24 d^7}+\frac {b^2 e^6 n^2 x}{36 d^6}-\frac {b^2 e^5 n^2 x^{4/3}}{48 d^5}+\frac {b^2 e^4 n^2 x^{5/3}}{60 d^4}-\frac {b^2 e^3 n^2 x^2}{72 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{12 d^9}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {b^2 e^9 n^2 \log (x)}{36 d^9}+\frac {\left (2 b e^3 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^4}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^4}+\frac {\left (2 b^2 e^2 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^7}{d (d-x)^7}-\frac {e^7}{d^2 (d-x)^6}-\frac {e^7}{d^3 (d-x)^5}-\frac {e^7}{d^4 (d-x)^4}-\frac {e^7}{d^5 (d-x)^3}-\frac {e^7}{d^6 (d-x)^2}-\frac {e^7}{d^7 (d-x)}-\frac {e^7}{d^7 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{21 d^2}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^6} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^3} \\ & = \frac {5 b^2 e^8 n^2 \sqrt [3]{x}}{28 d^8}-\frac {5 b^2 e^7 n^2 x^{2/3}}{56 d^7}+\frac {5 b^2 e^6 n^2 x}{84 d^6}-\frac {5 b^2 e^5 n^2 x^{4/3}}{112 d^5}+\frac {b^2 e^4 n^2 x^{5/3}}{28 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {5 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{28 d^9}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {5 b^2 e^9 n^2 \log (x)}{84 d^9}-\frac {\left (2 b e^4 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^5}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^5}-\frac {\left (b^2 e^3 n^2\right ) \text {Subst}\left (\int \left (\frac {e^6}{d (d-x)^6}+\frac {e^6}{d^2 (d-x)^5}+\frac {e^6}{d^3 (d-x)^4}+\frac {e^6}{d^4 (d-x)^3}+\frac {e^6}{d^5 (d-x)^2}+\frac {e^6}{d^6 (d-x)}+\frac {e^6}{d^6 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^3}+\frac {\left (2 b^2 e^4 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^5} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{15 d^4} \\ & = \frac {73 b^2 e^8 n^2 \sqrt [3]{x}}{252 d^8}-\frac {73 b^2 e^7 n^2 x^{2/3}}{504 d^7}+\frac {73 b^2 e^6 n^2 x}{756 d^6}-\frac {73 b^2 e^5 n^2 x^{4/3}}{1008 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {73 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{252 d^9}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {73 b^2 e^9 n^2 \log (x)}{756 d^9}+\frac {\left (2 b e^5 n\right ) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^6}-\frac {\left (2 b e^6 n\right ) \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 )}{3 d^6}+\frac {\left (2 b^2 e^4 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^5}{d (d-x)^5}-\frac {e^5}{d^2 (d-x)^4}-\frac {e^5}{d^3 (d-x)^3}-\frac {e^5}{d^4 (d-x)^2}-\frac {e^5}{d^5 (d-x)}-\frac {e^5}{d^5 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{15 d^4}-\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^4} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{6 d^5} \\ & = \frac {533 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac {533 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac {533 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {533 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{1260 d^9}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {533 b^2 e^9 n^2 \log (x)}{3780 d^9}-\frac {\left (2 b e^6 n\right ) \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 )}{3 d^7}+\frac {\left (2 b e^7 n\right ) \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 )}{3 d^7}-\frac {\left (b^2 e^5 n^2\right ) \text {Subst}\left (\int \left (\frac {e^4}{d (d-x)^4}+\frac {e^4}{d^2 (d-x)^3}+\frac {e^4}{d^3 (d-x)^2}+\frac {e^4}{d^4 (d-x)}+\frac {e^4}{d^4 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{6 d^5}+\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \frac {1}{x \left (-\frac {d}{e}+\frac {x}{e}\right )^3} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^6} \\ & = \frac {743 b^2 e^8 n^2 \sqrt [3]{x}}{1260 d^8}-\frac {743 b^2 e^7 n^2 x^{2/3}}{2520 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {743 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{1260 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {743 b^2 e^9 n^2 \log (x)}{3780 d^9}+\frac {\left (2 b e^7 n\right ) \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 )}{3 d^8}-\frac {\left (2 b e^8 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 )}{3 d^8}+\frac {\left (2 b^2 e^6 n^2\right ) \text {Subst}\left (\int \left (-\frac {e^3}{d (d-x)^3}-\frac {e^3}{d^2 (d-x)^2}-\frac {e^3}{d^3 (d-x)}-\frac {e^3}{d^3 x}\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{9 d^6}-\frac {\left (b^2 e^7 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 )}{3 d^7} \\ & = \frac {341 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {341 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 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 )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}-\frac {2 b e^9 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 )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {341 b^2 e^9 n^2 \log (x)}{1260 d^9}-\frac {\left (b^2 e^7 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 )}{3 d^7}+\frac {\left (2 b^2 e^8 n^2\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x}{e}} \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{3 d^9}+\frac {\left (2 b^2 e^9 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 )}{3 d^9} \\ & = \frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 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 )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}-\frac {2 b e^9 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 )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {2 b^2 e^9 n^2 \text {Li}_2\left (\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.05 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \left (x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b e n \left (-10080 a d e^7 \sqrt [3]{x}+17316 b d e^7 n \sqrt [3]{x}+5040 a d^2 e^6 x^{2/3}-6138 b d^2 e^6 n x^{2/3}-3360 a d^3 e^5 x+2972 b d^3 e^5 n x+2520 a d^4 e^4 x^{4/3}-1599 b d^4 e^4 n x^{4/3}-2016 a d^5 e^3 x^{5/3}+876 b d^5 e^3 n x^{5/3}+1680 a d^6 e^2 x^2-450 b d^6 e^2 n x^2-1440 a d^7 e x^{7/3}+180 b d^7 e n x^{7/3}+1260 a d^8 x^{8/3}-22356 b e^8 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )-10080 b d e^7 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+5040 b d^2 e^6 x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-3360 b d^3 e^5 x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+2520 b d^4 e^4 x^{4/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-2016 b d^5 e^3 x^{5/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1680 b d^6 e^2 x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-1440 b d^7 e x^{7/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1260 b d^8 x^{8/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+10080 a e^8 \log \left (e+d \sqrt [3]{x}\right )-5040 b e^8 n \log \left (e+d \sqrt [3]{x}\right )+10080 b e^8 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (e+d \sqrt [3]{x}\right )-5040 b e^8 n \log ^2\left (e+d \sqrt [3]{x}\right )+10080 b e^8 n \log \left (e+d \sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )-7452 b e^8 n \log (x)+10080 b e^8 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )}{5040 d^9}\right ) \]
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\[\int x^{2} {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}d x\]
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\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
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\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \]
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