Optimal. Leaf size=547 \[ \frac {4 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 i b^2 d^{9/2} n^2 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {8 b^2 d^{9/2} n^2 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3 \]
[Out]
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Rubi [A] time = 0.77, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {2458, 2457, 2476, 2448, 321, 205, 2455, 302, 2470, 12, 4920, 4854, 2402, 2315} \[ \frac {4 i b^2 d^{9/2} n^2 \text {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{3 e^{9/2}}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^{9/2}}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {8 b^2 d^{9/2} n^2 \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{3 e^{9/2}}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 205
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 2448
Rule 2455
Rule 2457
Rule 2458
Rule 2470
Rule 2476
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \, dx &=3 \operatorname {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b e n) \operatorname {Subst}\left (\int \frac {x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b e n) \operatorname {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{3} (4 b n) \operatorname {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )-\frac {\left (4 b d^4 n\right ) \operatorname {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (4 b d^5 n\right ) \operatorname {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 e^4}+\frac {\left (4 b d^3 n\right ) \operatorname {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (4 b d^2 n\right ) \operatorname {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {(4 b d n) \operatorname {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (4 b^2 d^4 n\right ) \operatorname {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}-\frac {1}{21} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {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 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac {1}{27} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {1}{21} \left (8 b^2 d n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {d^3}{e^4}+\frac {d^2 x^2}{e^3}-\frac {d x^4}{e^2}+\frac {x^6}{e}+\frac {d^4}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )-\frac {\left (8 b^2 d^{9/2} n^2\right ) \operatorname {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 e^{7/2}}+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \left (-\frac {d}{e^2}+\frac {x^2}{e}+\frac {d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \operatorname {Subst}\left (\int \left (\frac {d^2}{e^3}-\frac {d x^2}{e^2}+\frac {x^4}{e}-\frac {d^3}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{15 e}+\frac {1}{27} \left (8 b^2 e n^2\right ) \operatorname {Subst}\left (\int \left (\frac {d^4}{e^5}-\frac {d^3 x^2}{e^4}+\frac {d^2 x^4}{e^3}-\frac {d x^6}{e^2}+\frac {x^8}{e}-\frac {d^5}{e^5 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {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 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}-\frac {\left (8 b^2 d^5 n^2\right ) \operatorname {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{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 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-\frac {\left (8 b^2 d^4 n^2\right ) \operatorname {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 e^4}\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{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 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {\left (8 i b^2 d^{9/2} n^2\right ) \operatorname {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 e^{9/2}}\\ &=-\frac {4 a b d^4 n \sqrt [3]{x}}{3 e^4}+\frac {4504 b^2 d^4 n^2 \sqrt [3]{x}}{945 e^4}-\frac {1984 b^2 d^3 n^2 x}{2835 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3}}{4725 e^2}-\frac {128 b^2 d n^2 x^{7/3}}{1323 e}+\frac {8}{243} b^2 n^2 x^3-\frac {4504 b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{945 e^{9/2}}+\frac {4 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{3 e^{9/2}}+\frac {8 b^2 d^{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 e^{9/2}}-\frac {4 b^2 d^4 n \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{3 e^4}+\frac {4 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{9 e^3}-\frac {4 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{15 e^2}+\frac {4 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{21 e}-\frac {4}{27} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )+\frac {4 b d^{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 e^{9/2}}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {4 i b^2 d^{9/2} n^2 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{3 e^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 438, normalized size = 0.80 \[ \frac {\sqrt {e} \sqrt [3]{x} \left (99225 a^2 e^4 x^{8/3}-630 b \left (2 b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )-315 a e^4 x^{8/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )-1260 a b n \left (315 d^4-105 d^3 e x^{2/3}+63 d^2 e^2 x^{4/3}-45 d e^3 x^2+35 e^4 x^{8/3}\right )+99225 b^2 e^4 x^{8/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+8 b^2 n^2 \left (177345 d^4-26040 d^3 e x^{2/3}+9009 d^2 e^2 x^{4/3}-3600 d e^3 x^2+1225 e^4 x^{8/3}\right )\right )+1260 b d^{9/2} n \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right ) \left (315 a+315 b \log \left (c \left (d+e x^{2/3}\right )^n\right )+630 b n \log \left (\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )-1126 b n\right )+396900 i b^2 d^{9/2} n^2 \text {Li}_2\left (\frac {i \sqrt {d}+\sqrt {e} \sqrt [3]{x}}{\sqrt {e} \sqrt [3]{x}-i \sqrt {d}}\right )+396900 i b^2 d^{9/2} n^2 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{297675 e^{9/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right )^{2} + 2 \, a b x^{2} \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a^{2} x^{2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \left (b \ln \left (c \left (e \,x^{\frac {2}{3}}+d \right )^{n}\right )+a \right )^{2} x^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, b^{2} n^{2} x^{3} \log \left (e x^{\frac {2}{3}} + d\right )^{2} + \int \frac {9 \, {\left (b^{2} e \log \relax (c)^{2} + 2 \, a b e \log \relax (c) + a^{2} e\right )} x^{3} + 9 \, {\left (b^{2} d \log \relax (c)^{2} + 2 \, a b d \log \relax (c) + a^{2} d\right )} x^{\frac {7}{3}} - 2 \, {\left (2 \, b^{2} e n x^{3} - 9 \, {\left (b^{2} e \log \relax (c) + a b e\right )} x^{3} - 9 \, {\left (b^{2} d \log \relax (c) + a b d\right )} x^{\frac {7}{3}}\right )} n \log \left (e x^{\frac {2}{3}} + d\right )}{9 \, {\left (e x + d x^{\frac {1}{3}}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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