Optimal. Leaf size=794 \[ \frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (1-\frac {2 \sqrt {d}}{\sqrt {d}+i \sqrt {e} \sqrt [3]{x}}\right )}{315 e^{9/2}}+\frac {2 b d^5 n \text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^{2/3}},x\right )}{3 e^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.94, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx &=3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c \left (d+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+e x^{2/3}\right )^n\right )\right )^3-(2 b e n) \text {Subst}\left (\int \frac {x^{10} \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{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 )^3-(2 b e n) \text {Subst}\left (\int \left (\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^5}-\frac {d^3 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^4}+\frac {d^2 x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^3}-\frac {d x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e^2}+\frac {x^8 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{e}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{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 )^3-(2 b n) \text {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 {\left (2 b d^4 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (2 b d^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {(2 b d n) \text {Subst}\left (\int x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2 \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \frac {x^8 \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 {\left (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {x^2 \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 )}{e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {Subst}\left (\int \frac {x^4 \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 )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \frac {x^6 \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 )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \text {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 {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {1}{7} \left (8 b^2 d n^2\right ) \text {Subst}\left (\int \left (-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4}+\frac {d^2 x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^6 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^4 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )+\frac {\left (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \left (\frac {a+b \log \left (c \left (d+e x^2\right )^n\right )}{e}-\frac {d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {Subst}\left (\int \left (-\frac {d \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}+\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {Subst}\left (\int \left (\frac {d^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3}-\frac {d x^2 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^2}+\frac {x^4 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e}-\frac {d^3 \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )}{e^3 \left (d+e x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e}+\frac {1}{9} \left (8 b^2 e n^2\right ) \text {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 {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{9} \left (8 b^2 n^2\right ) \text {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 (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (8 b^2 d^4 n^2\right ) \text {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 (8 b^2 d^4 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (8 b^2 d^5 n^2\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 )}{9 e^4}-\frac {\left (8 b^2 d^5 n^2\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 )}{7 e^4}-\frac {\left (8 b^2 d^5 n^2\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 )}{5 e^4}-\frac {\left (8 b^2 d^5 n^2\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 e^4}-\frac {\left (8 b^2 d^5 n^2\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 )}{e^4}-\frac {\left (8 b^2 d^3 n^2\right ) \text {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 )}{9 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {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 )}{7 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {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 )}{5 e^3}-\frac {\left (8 b^2 d^3 n^2\right ) \text {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 (8 b^2 d^2 n^2\right ) \text {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 )}{9 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {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 )}{7 e^2}+\frac {\left (8 b^2 d^2 n^2\right ) \text {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 )}{5 e^2}-\frac {\left (8 b^2 d n^2\right ) \text {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 )}{9 e}-\frac {\left (8 b^2 d n^2\right ) \text {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 )}{7 e}\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (8 b^3 d^4 n^2\right ) \text {Subst}\left (\int \log \left (c \left (d+e x^2\right )^n\right ) \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \text {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {1}{49} \left (16 b^3 d n^3\right ) \text {Subst}\left (\int \frac {x^8}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )+\frac {\left (16 b^3 d^5 n^3\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 )}{9 e^3}+\frac {\left (16 b^3 d^5 n^3\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 )}{7 e^3}+\frac {\left (16 b^3 d^5 n^3\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 )}{5 e^3}+\frac {\left (16 b^3 d^5 n^3\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 e^3}+\frac {\left (16 b^3 d^5 n^3\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 )}{e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {Subst}\left (\int \frac {x^4}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^2}-\frac {\left (16 b^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \text {Subst}\left (\int \frac {x^6}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \text {Subst}\left (\int \frac {x^{10}}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {1}{63} \left (16 b^3 d n^3\right ) \text {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 {1}{49} \left (16 b^3 d n^3\right ) \text {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 (16 b^3 d^{9/2} n^3\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 )}{9 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\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 )}{7 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\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 )}{5 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\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 e^{7/2}}+\frac {\left (16 b^3 d^{9/2} n^3\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 )}{e^{7/2}}-\frac {\left (16 b^3 d^4 n^3\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^3}-\frac {\left (16 b^3 d^4 n^3\right ) \text {Subst}\left (\int \frac {x^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^3}+\frac {\left (16 b^3 d^3 n^3\right ) \text {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 )}{27 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {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 )}{21 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {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 )}{15 e^2}+\frac {\left (16 b^3 d^3 n^3\right ) \text {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 (16 b^3 d^2 n^3\right ) \text {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 )}{45 e}-\frac {\left (16 b^3 d^2 n^3\right ) \text {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 )}{35 e}-\frac {\left (16 b^3 d^2 n^3\right ) \text {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 )}{25 e}-\frac {1}{81} \left (16 b^3 e n^3\right ) \text {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 {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (16 b^3 d^4 n^3\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 )}{9 e^4}-\frac {\left (16 b^3 d^4 n^3\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 )}{7 e^4}-\frac {\left (16 b^3 d^4 n^3\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 )}{5 e^4}-\frac {\left (16 b^3 d^4 n^3\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 e^4}-\frac {\left (16 b^3 d^4 n^3\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 )}{e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{81 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{63 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{49 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{45 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{35 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{27 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{25 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{21 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{15 e^4}+2 \frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{9 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{7 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{5 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e^4}+\frac {\left (16 b^3 d^5 n^3\right ) \text {Subst}\left (\int \frac {1}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}+\frac {\left (16 b^3 d^4 n^3\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 )}{9 e^4}+\frac {\left (16 b^3 d^4 n^3\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 )}{7 e^4}+\frac {\left (16 b^3 d^4 n^3\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 )}{5 e^4}+\frac {\left (16 b^3 d^4 n^3\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 e^4}+\frac {\left (16 b^3 d^4 n^3\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 )}{e^4}\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}-\frac {\left (16 i b^3 d^{9/2} n^3\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 )}{9 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\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 )}{7 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\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 )}{5 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\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 e^{9/2}}-\frac {\left (16 i b^3 d^{9/2} n^3\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 )}{e^{9/2}}\\ &=\frac {4504 a b^2 d^4 n^2 \sqrt [3]{x}}{315 e^4}-\frac {3475504 b^3 d^4 n^3 \sqrt [3]{x}}{99225 e^4}+\frac {637984 b^3 d^3 n^3 x}{297675 e^3}-\frac {221344 b^3 d^2 n^3 x^{5/3}}{496125 e^2}+\frac {3088 b^3 d n^3 x^{7/3}}{27783 e}-\frac {16}{729} b^3 n^3 x^3+\frac {3475504 b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )}{99225 e^{9/2}}-\frac {4504 i b^3 d^{9/2} n^3 \tan ^{-1}\left (\frac {\sqrt {e} \sqrt [3]{x}}{\sqrt {d}}\right )^2}{315 e^{9/2}}-\frac {9008 b^3 d^{9/2} n^3 \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 )}{315 e^{9/2}}+\frac {4504 b^3 d^4 n^2 \sqrt [3]{x} \log \left (c \left (d+e x^{2/3}\right )^n\right )}{315 e^4}-\frac {1984 b^2 d^3 n^2 x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{945 e^3}+\frac {1144 b^2 d^2 n^2 x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{1575 e^2}-\frac {128 b^2 d n^2 x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{441 e}+\frac {8}{81} b^2 n^2 x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )-\frac {4504 b^2 d^{9/2} n^2 \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 )}{315 e^{9/2}}-\frac {2 b d^4 n \sqrt [3]{x} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{e^4}+\frac {2 b d^3 n x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{3 e^3}-\frac {2 b d^2 n x^{5/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{5 e^2}+\frac {2 b d n x^{7/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{7 e}-\frac {2}{9} b n x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3-\frac {4504 i b^3 d^{9/2} n^3 \text {Li}_2\left (1-\frac {2}{1+\frac {i \sqrt {e} \sqrt [3]{x}}{\sqrt {d}}}\right )}{315 e^{9/2}}+\frac {\left (2 b d^5 n\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+e x^2\right )^n\right )\right )^2}{d+e x^2} \, dx,x,\sqrt [3]{x}\right )}{e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] Leaf count is larger than twice the leaf count of optimal. \(3146\) vs. \(2(794)=1588\).
time = 8.19, size = 3146, normalized size = 3.96 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________