Integrand size = 20, antiderivative size = 438 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}+\frac {18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3} \]
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Time = 0.29 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2501, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac {18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3} \]
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2501
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (\frac {d^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}-\frac {2 d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}+\frac {(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^2}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^2}-\frac {(6 d) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^2}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^2} \\ & = \frac {3 \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}-\frac {(6 d) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = \frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {(3 b n) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}+\frac {(9 b d n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}-\frac {\left (9 b d^2 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = -\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}-\frac {\left (9 b^2 d n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^3}+\frac {\left (18 b^2 d^2 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = \frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (18 b^3 d^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^3} \\ & = \frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}+\frac {18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.83 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \left (-510 d^2+57 d e \sqrt [3]{x}-8 e^2 x^{2/3}\right ) \sqrt [3]{x}-6 a b^2 n^2 \left (23 d^3-66 d^2 e \sqrt [3]{x}+15 d e^2 x^{2/3}-4 e^3 x\right )+36 a^3 \left (d^3+e^3 x\right )-18 a^2 b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )+6 b \left (18 a^2 \left (d^2-d e \sqrt [3]{x}+e^2 x^{2/3}\right )-6 a b n \left (11 d^2-5 d e \sqrt [3]{x}+2 e^2 x^{2/3}\right )+b^2 n^2 \left (85 d^2-19 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )\right ) \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+18 b^2 \left (6 a \left (d^3+e^3 x\right )-b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+36 b^3 \left (d^3+e^3 x\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{36 e^3} \]
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\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}d x\]
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Time = 0.34 (sec) , antiderivative size = 690, normalized size of antiderivative = 1.58 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {36 \, b^{3} e^{3} x \log \left (c\right )^{3} + 36 \, {\left (b^{3} e^{3} n^{3} x + b^{3} d^{3} n^{3}\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{3} - 36 \, {\left (b^{3} e^{3} n - 3 \, a b^{2} e^{3}\right )} x \log \left (c\right )^{2} + 18 \, {\left (3 \, b^{3} d e^{2} n^{3} x^{\frac {2}{3}} - 6 \, b^{3} d^{2} e n^{3} x^{\frac {1}{3}} - 11 \, b^{3} d^{3} n^{3} + 6 \, a b^{2} d^{3} n^{2} - 2 \, {\left (b^{3} e^{3} n^{3} - 3 \, a b^{2} e^{3} n^{2}\right )} x + 6 \, {\left (b^{3} e^{3} n^{2} x + b^{3} d^{3} n^{2}\right )} \log \left (c\right )\right )} \log \left (e x^{\frac {1}{3}} + d\right )^{2} + 12 \, {\left (2 \, b^{3} e^{3} n^{2} - 6 \, a b^{2} e^{3} n + 9 \, a^{2} b e^{3}\right )} x \log \left (c\right ) - 4 \, {\left (2 \, b^{3} e^{3} n^{3} - 6 \, a b^{2} e^{3} n^{2} + 9 \, a^{2} b e^{3} n - 9 \, a^{3} e^{3}\right )} x + 6 \, {\left (85 \, b^{3} d^{3} n^{3} - 66 \, a b^{2} d^{3} n^{2} + 18 \, a^{2} b d^{3} n + 18 \, {\left (b^{3} e^{3} n x + b^{3} d^{3} n\right )} \log \left (c\right )^{2} + 2 \, {\left (2 \, b^{3} e^{3} n^{3} - 6 \, a b^{2} e^{3} n^{2} + 9 \, a^{2} b e^{3} n\right )} x - 6 \, {\left (11 \, b^{3} d^{3} n^{2} - 6 \, a b^{2} d^{3} n + 2 \, {\left (b^{3} e^{3} n^{2} - 3 \, a b^{2} e^{3} n\right )} x\right )} \log \left (c\right ) - 3 \, {\left (5 \, b^{3} d e^{2} n^{3} - 6 \, b^{3} d e^{2} n^{2} \log \left (c\right ) - 6 \, a b^{2} d e^{2} n^{2}\right )} x^{\frac {2}{3}} + 6 \, {\left (11 \, b^{3} d^{2} e n^{3} - 6 \, b^{3} d^{2} e n^{2} \log \left (c\right ) - 6 \, a b^{2} d^{2} e n^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (e x^{\frac {1}{3}} + d\right ) + 3 \, {\left (19 \, b^{3} d e^{2} n^{3} + 18 \, b^{3} d e^{2} n \log \left (c\right )^{2} - 30 \, a b^{2} d e^{2} n^{2} + 18 \, a^{2} b d e^{2} n - 6 \, {\left (5 \, b^{3} d e^{2} n^{2} - 6 \, a b^{2} d e^{2} n\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 6 \, {\left (85 \, b^{3} d^{2} e n^{3} + 18 \, b^{3} d^{2} e n \log \left (c\right )^{2} - 66 \, a b^{2} d^{2} e n^{2} + 18 \, a^{2} b d^{2} e n - 6 \, {\left (11 \, b^{3} d^{2} e n^{2} - 6 \, a b^{2} d^{2} e n\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}}{36 \, e^{3}} \]
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\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.04 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {1}{2} \, {\left (e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} + 6 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )\right )} a^{2} b + \frac {1}{6} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + 18 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {{\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{3}}\right )} a b^{2} + \frac {1}{36} \, {\left (18 \, e n {\left (\frac {6 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{4}} - \frac {2 \, e^{2} x - 3 \, d e x^{\frac {2}{3}} + 6 \, d^{2} x^{\frac {1}{3}}}{e^{3}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + 36 \, x \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{3} + e n {\left (\frac {{\left (36 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{3} + 198 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 8 \, e^{3} x + 510 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 57 \, d e^{2} x^{\frac {2}{3}} - 510 \, d^{2} e x^{\frac {1}{3}}\right )} n^{2}}{e^{4}} - \frac {6 \, {\left (18 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4 \, e^{3} x + 66 \, d^{3} \log \left (e x^{\frac {1}{3}} + d\right ) + 15 \, d e^{2} x^{\frac {2}{3}} - 66 \, d^{2} e x^{\frac {1}{3}}\right )} n \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{e^{4}}\right )}\right )} b^{3} + a^{3} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 1072 vs. \(2 (384) = 768\).
Time = 0.32 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.45 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
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Time = 1.88 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.27 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=x\,\left (a^3-a^2\,b\,n+\frac {2\,a\,b^2\,n^2}{3}-\frac {2\,b^3\,n^3}{9}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{2\,e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{4\,e}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d^3}{e^3}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{2\,e^3}-x^{2/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}\right )+b^2\,x\,\left (3\,a-b\,n\right )+\frac {d\,x^{1/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {9\,a\,b^2\,d}{e}\right )}{e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{e^2}\right )+\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (18\,a^2\,b\,d^3\,n-66\,a\,b^2\,d^3\,n^2+85\,b^3\,d^3\,n^3\right )}{6\,e^3}+\frac {\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {x^{1/3}\,\left (\frac {d\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+6\,b^3\,d^2\,n^2\right )}{e}-\frac {x^{2/3}\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{2\,e}+\frac {b\,e\,x\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3}\right )}{e} \]
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