Optimal. Leaf size=449 \[ \frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {9 b^3 d^2 n^3 x^{2/3}}{e^2}+\frac {9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3} \]
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Rubi [A]
time = 0.30, antiderivative size = 449, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {2504, 2448,
2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} \frac {b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac {9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac {9 b^3 d^2 n^3 x^{2/3}}{e^2}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2333
Rule 2341
Rule 2342
Rule 2436
Rule 2437
Rule 2448
Rule 2504
Rubi steps
\begin {align*} \int x \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx &=\frac {3}{2} \text {Subst}\left (\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\right )\\ &=\frac {3}{2} \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,x^{2/3}\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,x^{2/3}\right )}{2 e^2}-\frac {(3 d) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,x^{2/3}\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,x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 e^3}-\frac {(3 d) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e x^{2/3}\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 x^{2/3}\right )}{2 e^3}\\ &=\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 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 x^{2/3}\right )}{2 e^3}\\ &=-\frac {9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (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 x^{2/3}\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 x^{2/3}\right )}{2 e^3}+\frac {\left (9 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 x^{2/3}\right )}{e^3}\\ &=\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (9 b^3 d^2 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x^{2/3}\right )}{e^3}\\ &=\frac {9 b^3 d n^3 \left (d+e x^{2/3}\right )^2}{8 e^3}-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^3}{9 e^3}+\frac {9 a b^2 d^2 n^2 x^{2/3}}{e^2}-\frac {9 b^3 d^2 n^3 x^{2/3}}{e^2}+\frac {9 b^3 d^2 n^2 \left (d+e x^{2/3}\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )}{e^3}-\frac {9 b^2 d n^2 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 e^3}+\frac {b^2 n^2 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {9 b d n \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 e^3}-\frac {b n \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 e^3}+\frac {3 d^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}-\frac {3 d \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}+\frac {\left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 e^3}\\ \end {align*}
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Mathematica [A]
time = 0.29, size = 428, normalized size = 0.95 \begin {gather*} \frac {36 a^3 d^3-198 a^2 b d^3 n-108 a^2 b d^2 e n x^{2/3}+396 a b^2 d^2 e n^2 x^{2/3}-510 b^3 d^2 e n^3 x^{2/3}+54 a^2 b d e^2 n x^{4/3}-90 a b^2 d e^2 n^2 x^{4/3}+57 b^3 d e^2 n^3 x^{4/3}+36 a^3 e^3 x^2-36 a^2 b e^3 n x^2+24 a b^2 e^3 n^2 x^2-8 b^3 e^3 n^3 x^2+114 b^3 d^3 n^3 \log \left (d+e x^{2/3}\right )+6 b \left (18 a^2 \left (d^3+e^3 x^2\right )-6 a b n \left (11 d^3+6 d^2 e x^{2/3}-3 d e^2 x^{4/3}+2 e^3 x^2\right )+b^2 n^2 \left (66 d^3+66 d^2 e x^{2/3}-15 d e^2 x^{4/3}+4 e^3 x^2\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+18 b^2 \left (6 a \left (d^3+e^3 x^2\right )-b n \left (11 d^3+6 d^2 e x^{2/3}-3 d e^2 x^{4/3}+2 e^3 x^2\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+36 b^3 \left (d^3+e^3 x^2\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )}{72 e^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )^{3}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 489, normalized size = 1.09 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + \frac {1}{4} \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} a^{2} b n e + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{12} \, {\left ({\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 66 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right ) - 66 \, d^{2} x^{\frac {2}{3}} e + 15 \, d x^{\frac {4}{3}} e^{2} - 4 \, x^{2} e^{3}\right )} n^{2} e^{\left (-3\right )} - 6 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )\right )} a b^{2} + \frac {1}{72} \, {\left (18 \, {\left (6 \, d^{3} e^{\left (-4\right )} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} n e \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )^{2} + {\left ({\left (36 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{3} + 198 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 510 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right ) - 510 \, d^{2} x^{\frac {2}{3}} e + 57 \, d x^{\frac {4}{3}} e^{2} - 8 \, x^{2} e^{3}\right )} n^{2} e^{\left (-4\right )} - 6 \, {\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 66 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right ) - 66 \, d^{2} x^{\frac {2}{3}} e + 15 \, d x^{\frac {4}{3}} e^{2} - 4 \, x^{2} e^{3}\right )} n e^{\left (-4\right )} \log \left ({\left (x^{\frac {2}{3}} e + d\right )}^{n} c\right )\right )} n e\right )} b^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.46, size = 671, normalized size = 1.49 \begin {gather*} \frac {1}{72} \, {\left (36 \, b^{3} x^{2} e^{3} \log \left (c\right )^{3} - 36 \, {\left (b^{3} n - 3 \, a b^{2}\right )} x^{2} e^{3} \log \left (c\right )^{2} + 12 \, {\left (2 \, b^{3} n^{2} - 6 \, a b^{2} n + 9 \, a^{2} b\right )} x^{2} e^{3} \log \left (c\right ) - 4 \, {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n - 9 \, a^{3}\right )} x^{2} e^{3} + 36 \, {\left (b^{3} d^{3} n^{3} + b^{3} n^{3} x^{2} e^{3}\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{3} - 18 \, {\left (6 \, b^{3} d^{2} n^{3} x^{\frac {2}{3}} e - 3 \, b^{3} d n^{3} x^{\frac {4}{3}} e^{2} + 11 \, b^{3} d^{3} n^{3} - 6 \, a b^{2} d^{3} n^{2} + 2 \, {\left (b^{3} n^{3} - 3 \, a b^{2} n^{2}\right )} x^{2} e^{3} - 6 \, {\left (b^{3} d^{3} n^{2} + b^{3} n^{2} x^{2} e^{3}\right )} \log \left (c\right )\right )} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 6 \, {\left (85 \, b^{3} d^{3} n^{3} - 66 \, a b^{2} d^{3} n^{2} + 18 \, a^{2} b d^{3} n + 2 \, {\left (2 \, b^{3} n^{3} - 6 \, a b^{2} n^{2} + 9 \, a^{2} b n\right )} x^{2} e^{3} + 18 \, {\left (b^{3} d^{3} n + b^{3} n x^{2} e^{3}\right )} \log \left (c\right )^{2} - 6 \, {\left (11 \, b^{3} d^{3} n^{2} - 6 \, a b^{2} d^{3} n + 2 \, {\left (b^{3} n^{2} - 3 \, a b^{2} n\right )} x^{2} e^{3}\right )} \log \left (c\right ) - 6 \, {\left (6 \, b^{3} d^{2} n^{2} e \log \left (c\right ) - {\left (11 \, b^{3} d^{2} n^{3} - 6 \, a b^{2} d^{2} n^{2}\right )} e\right )} x^{\frac {2}{3}} + 3 \, {\left (6 \, b^{3} d n^{2} x e^{2} \log \left (c\right ) - {\left (5 \, b^{3} d n^{3} - 6 \, a b^{2} d n^{2}\right )} x e^{2}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {2}{3}} e + d\right ) - 6 \, {\left (18 \, b^{3} d^{2} n e \log \left (c\right )^{2} - 6 \, {\left (11 \, b^{3} d^{2} n^{2} - 6 \, a b^{2} d^{2} n\right )} e \log \left (c\right ) + {\left (85 \, b^{3} d^{2} n^{3} - 66 \, a b^{2} d^{2} n^{2} + 18 \, a^{2} b d^{2} n\right )} e\right )} x^{\frac {2}{3}} + 3 \, {\left (18 \, b^{3} d n x e^{2} \log \left (c\right )^{2} - 6 \, {\left (5 \, b^{3} d n^{2} - 6 \, a b^{2} d n\right )} x e^{2} \log \left (c\right ) + {\left (19 \, b^{3} d n^{3} - 30 \, a b^{2} d n^{2} + 18 \, a^{2} b d n\right )} x e^{2}\right )} x^{\frac {1}{3}}\right )} e^{\left (-3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A]
time = 4.91, size = 778, normalized size = 1.73 \begin {gather*} \frac {1}{2} \, b^{3} x^{2} \log \left (c\right )^{3} + \frac {1}{72} \, {\left (36 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right )^{3} + {\left (36 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{3} - 36 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 162 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 324 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + 24 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} \log \left (x^{\frac {2}{3}} e + d\right ) - 162 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d \log \left (x^{\frac {2}{3}} e + d\right ) + 648 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2} \log \left (x^{\frac {2}{3}} e + d\right ) - 8 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} + 81 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d - 648 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2}\right )} e^{\left (-3\right )}\right )} b^{3} n^{3} + \frac {1}{12} \, {\left (18 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + {\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 12 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} \log \left (x^{\frac {2}{3}} e + d\right ) + 54 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d \log \left (x^{\frac {2}{3}} e + d\right ) - 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + 4 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} - 27 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d + 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2}\right )} e^{\left (-3\right )}\right )} b^{3} n^{2} \log \left (c\right ) + \frac {1}{4} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} b^{3} n \log \left (c\right )^{2} + \frac {3}{2} \, a b^{2} x^{2} \log \left (c\right )^{2} + \frac {1}{12} \, {\left (18 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right )^{2} + {\left (18 \, d^{3} \log \left (x^{\frac {2}{3}} e + d\right )^{2} - 12 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} \log \left (x^{\frac {2}{3}} e + d\right ) + 54 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d \log \left (x^{\frac {2}{3}} e + d\right ) - 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + 4 \, {\left (x^{\frac {2}{3}} e + d\right )}^{3} - 27 \, {\left (x^{\frac {2}{3}} e + d\right )}^{2} d + 108 \, {\left (x^{\frac {2}{3}} e + d\right )} d^{2}\right )} e^{\left (-3\right )}\right )} a b^{2} n^{2} + \frac {1}{2} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} a b^{2} n \log \left (c\right ) + \frac {3}{2} \, a^{2} b x^{2} \log \left (c\right ) + \frac {1}{4} \, {\left (6 \, x^{2} \log \left (x^{\frac {2}{3}} e + d\right ) + {\left (6 \, d^{3} e^{\left (-4\right )} \log \left ({\left | x^{\frac {2}{3}} e + d \right |}\right ) + {\left (3 \, d x^{\frac {4}{3}} e - 2 \, x^{2} e^{2} - 6 \, d^{2} x^{\frac {2}{3}}\right )} e^{\left (-3\right )}\right )} e\right )} a^{2} b n + \frac {1}{2} \, a^{3} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.72, size = 575, normalized size = 1.28 \begin {gather*} {\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^3\,\left (\frac {b^3\,x^2}{2}+\frac {b^3\,d^3}{2\,e^3}\right )-x^{4/3}\,\left (\frac {d\,\left (\frac {3\,a^3}{2}-\frac {3\,a^2\,b\,n}{2}+a\,b^2\,n^2-\frac {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 )}{8\,e}\right )+{\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )}^2\,\left (\frac {b^2\,x^2\,\left (3\,a-b\,n\right )}{2}-\frac {x^{4/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}\right )}{2}+\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{4\,e^3}+\frac {d\,x^{2/3}\,\left (\frac {6\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {18\,a\,b^2\,d}{e}\right )}{4\,e}\right )+x^{2/3}\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a^3}{2}-\frac {3\,a^2\,b\,n}{2}+a\,b^2\,n^2-\frac {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 )}{4\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{2\,e^2}\right )+x^2\,\left (\frac {a^3}{2}-\frac {a^2\,b\,n}{2}+\frac {a\,b^2\,n^2}{3}-\frac {b^3\,n^3}{9}\right )+\frac {\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\,\left (\frac {x^{2/3}\,\left (\frac {d\,\left (2\,b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-6\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+12\,b^3\,d^2\,n^2\right )}{2\,e}-\frac {x^{4/3}\,\left (2\,b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-6\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{4\,e}+\frac {b\,e\,x^2\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3}\right )}{2\,e}+\frac {\ln \left (d+e\,x^{2/3}\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 )}{12\,e^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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