3.5.57 \(\int x^2 (a+b \log (c (d+e \sqrt [3]{x})^n))^3 \, dx\) [457]

Optimal. Leaf size=1357 \[ \frac {9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac {63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac {252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac {7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac {72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac {18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac {18 b^3 d^8 n^3 \sqrt [3]{x}}{e^8}+\frac {18 b^3 d^8 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^9}-\frac {18 b^2 d^7 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 )}{e^9}+\frac {56 b^2 d^6 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^9}-\frac {63 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^9}+\frac {252 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^9}-\frac {14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}+\frac {72 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^9}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{32 e^9}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{81 e^9}-\frac {9 b d^8 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^9}+\frac {18 b d^7 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}{e^9}-\frac {28 b d^6 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^9}+\frac {63 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^9}-\frac {126 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^9}+\frac {14 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^9}-\frac {36 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^9}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^9}-\frac {b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{9 e^9}+\frac {3 d^8 \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^9}-\frac {12 d^7 \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^9}+\frac {28 d^6 \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^9}-\frac {42 d^5 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {42 d^4 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {28 d^3 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {12 d^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {\left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^9} \]

[Out]

1/3*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^9-9*b*d^8*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^
9+18*b*d^7*n*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9-28*b*d^6*n*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/
3))^n))^2/e^9+63/2*b*d^5*n*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9-126/5*b*d^4*n*(d+e*x^(1/3))^5*(a+
b*ln(c*(d+e*x^(1/3))^n))^2/e^9+14*b*d^3*n*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9-36/7*b*d^2*n*(d+e*
x^(1/3))^7*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9+9/8*b*d*n*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9+18*b^
3*d^8*n^2*(d+e*x^(1/3))*ln(c*(d+e*x^(1/3))^n)/e^9-18*b^2*d^7*n^2*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))/e
^9+56/3*b^2*d^6*n^2*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9-63/4*b^2*d^5*n^2*(d+e*x^(1/3))^4*(a+b*ln(c
*(d+e*x^(1/3))^n))/e^9+252/25*b^2*d^4*n^2*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9-14/3*b^2*d^3*n^2*(d+
e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+72/49*b^2*d^2*n^2*(d+e*x^(1/3))^7*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9
-9/32*b^2*d*n^2*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))/e^9+2/81*b^2*n^2*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*x^
(1/3))^n))/e^9-1/9*b*n*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^9-252/125*b^3*d^4*n^3*(d+e*x^(1/3))^5/e
^9+7/9*b^3*d^3*n^3*(d+e*x^(1/3))^6/e^9-72/343*b^3*d^2*n^3*(d+e*x^(1/3))^7/e^9+9/256*b^3*d*n^3*(d+e*x^(1/3))^8/
e^9-18*b^3*d^8*n^3*x^(1/3)/e^8+18*a*b^2*d^8*n^2*x^(1/3)/e^8+3*d^8*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))^3/
e^9-12*d^7*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^9+28*d^6*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n)
)^3/e^9-42*d^5*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^9+42*d^4*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3)
)^n))^3/e^9-28*d^3*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^9+12*d^2*(d+e*x^(1/3))^7*(a+b*ln(c*(d+e*x^(
1/3))^n))^3/e^9-3*d*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^9+9*b^3*d^7*n^3*(d+e*x^(1/3))^2/e^9-56/9*b
^3*d^6*n^3*(d+e*x^(1/3))^3/e^9+63/16*b^3*d^5*n^3*(d+e*x^(1/3))^4/e^9-2/729*b^3*n^3*(d+e*x^(1/3))^9/e^9

________________________________________________________________________________________

Rubi [A]
time = 1.02, antiderivative size = 1357, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \begin {gather*} -\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^9}{3 e^9}-\frac {b n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^9}{9 e^9}+\frac {2 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^9}{81 e^9}+\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac {3 d \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^8}{e^9}+\frac {9 b d n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^8}{8 e^9}-\frac {9 b^2 d n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^8}{32 e^9}-\frac {72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac {12 d^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^7}{e^9}-\frac {36 b d^2 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^7}{7 e^9}+\frac {72 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^7}{49 e^9}+\frac {7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac {28 d^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^6}{e^9}+\frac {14 b d^3 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^6}{e^9}-\frac {14 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^6}{3 e^9}-\frac {252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac {42 d^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^5}{e^9}-\frac {126 b d^4 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^5}{5 e^9}+\frac {252 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^5}{25 e^9}+\frac {63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac {42 d^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^4}{e^9}+\frac {63 b d^5 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^4}{2 e^9}-\frac {63 b^2 d^5 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^4}{4 e^9}-\frac {56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac {28 d^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^3}{e^9}-\frac {28 b d^6 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^3}{e^9}+\frac {56 b^2 d^6 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^3}{3 e^9}+\frac {9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {12 d^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac {18 b d^7 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {18 b^2 d^7 n^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right ) \left (d+e \sqrt [3]{x}\right )^2}{e^9}+\frac {3 d^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac {9 b d^8 n \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2 \left (d+e \sqrt [3]{x}\right )}{e^9}+\frac {18 b^3 d^8 n^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \left (d+e \sqrt [3]{x}\right )}{e^9}-\frac {18 b^3 d^8 n^3 \sqrt [3]{x}}{e^8}+\frac {18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3,x]

[Out]

(9*b^3*d^7*n^3*(d + e*x^(1/3))^2)/e^9 - (56*b^3*d^6*n^3*(d + e*x^(1/3))^3)/(9*e^9) + (63*b^3*d^5*n^3*(d + e*x^
(1/3))^4)/(16*e^9) - (252*b^3*d^4*n^3*(d + e*x^(1/3))^5)/(125*e^9) + (7*b^3*d^3*n^3*(d + e*x^(1/3))^6)/(9*e^9)
- (72*b^3*d^2*n^3*(d + e*x^(1/3))^7)/(343*e^9) + (9*b^3*d*n^3*(d + e*x^(1/3))^8)/(256*e^9) - (2*b^3*n^3*(d +
e*x^(1/3))^9)/(729*e^9) + (18*a*b^2*d^8*n^2*x^(1/3))/e^8 - (18*b^3*d^8*n^3*x^(1/3))/e^8 + (18*b^3*d^8*n^2*(d +
e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^9 - (18*b^2*d^7*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])
)/e^9 + (56*b^2*d^6*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) - (63*b^2*d^5*n^2*(d + e*x
^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^9) + (252*b^2*d^4*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x
^(1/3))^n]))/(25*e^9) - (14*b^2*d^3*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(3*e^9) + (72*b^2*
d^2*n^2*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n]))/(49*e^9) - (9*b^2*d*n^2*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n]))/(32*e^9) + (2*b^2*n^2*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(81*e^9) -
(9*b*d^8*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (18*b*d^7*n*(d + e*x^(1/3))^2*(a + b*Log
[c*(d + e*x^(1/3))^n])^2)/e^9 - (28*b*d^6*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (63*b*
d^5*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^9) - (126*b*d^4*n*(d + e*x^(1/3))^5*(a + b*Lo
g[c*(d + e*x^(1/3))^n])^2)/(5*e^9) + (14*b*d^3*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 - (
36*b*d^2*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^9) + (9*b*d*n*(d + e*x^(1/3))^8*(a + b*L
og[c*(d + e*x^(1/3))^n])^2)/(8*e^9) - (b*n*(d + e*x^(1/3))^9*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(9*e^9) + (3*
d^8*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (12*d^7*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^
(1/3))^n])^3)/e^9 + (28*d^6*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 - (42*d^5*(d + e*x^(1/3)
)^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (42*d^4*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^
9 - (28*d^3*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + (12*d^2*(d + e*x^(1/3))^7*(a + b*Log[c
*(d + e*x^(1/3))^n])^3)/e^9 - (3*d*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^9 + ((d + e*x^(1/3)
)^9*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(3*e^9)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[
e*f - d*g, 0] && IGtQ[q, 0]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rubi steps

\begin {align*} \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx &=3 \text {Subst}\left (\int x^8 \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^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac {8 d^7 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac {28 d^6 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac {56 d^5 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac {70 d^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac {56 d^3 (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac {28 d^2 (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}-\frac {8 d (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}+\frac {(d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^8}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=\frac {3 \text {Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}-\frac {(24 d) \text {Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}+\frac {\left (84 d^2\right ) \text {Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}-\frac {\left (168 d^3\right ) \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}+\frac {\left (210 d^4\right ) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}-\frac {\left (168 d^5\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^8}+\frac {\left (84 d^6\right ) \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^8}-\frac {\left (24 d^7\right ) \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^8}+\frac {\left (3 d^8\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^8}\\ &=\frac {3 \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {(24 d) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (84 d^2\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {\left (168 d^3\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (210 d^4\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {\left (168 d^5\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (84 d^6\right ) \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^9}-\frac {\left (24 d^7\right ) \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^9}+\frac {\left (3 d^8\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^9}\\ &=\frac {3 d^8 \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^9}-\frac {12 d^7 \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^9}+\frac {28 d^6 \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^9}-\frac {42 d^5 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {42 d^4 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {28 d^3 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {12 d^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {\left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^9}-\frac {(b n) \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {(9 b d n) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {\left (36 b d^2 n\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (84 b d^3 n\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {\left (126 b d^4 n\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (126 b d^5 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}-\frac {\left (84 b d^6 n\right ) \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^9}+\frac {\left (36 b d^7 n\right ) \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^9}-\frac {\left (9 b d^8 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^9}\\ &=-\frac {9 b d^8 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^9}+\frac {18 b d^7 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}{e^9}-\frac {28 b d^6 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^9}+\frac {63 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^9}-\frac {126 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^9}+\frac {14 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^9}-\frac {36 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^9}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^9}-\frac {b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{9 e^9}+\frac {3 d^8 \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^9}-\frac {12 d^7 \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^9}+\frac {28 d^6 \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^9}-\frac {42 d^5 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {42 d^4 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {28 d^3 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {12 d^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {\left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^9}+\frac {\left (2 b^2 n^2\right ) \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{9 e^9}-\frac {\left (9 b^2 d n^2\right ) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{4 e^9}+\frac {\left (72 b^2 d^2 n^2\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{7 e^9}-\frac {\left (28 b^2 d^3 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (252 b^2 d^4 n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{5 e^9}-\frac {\left (63 b^2 d^5 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}+\frac {\left (56 b^2 d^6 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^9}-\frac {\left (36 b^2 d^7 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^9}+\frac {\left (18 b^2 d^8 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^9}\\ &=\frac {9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac {63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac {252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac {7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac {72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac {18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac {18 b^2 d^7 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 )}{e^9}+\frac {56 b^2 d^6 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^9}-\frac {63 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^9}+\frac {252 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^9}-\frac {14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}+\frac {72 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^9}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{32 e^9}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{81 e^9}-\frac {9 b d^8 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^9}+\frac {18 b d^7 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}{e^9}-\frac {28 b d^6 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^9}+\frac {63 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^9}-\frac {126 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^9}+\frac {14 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^9}-\frac {36 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^9}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^9}-\frac {b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{9 e^9}+\frac {3 d^8 \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^9}-\frac {12 d^7 \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^9}+\frac {28 d^6 \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^9}-\frac {42 d^5 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {42 d^4 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {28 d^3 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {12 d^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {\left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^9}+\frac {\left (18 b^3 d^8 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e \sqrt [3]{x}\right )}{e^9}\\ &=\frac {9 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {56 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^9}+\frac {63 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac {252 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac {7 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{9 e^9}-\frac {72 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^9}+\frac {18 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}-\frac {18 b^3 d^8 n^3 \sqrt [3]{x}}{e^8}+\frac {18 b^3 d^8 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^9}-\frac {18 b^2 d^7 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 )}{e^9}+\frac {56 b^2 d^6 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^9}-\frac {63 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{4 e^9}+\frac {252 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{25 e^9}-\frac {14 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^9}+\frac {72 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^9}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{32 e^9}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{81 e^9}-\frac {9 b d^8 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^9}+\frac {18 b d^7 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}{e^9}-\frac {28 b d^6 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^9}+\frac {63 b d^5 n \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^9}-\frac {126 b d^4 n \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{5 e^9}+\frac {14 b d^3 n \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^9}-\frac {36 b d^2 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^9}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{8 e^9}-\frac {b n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{9 e^9}+\frac {3 d^8 \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^9}-\frac {12 d^7 \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^9}+\frac {28 d^6 \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^9}-\frac {42 d^5 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {42 d^4 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {28 d^3 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {12 d^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^9}+\frac {\left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^9}\\ \end {align*}

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Mathematica [A]
time = 0.71, size = 895, normalized size = 0.66 \begin {gather*} \frac {2667168000 b^3 d^9 n^3 \log ^3\left (d+e \sqrt [3]{x}\right )+3175200 b^2 d^9 n^2 \log ^2\left (d+e \sqrt [3]{x}\right ) \left (-2520 a+7129 b n-2520 b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+2520 b d^9 n \log \left (d+e \sqrt [3]{x}\right ) \left (3175200 a^2-17965080 a b n+30300391 b^2 n^2+2520 b (2520 a-7129 b n) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )+e \sqrt [3]{x} \left (2667168000 a^3 e^8 x^{8/3}+b^3 n^3 \left (-76356985320 d^8+15542491860 d^7 e \sqrt [3]{x}-5483495640 d^6 e^2 x^{2/3}[phone] d^5 e^3 x-1075607064 d^4 e^4 x^{4/3}[phone] d^3 e^5 x^{5/3}-219465000 d^2 e^6 x^2+83734875 d e^7 x^{7/3}-21952000 e^8 x^{8/3}\right )-3175200 a^2 b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+840 d^6 e^2 x^{2/3}-630 d^5 e^3 x+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )+2520 a b^2 n^2 \left (17965080 d^8-5807340 d^7 e \sqrt [3]{x}+2813160 d^6 e^2 x^{2/3}-1580670 d^5 e^3 x+947016 d^4 e^4 x^{4/3}-577500 d^3 e^5 x^{5/3}+343800 d^2 e^6 x^2-187425 d e^7 x^{7/3}+78400 e^8 x^{8/3}\right )+2520 b \left (3175200 a^2 e^8 x^{8/3}-2520 a b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+840 d^6 e^2 x^{2/3}-630 d^5 e^3 x+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )+b^2 n^2 \left (17965080 d^8-5807340 d^7 e \sqrt [3]{x}+2813160 d^6 e^2 x^{2/3}-1580670 d^5 e^3 x+947016 d^4 e^4 x^{4/3}-577500 d^3 e^5 x^{5/3}+343800 d^2 e^6 x^2-187425 d e^7 x^{7/3}+78400 e^8 x^{8/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-3175200 b^2 \left (-2520 a e^8 x^{8/3}+b n \left (2520 d^8-1260 d^7 e \sqrt [3]{x}+840 d^6 e^2 x^{2/3}-630 d^5 e^3 x+504 d^4 e^4 x^{4/3}-420 d^3 e^5 x^{5/3}+360 d^2 e^6 x^2-315 d e^7 x^{7/3}+280 e^8 x^{8/3}\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )[phone] b^3 e^8 x^{8/3} \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{8001504000 e^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3,x]

[Out]

(2667168000*b^3*d^9*n^3*Log[d + e*x^(1/3)]^3 + 3175200*b^2*d^9*n^2*Log[d + e*x^(1/3)]^2*(-2520*a + 7129*b*n -
2520*b*Log[c*(d + e*x^(1/3))^n]) + 2520*b*d^9*n*Log[d + e*x^(1/3)]*(3175200*a^2 - 17965080*a*b*n + 30300391*b^
2*n^2 + 2520*b*(2520*a - 7129*b*n)*Log[c*(d + e*x^(1/3))^n] + 3175200*b^2*Log[c*(d + e*x^(1/3))^n]^2) + e*x^(1
/3)*(2667168000*a^3*e^8*x^(8/3) + b^3*n^3*(-76356985320*d^8 + 15542491860*d^7*e*x^(1/3) - 5483495640*d^6*e^2*x
^(2/3) + 2340330930*d^5*e^3*x - 1075607064*d^4*e^4*x^(4/3) + 498592500*d^3*e^5*x^(5/3) - 219465000*d^2*e^6*x^2
+ 83734875*d*e^7*x^(7/3) - 21952000*e^8*x^(8/3)) - 3175200*a^2*b*n*(2520*d^8 - 1260*d^7*e*x^(1/3) + 840*d^6*e
^2*x^(2/3) - 630*d^5*e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315*d*e^7*x^(7/3) +
280*e^8*x^(8/3)) + 2520*a*b^2*n^2*(17965080*d^8 - 5807340*d^7*e*x^(1/3) + 2813160*d^6*e^2*x^(2/3) - 1580670*d
^5*e^3*x + 947016*d^4*e^4*x^(4/3) - 577500*d^3*e^5*x^(5/3) + 343800*d^2*e^6*x^2 - 187425*d*e^7*x^(7/3) + 78400
*e^8*x^(8/3)) + 2520*b*(3175200*a^2*e^8*x^(8/3) - 2520*a*b*n*(2520*d^8 - 1260*d^7*e*x^(1/3) + 840*d^6*e^2*x^(2
/3) - 630*d^5*e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315*d*e^7*x^(7/3) + 280*e^
8*x^(8/3)) + b^2*n^2*(17965080*d^8 - 5807340*d^7*e*x^(1/3) + 2813160*d^6*e^2*x^(2/3) - 1580670*d^5*e^3*x + 947
016*d^4*e^4*x^(4/3) - 577500*d^3*e^5*x^(5/3) + 343800*d^2*e^6*x^2 - 187425*d*e^7*x^(7/3) + 78400*e^8*x^(8/3)))
*Log[c*(d + e*x^(1/3))^n] - 3175200*b^2*(-2520*a*e^8*x^(8/3) + b*n*(2520*d^8 - 1260*d^7*e*x^(1/3) + 840*d^6*e^
2*x^(2/3) - 630*d^5*e^3*x + 504*d^4*e^4*x^(4/3) - 420*d^3*e^5*x^(5/3) + 360*d^2*e^6*x^2 - 315*d*e^7*x^(7/3) +
280*e^8*x^(8/3)))*Log[c*(d + e*x^(1/3))^n]^2 + 2667168000*b^3*e^8*x^(8/3)*Log[c*(d + e*x^(1/3))^n]^3))/(800150
4000*e^9)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3,x)

[Out]

int(x^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3,x)

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Maxima [A]
time = 0.31, size = 835, normalized size = 0.62 \begin {gather*} \frac {1}{3} \, b^{3} x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{3} + a b^{2} x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{2} + a^{2} b x^{3} \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right ) + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{2520} \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} a^{2} b n e - \frac {1}{3175200} \, {\left ({\left (3175200 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 17965080 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right ) - 17965080 \, d^{8} x^{\frac {1}{3}} e + 5807340 \, d^{7} x^{\frac {2}{3}} e^{2} - 2813160 \, d^{6} x e^{3} + 1580670 \, d^{5} x^{\frac {4}{3}} e^{4} - 947016 \, d^{4} x^{\frac {5}{3}} e^{5} + 577500 \, d^{3} x^{2} e^{6} - 343800 \, d^{2} x^{\frac {7}{3}} e^{7} + 187425 \, d x^{\frac {8}{3}} e^{8} - 78400 \, x^{3} e^{9}\right )} n^{2} e^{\left (-9\right )} - 2520 \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} n e \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )\right )} a b^{2} + \frac {1}{8001504000} \, {\left (3175200 \, {\left (2520 \, d^{9} e^{\left (-10\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + {\left (1260 \, d^{7} x^{\frac {2}{3}} e - 2520 \, d^{8} x^{\frac {1}{3}} - 840 \, d^{6} x e^{2} + 630 \, d^{5} x^{\frac {4}{3}} e^{3} - 504 \, d^{4} x^{\frac {5}{3}} e^{4} + 420 \, d^{3} x^{2} e^{5} - 360 \, d^{2} x^{\frac {7}{3}} e^{6} + 315 \, d x^{\frac {8}{3}} e^{7} - 280 \, x^{3} e^{8}\right )} e^{\left (-9\right )}\right )} n e \log \left ({\left (x^{\frac {1}{3}} e + d\right )}^{n} c\right )^{2} + {\left ({\left (2667168000 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right )^{3} + 22636000800 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 76356985320 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right ) - 76356985320 \, d^{8} x^{\frac {1}{3}} e + 15542491860 \, d^{7} x^{\frac {2}{3}} e^{2} - 5483495640 \, d^{6} x e^{3} + 2340330930 \, d^{5} x^{\frac {4}{3}} e^{4} - 1075607064 \, d^{4} x^{\frac {5}{3}} e^{5} + 498592500 \, d^{3} x^{2} e^{6} - 219465000 \, d^{2} x^{\frac {7}{3}} e^{7} + 83734875 \, d x^{\frac {8}{3}} e^{8} - 21952000 \, x^{3} e^{9}\right )} n^{2} e^{\left (-10\right )} - 2520 \, {\left (3175200 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right )^{2} + 17965080 \, d^{9} \log \left (x^{\frac {1}{3}} e + d\right ) - 17965080 \, d^{8} x^{\frac {1}{3}} e + 5807340 \, d^{7} x^{\frac {2}{3}} e^{2} - 2813160 \, d^{6} x e^{3} + 1580670 \, d^{5} x^{\frac {4}{3}} e^{4} - 947016 \, d^{4} x^{\frac {5}{3}} e^{5} + 577500 \, d^{3} x^{2} e^{6} - 343800 \, d^{2} x^{\frac {7}{3}} e^{7} + 187425 \, d x^{\frac {8}{3}} e^{8} - 78400 \, x^{3} e^{9}\right )} n e^{\left (-10\right )} \log \left ({\left (x^{\frac {1}{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.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^3,x, algorithm="maxima")

[Out]

1/3*b^3*x^3*log((x^(1/3)*e + d)^n*c)^3 + a*b^2*x^3*log((x^(1/3)*e + d)^n*c)^2 + a^2*b*x^3*log((x^(1/3)*e + d)^
n*c) + 1/3*a^3*x^3 + 1/2520*(2520*d^9*e^(-10)*log(x^(1/3)*e + d) + (1260*d^7*x^(2/3)*e - 2520*d^8*x^(1/3) - 84
0*d^6*x*e^2 + 630*d^5*x^(4/3)*e^3 - 504*d^4*x^(5/3)*e^4 + 420*d^3*x^2*e^5 - 360*d^2*x^(7/3)*e^6 + 315*d*x^(8/3
)*e^7 - 280*x^3*e^8)*e^(-9))*a^2*b*n*e - 1/3175200*((3175200*d^9*log(x^(1/3)*e + d)^2 + 17965080*d^9*log(x^(1/
3)*e + d) - 17965080*d^8*x^(1/3)*e + 5807340*d^7*x^(2/3)*e^2 - 2813160*d^6*x*e^3 + 1580670*d^5*x^(4/3)*e^4 - 9
47016*d^4*x^(5/3)*e^5 + 577500*d^3*x^2*e^6 - 343800*d^2*x^(7/3)*e^7 + 187425*d*x^(8/3)*e^8 - 78400*x^3*e^9)*n^
2*e^(-9) - 2520*(2520*d^9*e^(-10)*log(x^(1/3)*e + d) + (1260*d^7*x^(2/3)*e - 2520*d^8*x^(1/3) - 840*d^6*x*e^2
+ 630*d^5*x^(4/3)*e^3 - 504*d^4*x^(5/3)*e^4 + 420*d^3*x^2*e^5 - 360*d^2*x^(7/3)*e^6 + 315*d*x^(8/3)*e^7 - 280*
x^3*e^8)*e^(-9))*n*e*log((x^(1/3)*e + d)^n*c))*a*b^2 + 1/8001504000*(3175200*(2520*d^9*e^(-10)*log(x^(1/3)*e +
d) + (1260*d^7*x^(2/3)*e - 2520*d^8*x^(1/3) - 840*d^6*x*e^2 + 630*d^5*x^(4/3)*e^3 - 504*d^4*x^(5/3)*e^4 + 420
*d^3*x^2*e^5 - 360*d^2*x^(7/3)*e^6 + 315*d*x^(8/3)*e^7 - 280*x^3*e^8)*e^(-9))*n*e*log((x^(1/3)*e + d)^n*c)^2 +
((2667168000*d^9*log(x^(1/3)*e + d)^3 + 22636000800*d^9*log(x^(1/3)*e + d)^2 + 76356985320*d^9*log(x^(1/3)*e
+ d) - 76356985320*d^8*x^(1/3)*e + 15542491860*d^7*x^(2/3)*e^2 - 5483495640*d^6*x*e^3 + 2340330930*d^5*x^(4/3)
*e^4 - 1075607064*d^4*x^(5/3)*e^5 + 498592500*d^3*x^2*e^6 - 219465000*d^2*x^(7/3)*e^7 + 83734875*d*x^(8/3)*e^8
- 21952000*x^3*e^9)*n^2*e^(-10) - 2520*(3175200*d^9*log(x^(1/3)*e + d)^2 + 17965080*d^9*log(x^(1/3)*e + d) -
17965080*d^8*x^(1/3)*e + 5807340*d^7*x^(2/3)*e^2 - 2813160*d^6*x*e^3 + 1580670*d^5*x^(4/3)*e^4 - 947016*d^4*x^
(5/3)*e^5 + 577500*d^3*x^2*e^6 - 343800*d^2*x^(7/3)*e^7 + 187425*d*x^(8/3)*e^8 - 78400*x^3*e^9)*n*e^(-10)*log(
(x^(1/3)*e + d)^n*c))*n*e)*b^3

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Fricas [A]
time = 0.46, size = 1544, normalized size = 1.14 \begin {gather*} \frac {1}{8001504000} \, {\left (2667168000 \, b^{3} x^{3} e^{9} \log \left (c\right )^{3} - 10976000 \, {\left (2 \, b^{3} n^{3} - 18 \, a b^{2} n^{2} + 81 \, a^{2} b n - 243 \, a^{3}\right )} x^{3} e^{9} + 10500 \, {\left (47485 \, b^{3} d^{3} n^{3} - 138600 \, a b^{2} d^{3} n^{2} + 127008 \, a^{2} b d^{3} n\right )} x^{2} e^{6} + 2667168000 \, {\left (b^{3} d^{9} n^{3} + b^{3} n^{3} x^{3} e^{9}\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{3} - 840 \, {\left (6527971 \, b^{3} d^{6} n^{3} - 8439480 \, a b^{2} d^{6} n^{2} + 3175200 \, a^{2} b d^{6} n\right )} x e^{3} - 3175200 \, {\left (7129 \, b^{3} d^{9} n^{3} - 2520 \, a b^{2} d^{9} n^{2} + 840 \, b^{3} d^{6} n^{3} x e^{3} - 420 \, b^{3} d^{3} n^{3} x^{2} e^{6} + 280 \, {\left (b^{3} n^{3} - 9 \, a b^{2} n^{2}\right )} x^{3} e^{9} - 2520 \, {\left (b^{3} d^{9} n^{2} + b^{3} n^{2} x^{3} e^{9}\right )} \log \left (c\right ) - 63 \, {\left (20 \, b^{3} d^{7} n^{3} e^{2} - 8 \, b^{3} d^{4} n^{3} x e^{5} + 5 \, b^{3} d n^{3} x^{2} e^{8}\right )} x^{\frac {2}{3}} + 90 \, {\left (28 \, b^{3} d^{8} n^{3} e - 7 \, b^{3} d^{5} n^{3} x e^{4} + 4 \, b^{3} d^{2} n^{3} x^{2} e^{7}\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}} e + d\right )^{2} - 444528000 \, {\left (6 \, b^{3} d^{6} n x e^{3} - 3 \, b^{3} d^{3} n x^{2} e^{6} + 2 \, {\left (b^{3} n - 9 \, a b^{2}\right )} x^{3} e^{9}\right )} \log \left (c\right )^{2} + 2520 \, {\left (30300391 \, b^{3} d^{9} n^{3} - 17965080 \, a b^{2} d^{9} n^{2} + 3175200 \, a^{2} b d^{9} n + 39200 \, {\left (2 \, b^{3} n^{3} - 18 \, a b^{2} n^{2} + 81 \, a^{2} b n\right )} x^{3} e^{9} - 2100 \, {\left (275 \, b^{3} d^{3} n^{3} - 504 \, a b^{2} d^{3} n^{2}\right )} x^{2} e^{6} + 840 \, {\left (3349 \, b^{3} d^{6} n^{3} - 2520 \, a b^{2} d^{6} n^{2}\right )} x e^{3} + 3175200 \, {\left (b^{3} d^{9} n + b^{3} n x^{3} e^{9}\right )} \log \left (c\right )^{2} - 2520 \, {\left (7129 \, b^{3} d^{9} n^{2} - 2520 \, a b^{2} d^{9} n + 840 \, b^{3} d^{6} n^{2} x e^{3} - 420 \, b^{3} d^{3} n^{2} x^{2} e^{6} + 280 \, {\left (b^{3} n^{2} - 9 \, a b^{2} n\right )} x^{3} e^{9}\right )} \log \left (c\right ) - 63 \, {\left (175 \, {\left (17 \, b^{3} d n^{3} - 72 \, a b^{2} d n^{2}\right )} x^{2} e^{8} - 8 \, {\left (1879 \, b^{3} d^{4} n^{3} - 2520 \, a b^{2} d^{4} n^{2}\right )} x e^{5} + 20 \, {\left (4609 \, b^{3} d^{7} n^{3} - 2520 \, a b^{2} d^{7} n^{2}\right )} e^{2} - 2520 \, {\left (20 \, b^{3} d^{7} n^{2} e^{2} - 8 \, b^{3} d^{4} n^{2} x e^{5} + 5 \, b^{3} d n^{2} x^{2} e^{8}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} + 90 \, {\left (20 \, {\left (191 \, b^{3} d^{2} n^{3} - 504 \, a b^{2} d^{2} n^{2}\right )} x^{2} e^{7} - 7 \, {\left (2509 \, b^{3} d^{5} n^{3} - 2520 \, a b^{2} d^{5} n^{2}\right )} x e^{4} + 28 \, {\left (7129 \, b^{3} d^{8} n^{3} - 2520 \, a b^{2} d^{8} n^{2}\right )} e - 2520 \, {\left (28 \, b^{3} d^{8} n^{2} e - 7 \, b^{3} d^{5} n^{2} x e^{4} + 4 \, b^{3} d^{2} n^{2} x^{2} e^{7}\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} \log \left (x^{\frac {1}{3}} e + d\right ) + 352800 \, {\left (280 \, {\left (2 \, b^{3} n^{2} - 18 \, a b^{2} n + 81 \, a^{2} b\right )} x^{3} e^{9} - 15 \, {\left (275 \, b^{3} d^{3} n^{2} - 504 \, a b^{2} d^{3} n\right )} x^{2} e^{6} + 6 \, {\left (3349 \, b^{3} d^{6} n^{2} - 2520 \, a b^{2} d^{6} n\right )} x e^{3}\right )} \log \left (c\right ) + 63 \, {\left (6125 \, {\left (217 \, b^{3} d n^{3} - 1224 \, a b^{2} d n^{2} + 2592 \, a^{2} b d n\right )} x^{2} e^{8} - 8 \, {\left (2134141 \, b^{3} d^{4} n^{3} - 4735080 \, a b^{2} d^{4} n^{2} + 3175200 \, a^{2} b d^{4} n\right )} x e^{5} + 3175200 \, {\left (20 \, b^{3} d^{7} n e^{2} - 8 \, b^{3} d^{4} n x e^{5} + 5 \, b^{3} d n x^{2} e^{8}\right )} \log \left (c\right )^{2} + 20 \, {\left (12335311 \, b^{3} d^{7} n^{3} - 11614680 \, a b^{2} d^{7} n^{2} + 3175200 \, a^{2} b d^{7} n\right )} e^{2} - 2520 \, {\left (175 \, {\left (17 \, b^{3} d n^{2} - 72 \, a b^{2} d n\right )} x^{2} e^{8} - 8 \, {\left (1879 \, b^{3} d^{4} n^{2} - 2520 \, a b^{2} d^{4} n\right )} x e^{5} + 20 \, {\left (4609 \, b^{3} d^{7} n^{2} - 2520 \, a b^{2} d^{7} n\right )} e^{2}\right )} \log \left (c\right )\right )} x^{\frac {2}{3}} - 90 \, {\left (100 \, {\left (24385 \, b^{3} d^{2} n^{3} - 96264 \, a b^{2} d^{2} n^{2} + 127008 \, a^{2} b d^{2} n\right )} x^{2} e^{7} - 7 \, {\left (3714811 \, b^{3} d^{5} n^{3} - 6322680 \, a b^{2} d^{5} n^{2} + 3175200 \, a^{2} b d^{5} n\right )} x e^{4} + 3175200 \, {\left (28 \, b^{3} d^{8} n e - 7 \, b^{3} d^{5} n x e^{4} + 4 \, b^{3} d^{2} n x^{2} e^{7}\right )} \log \left (c\right )^{2} + 28 \, {\left (30300391 \, b^{3} d^{8} n^{3} - 17965080 \, a b^{2} d^{8} n^{2} + 3175200 \, a^{2} b d^{8} n\right )} e - 2520 \, {\left (20 \, {\left (191 \, b^{3} d^{2} n^{2} - 504 \, a b^{2} d^{2} n\right )} x^{2} e^{7} - 7 \, {\left (2509 \, b^{3} d^{5} n^{2} - 2520 \, a b^{2} d^{5} n\right )} x e^{4} + 28 \, {\left (7129 \, b^{3} d^{8} n^{2} - 2520 \, a b^{2} d^{8} n\right )} e\right )} \log \left (c\right )\right )} x^{\frac {1}{3}}\right )} e^{\left (-9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^3,x, algorithm="fricas")

[Out]

1/8001504000*(2667168000*b^3*x^3*e^9*log(c)^3 - 10976000*(2*b^3*n^3 - 18*a*b^2*n^2 + 81*a^2*b*n - 243*a^3)*x^3
*e^9 + 10500*(47485*b^3*d^3*n^3 - 138600*a*b^2*d^3*n^2 + 127008*a^2*b*d^3*n)*x^2*e^6 + 2667168000*(b^3*d^9*n^3
+ b^3*n^3*x^3*e^9)*log(x^(1/3)*e + d)^3 - 840*(6527971*b^3*d^6*n^3 - 8439480*a*b^2*d^6*n^2 + 3175200*a^2*b*d^
6*n)*x*e^3 - 3175200*(7129*b^3*d^9*n^3 - 2520*a*b^2*d^9*n^2 + 840*b^3*d^6*n^3*x*e^3 - 420*b^3*d^3*n^3*x^2*e^6
+ 280*(b^3*n^3 - 9*a*b^2*n^2)*x^3*e^9 - 2520*(b^3*d^9*n^2 + b^3*n^2*x^3*e^9)*log(c) - 63*(20*b^3*d^7*n^3*e^2 -
8*b^3*d^4*n^3*x*e^5 + 5*b^3*d*n^3*x^2*e^8)*x^(2/3) + 90*(28*b^3*d^8*n^3*e - 7*b^3*d^5*n^3*x*e^4 + 4*b^3*d^2*n
^3*x^2*e^7)*x^(1/3))*log(x^(1/3)*e + d)^2 - 444528000*(6*b^3*d^6*n*x*e^3 - 3*b^3*d^3*n*x^2*e^6 + 2*(b^3*n - 9*
a*b^2)*x^3*e^9)*log(c)^2 + 2520*(30300391*b^3*d^9*n^3 - 17965080*a*b^2*d^9*n^2 + 3175200*a^2*b*d^9*n + 39200*(
2*b^3*n^3 - 18*a*b^2*n^2 + 81*a^2*b*n)*x^3*e^9 - 2100*(275*b^3*d^3*n^3 - 504*a*b^2*d^3*n^2)*x^2*e^6 + 840*(334
9*b^3*d^6*n^3 - 2520*a*b^2*d^6*n^2)*x*e^3 + 3175200*(b^3*d^9*n + b^3*n*x^3*e^9)*log(c)^2 - 2520*(7129*b^3*d^9*
n^2 - 2520*a*b^2*d^9*n + 840*b^3*d^6*n^2*x*e^3 - 420*b^3*d^3*n^2*x^2*e^6 + 280*(b^3*n^2 - 9*a*b^2*n)*x^3*e^9)*
log(c) - 63*(175*(17*b^3*d*n^3 - 72*a*b^2*d*n^2)*x^2*e^8 - 8*(1879*b^3*d^4*n^3 - 2520*a*b^2*d^4*n^2)*x*e^5 + 2
0*(4609*b^3*d^7*n^3 - 2520*a*b^2*d^7*n^2)*e^2 - 2520*(20*b^3*d^7*n^2*e^2 - 8*b^3*d^4*n^2*x*e^5 + 5*b^3*d*n^2*x
^2*e^8)*log(c))*x^(2/3) + 90*(20*(191*b^3*d^2*n^3 - 504*a*b^2*d^2*n^2)*x^2*e^7 - 7*(2509*b^3*d^5*n^3 - 2520*a*
b^2*d^5*n^2)*x*e^4 + 28*(7129*b^3*d^8*n^3 - 2520*a*b^2*d^8*n^2)*e - 2520*(28*b^3*d^8*n^2*e - 7*b^3*d^5*n^2*x*e
^4 + 4*b^3*d^2*n^2*x^2*e^7)*log(c))*x^(1/3))*log(x^(1/3)*e + d) + 352800*(280*(2*b^3*n^2 - 18*a*b^2*n + 81*a^2
*b)*x^3*e^9 - 15*(275*b^3*d^3*n^2 - 504*a*b^2*d^3*n)*x^2*e^6 + 6*(3349*b^3*d^6*n^2 - 2520*a*b^2*d^6*n)*x*e^3)*
log(c) + 63*(6125*(217*b^3*d*n^3 - 1224*a*b^2*d*n^2 + 2592*a^2*b*d*n)*x^2*e^8 - 8*(2134141*b^3*d^4*n^3 - 47350
80*a*b^2*d^4*n^2 + 3175200*a^2*b*d^4*n)*x*e^5 + 3175200*(20*b^3*d^7*n*e^2 - 8*b^3*d^4*n*x*e^5 + 5*b^3*d*n*x^2*
e^8)*log(c)^2 + 20*(12335311*b^3*d^7*n^3 - 11614680*a*b^2*d^7*n^2 + 3175200*a^2*b*d^7*n)*e^2 - 2520*(175*(17*b
^3*d*n^2 - 72*a*b^2*d*n)*x^2*e^8 - 8*(1879*b^3*d^4*n^2 - 2520*a*b^2*d^4*n)*x*e^5 + 20*(4609*b^3*d^7*n^2 - 2520
*a*b^2*d^7*n)*e^2)*log(c))*x^(2/3) - 90*(100*(24385*b^3*d^2*n^3 - 96264*a*b^2*d^2*n^2 + 127008*a^2*b*d^2*n)*x^
2*e^7 - 7*(3714811*b^3*d^5*n^3 - 6322680*a*b^2*d^5*n^2 + 3175200*a^2*b*d^5*n)*x*e^4 + 3175200*(28*b^3*d^8*n*e
- 7*b^3*d^5*n*x*e^4 + 4*b^3*d^2*n*x^2*e^7)*log(c)^2 + 28*(30300391*b^3*d^8*n^3 - 17965080*a*b^2*d^8*n^2 + 3175
200*a^2*b*d^8*n)*e - 2520*(20*(191*b^3*d^2*n^2 - 504*a*b^2*d^2*n)*x^2*e^7 - 7*(2509*b^3*d^5*n^2 - 2520*a*b^2*d
^5*n)*x*e^4 + 28*(7129*b^3*d^8*n^2 - 2520*a*b^2*d^8*n)*e)*log(c))*x^(1/3))*e^(-9)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*(d+e*x**(1/3))**n))**3,x)

[Out]

Integral(x**2*(a + b*log(c*(d + e*x**(1/3))**n))**3, x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3333 vs. \(2 (1214) = 2428\).
time = 3.78, size = 3333, normalized size = 2.46 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*(d+e*x^(1/3))^n))^3,x, algorithm="giac")

[Out]

1/8001504000*(2667168000*b^3*x^3*e*log(c)^3 + 8001504000*a*b^2*x^3*e*log(c)^2 + 8001504000*a^2*b*x^3*e*log(c)
+ (2667168000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^3 - 24004512000*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/
3)*e + d)^3 + 96018048000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^3 - 224042112000*(x^(1/3)*e + d)^6*d
^3*e^(-8)*log(x^(1/3)*e + d)^3 + 336063168000*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^3 - 336063168000
*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^3 + 224042112000*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e +
d)^3 - 96018048000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^3 + 24004512000*(x^(1/3)*e + d)*d^8*e^(-8)
*log(x^(1/3)*e + d)^3 - 889056000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 + 9001692000*(x^(1/3)*e + d)^8
*d*e^(-8)*log(x^(1/3)*e + d)^2 - 41150592000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 + 112021056000*
(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d)^2 - 201637900800*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e +
d)^2 + 252047376000*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^2 - 224042112000*(x^(1/3)*e + d)^3*d^6*e^(
-8)*log(x^(1/3)*e + d)^2 + 144027072000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 - 72013536000*(x^(1/
3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d)^2 + 197568000*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 2250423000
*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d) + 11757312000*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) -
37340352000*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e + d) + 80655160320*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^
(1/3)*e + d) - 126023688000*(x^(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d) + 149361408000*(x^(1/3)*e + d)^3*d
^6*e^(-8)*log(x^(1/3)*e + d) - 144027072000*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) + 144027072000*(x^
(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 21952000*(x^(1/3)*e + d)^9*e^(-8) + 281302875*(x^(1/3)*e + d)^8*d
*e^(-8) - 1679616000*(x^(1/3)*e + d)^7*d^2*e^(-8) + 6223392000*(x^(1/3)*e + d)^6*d^3*e^(-8) - 16131032064*(x^(
1/3)*e + d)^5*d^4*e^(-8) + 31505922000*(x^(1/3)*e + d)^4*d^5*e^(-8) - 49787136000*(x^(1/3)*e + d)^3*d^6*e^(-8)
+ 72013536000*(x^(1/3)*e + d)^2*d^7*e^(-8) - 144027072000*(x^(1/3)*e + d)*d^8*e^(-8))*b^3*n^3 + 2667168000*a^
3*x^3*e + 2520*(3175200*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 - 28576800*(x^(1/3)*e + d)^8*d*e^(-8)*lo
g(x^(1/3)*e + d)^2 + 114307200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 - 266716800*(x^(1/3)*e + d)^6
*d^3*e^(-8)*log(x^(1/3)*e + d)^2 + 400075200*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^2 - 400075200*(x^
(1/3)*e + d)^4*d^5*e^(-8)*log(x^(1/3)*e + d)^2 + 266716800*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d)^2 -
114307200*(x^(1/3)*e + d)^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 + 28576800*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)
*e + d)^2 - 705600*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) + 7144200*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3
)*e + d) - 32659200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) + 88905600*(x^(1/3)*e + d)^6*d^3*e^(-8)*lo
g(x^(1/3)*e + d) - 160030080*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) + 200037600*(x^(1/3)*e + d)^4*d^5
*e^(-8)*log(x^(1/3)*e + d) - 177811200*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) + 114307200*(x^(1/3)*e
+ d)^2*d^7*e^(-8)*log(x^(1/3)*e + d) - 57153600*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) + 78400*(x^(1/3)
*e + d)^9*e^(-8) - 893025*(x^(1/3)*e + d)^8*d*e^(-8) + 4665600*(x^(1/3)*e + d)^7*d^2*e^(-8) - 14817600*(x^(1/3
)*e + d)^6*d^3*e^(-8) + 32006016*(x^(1/3)*e + d)^5*d^4*e^(-8) - 50009400*(x^(1/3)*e + d)^4*d^5*e^(-8) + 592704
00*(x^(1/3)*e + d)^3*d^6*e^(-8) - 57153600*(x^(1/3)*e + d)^2*d^7*e^(-8) + 57153600*(x^(1/3)*e + d)*d^8*e^(-8))
*b^3*n^2*log(c) + 3175200*(2520*(x^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d) - 22680*(x^(1/3)*e + d)^8*d*e^(-8)
*log(x^(1/3)*e + d) + 90720*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d) - 211680*(x^(1/3)*e + d)^6*d^3*e^(
-8)*log(x^(1/3)*e + d) + 317520*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d) - 317520*(x^(1/3)*e + d)^4*d^5
*e^(-8)*log(x^(1/3)*e + d) + 211680*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d) - 90720*(x^(1/3)*e + d)^2*
d^7*e^(-8)*log(x^(1/3)*e + d) + 22680*(x^(1/3)*e + d)*d^8*e^(-8)*log(x^(1/3)*e + d) - 280*(x^(1/3)*e + d)^9*e^
(-8) + 2835*(x^(1/3)*e + d)^8*d*e^(-8) - 12960*(x^(1/3)*e + d)^7*d^2*e^(-8) + 35280*(x^(1/3)*e + d)^6*d^3*e^(-
8) - 63504*(x^(1/3)*e + d)^5*d^4*e^(-8) + 79380*(x^(1/3)*e + d)^4*d^5*e^(-8) - 70560*(x^(1/3)*e + d)^3*d^6*e^(
-8) + 45360*(x^(1/3)*e + d)^2*d^7*e^(-8) - 22680*(x^(1/3)*e + d)*d^8*e^(-8))*b^3*n*log(c)^2 + 2520*(3175200*(x
^(1/3)*e + d)^9*e^(-8)*log(x^(1/3)*e + d)^2 - 28576800*(x^(1/3)*e + d)^8*d*e^(-8)*log(x^(1/3)*e + d)^2 + 11430
7200*(x^(1/3)*e + d)^7*d^2*e^(-8)*log(x^(1/3)*e + d)^2 - 266716800*(x^(1/3)*e + d)^6*d^3*e^(-8)*log(x^(1/3)*e
+ d)^2 + 400075200*(x^(1/3)*e + d)^5*d^4*e^(-8)*log(x^(1/3)*e + d)^2 - 400075200*(x^(1/3)*e + d)^4*d^5*e^(-8)*
log(x^(1/3)*e + d)^2 + 266716800*(x^(1/3)*e + d)^3*d^6*e^(-8)*log(x^(1/3)*e + d)^2 - 114307200*(x^(1/3)*e + d)
^2*d^7*e^(-8)*log(x^(1/3)*e + d)^2 + 28576800*(...

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Mupad [B]
time = 8.25, size = 1386, normalized size = 1.02 \begin {gather*} \frac {a^3\,x^3}{3}+\frac {b^3\,x^3\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{3}-\frac {2\,b^3\,n^3\,x^3}{729}+a\,b^2\,x^3\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2-\frac {b^3\,n\,x^3\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{9}+\frac {2\,b^3\,n^2\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{81}+\frac {2\,a\,b^2\,n^2\,x^3}{81}+\frac {b^3\,d^9\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{3\,e^9}+a^2\,b\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )-\frac {a^2\,b\,n\,x^3}{9}-\frac {2\,a\,b^2\,n\,x^3\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{9}+\frac {30300391\,b^3\,d^9\,n^3\,\ln \left (d+e\,x^{1/3}\right )}{3175200\,e^9}+\frac {47485\,b^3\,d^3\,n^3\,x^2}{762048\,e^3}-\frac {24385\,b^3\,d^2\,n^3\,x^{7/3}}{889056\,e^2}-\frac {2134141\,b^3\,d^4\,n^3\,x^{5/3}}{15876000\,e^4}+\frac {3714811\,b^3\,d^5\,n^3\,x^{4/3}}{12700800\,e^5}+\frac {12335311\,b^3\,d^7\,n^3\,x^{2/3}}{6350400\,e^7}-\frac {30300391\,b^3\,d^8\,n^3\,x^{1/3}}{3175200\,e^8}+\frac {a\,b^2\,d^9\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{e^9}-\frac {7129\,b^3\,d^9\,n\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2520\,e^9}+\frac {217\,b^3\,d\,n^3\,x^{8/3}}{20736\,e}-\frac {6527971\,b^3\,d^6\,n^3\,x}{9525600\,e^6}+\frac {a^2\,b\,d^9\,n\,\ln \left (d+e\,x^{1/3}\right )}{e^9}+\frac {b^3\,d\,n\,x^{8/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{8\,e}-\frac {17\,b^3\,d\,n^2\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{288\,e}-\frac {b^3\,d^6\,n\,x\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{3\,e^6}+\frac {3349\,b^3\,d^6\,n^2\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3780\,e^6}+\frac {a^2\,b\,d^3\,n\,x^2}{6\,e^3}-\frac {17\,a\,b^2\,d\,n^2\,x^{8/3}}{288\,e}+\frac {3349\,a\,b^2\,d^6\,n^2\,x}{3780\,e^6}-\frac {a^2\,b\,d^2\,n\,x^{7/3}}{7\,e^2}-\frac {a^2\,b\,d^4\,n\,x^{5/3}}{5\,e^4}+\frac {a^2\,b\,d^5\,n\,x^{4/3}}{4\,e^5}+\frac {a^2\,b\,d^7\,n\,x^{2/3}}{2\,e^7}-\frac {a^2\,b\,d^8\,n\,x^{1/3}}{e^8}-\frac {7129\,a\,b^2\,d^9\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{1260\,e^9}+\frac {b^3\,d^3\,n\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{6\,e^3}-\frac {275\,b^3\,d^3\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{1512\,e^3}-\frac {b^3\,d^2\,n\,x^{7/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{7\,e^2}+\frac {191\,b^3\,d^2\,n^2\,x^{7/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{1764\,e^2}-\frac {b^3\,d^4\,n\,x^{5/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{5\,e^4}+\frac {1879\,b^3\,d^4\,n^2\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{6300\,e^4}+\frac {b^3\,d^5\,n\,x^{4/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{4\,e^5}-\frac {2509\,b^3\,d^5\,n^2\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5040\,e^5}+\frac {b^3\,d^7\,n\,x^{2/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^7}-\frac {4609\,b^3\,d^7\,n^2\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2520\,e^7}-\frac {b^3\,d^8\,n\,x^{1/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{e^8}+\frac {7129\,b^3\,d^8\,n^2\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{1260\,e^8}-\frac {275\,a\,b^2\,d^3\,n^2\,x^2}{1512\,e^3}+\frac {191\,a\,b^2\,d^2\,n^2\,x^{7/3}}{1764\,e^2}+\frac {1879\,a\,b^2\,d^4\,n^2\,x^{5/3}}{6300\,e^4}-\frac {2509\,a\,b^2\,d^5\,n^2\,x^{4/3}}{5040\,e^5}-\frac {4609\,a\,b^2\,d^7\,n^2\,x^{2/3}}{2520\,e^7}+\frac {7129\,a\,b^2\,d^8\,n^2\,x^{1/3}}{1260\,e^8}+\frac {a^2\,b\,d\,n\,x^{8/3}}{8\,e}-\frac {a^2\,b\,d^6\,n\,x}{3\,e^6}+\frac {a\,b^2\,d\,n\,x^{8/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e}-\frac {2\,a\,b^2\,d^6\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^6}+\frac {a\,b^2\,d^3\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{3\,e^3}-\frac {2\,a\,b^2\,d^2\,n\,x^{7/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{7\,e^2}-\frac {2\,a\,b^2\,d^4\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e^4}+\frac {a\,b^2\,d^5\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^5}+\frac {a\,b^2\,d^7\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^7}-\frac {2\,a\,b^2\,d^8\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*log(c*(d + e*x^(1/3))^n))^3,x)

[Out]

(a^3*x^3)/3 + (b^3*x^3*log(c*(d + e*x^(1/3))^n)^3)/3 - (2*b^3*n^3*x^3)/729 + a*b^2*x^3*log(c*(d + e*x^(1/3))^n
)^2 - (b^3*n*x^3*log(c*(d + e*x^(1/3))^n)^2)/9 + (2*b^3*n^2*x^3*log(c*(d + e*x^(1/3))^n))/81 + (2*a*b^2*n^2*x^
3)/81 + (b^3*d^9*log(c*(d + e*x^(1/3))^n)^3)/(3*e^9) + a^2*b*x^3*log(c*(d + e*x^(1/3))^n) - (a^2*b*n*x^3)/9 -
(2*a*b^2*n*x^3*log(c*(d + e*x^(1/3))^n))/9 + (30300391*b^3*d^9*n^3*log(d + e*x^(1/3)))/(3175200*e^9) + (47485*
b^3*d^3*n^3*x^2)/(762048*e^3) - (24385*b^3*d^2*n^3*x^(7/3))/(889056*e^2) - (2134141*b^3*d^4*n^3*x^(5/3))/(1587
6000*e^4) + (3714811*b^3*d^5*n^3*x^(4/3))/(12700800*e^5) + (12335311*b^3*d^7*n^3*x^(2/3))/(6350400*e^7) - (303
00391*b^3*d^8*n^3*x^(1/3))/(3175200*e^8) + (a*b^2*d^9*log(c*(d + e*x^(1/3))^n)^2)/e^9 - (7129*b^3*d^9*n*log(c*
(d + e*x^(1/3))^n)^2)/(2520*e^9) + (217*b^3*d*n^3*x^(8/3))/(20736*e) - (6527971*b^3*d^6*n^3*x)/(9525600*e^6) +
(a^2*b*d^9*n*log(d + e*x^(1/3)))/e^9 + (b^3*d*n*x^(8/3)*log(c*(d + e*x^(1/3))^n)^2)/(8*e) - (17*b^3*d*n^2*x^(
8/3)*log(c*(d + e*x^(1/3))^n))/(288*e) - (b^3*d^6*n*x*log(c*(d + e*x^(1/3))^n)^2)/(3*e^6) + (3349*b^3*d^6*n^2*
x*log(c*(d + e*x^(1/3))^n))/(3780*e^6) + (a^2*b*d^3*n*x^2)/(6*e^3) - (17*a*b^2*d*n^2*x^(8/3))/(288*e) + (3349*
a*b^2*d^6*n^2*x)/(3780*e^6) - (a^2*b*d^2*n*x^(7/3))/(7*e^2) - (a^2*b*d^4*n*x^(5/3))/(5*e^4) + (a^2*b*d^5*n*x^(
4/3))/(4*e^5) + (a^2*b*d^7*n*x^(2/3))/(2*e^7) - (a^2*b*d^8*n*x^(1/3))/e^8 - (7129*a*b^2*d^9*n^2*log(d + e*x^(1
/3)))/(1260*e^9) + (b^3*d^3*n*x^2*log(c*(d + e*x^(1/3))^n)^2)/(6*e^3) - (275*b^3*d^3*n^2*x^2*log(c*(d + e*x^(1
/3))^n))/(1512*e^3) - (b^3*d^2*n*x^(7/3)*log(c*(d + e*x^(1/3))^n)^2)/(7*e^2) + (191*b^3*d^2*n^2*x^(7/3)*log(c*
(d + e*x^(1/3))^n))/(1764*e^2) - (b^3*d^4*n*x^(5/3)*log(c*(d + e*x^(1/3))^n)^2)/(5*e^4) + (1879*b^3*d^4*n^2*x^
(5/3)*log(c*(d + e*x^(1/3))^n))/(6300*e^4) + (b^3*d^5*n*x^(4/3)*log(c*(d + e*x^(1/3))^n)^2)/(4*e^5) - (2509*b^
3*d^5*n^2*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(5040*e^5) + (b^3*d^7*n*x^(2/3)*log(c*(d + e*x^(1/3))^n)^2)/(2*e^7
) - (4609*b^3*d^7*n^2*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(2520*e^7) - (b^3*d^8*n*x^(1/3)*log(c*(d + e*x^(1/3))^
n)^2)/e^8 + (7129*b^3*d^8*n^2*x^(1/3)*log(c*(d + e*x^(1/3))^n))/(1260*e^8) - (275*a*b^2*d^3*n^2*x^2)/(1512*e^3
) + (191*a*b^2*d^2*n^2*x^(7/3))/(1764*e^2) + (1879*a*b^2*d^4*n^2*x^(5/3))/(6300*e^4) - (2509*a*b^2*d^5*n^2*x^(
4/3))/(5040*e^5) - (4609*a*b^2*d^7*n^2*x^(2/3))/(2520*e^7) + (7129*a*b^2*d^8*n^2*x^(1/3))/(1260*e^8) + (a^2*b*
d*n*x^(8/3))/(8*e) - (a^2*b*d^6*n*x)/(3*e^6) + (a*b^2*d*n*x^(8/3)*log(c*(d + e*x^(1/3))^n))/(4*e) - (2*a*b^2*d
^6*n*x*log(c*(d + e*x^(1/3))^n))/(3*e^6) + (a*b^2*d^3*n*x^2*log(c*(d + e*x^(1/3))^n))/(3*e^3) - (2*a*b^2*d^2*n
*x^(7/3)*log(c*(d + e*x^(1/3))^n))/(7*e^2) - (2*a*b^2*d^4*n*x^(5/3)*log(c*(d + e*x^(1/3))^n))/(5*e^4) + (a*b^2
*d^5*n*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(2*e^5) + (a*b^2*d^7*n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/e^7 - (2*a*b
^2*d^8*n*x^(1/3)*log(c*(d + e*x^(1/3))^n))/e^8

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