Integrand size = 20, antiderivative size = 438 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {9 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^3}+\frac {18 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {18 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}+\frac {18 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac {9 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{3 e^3}-\frac {9 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {3 d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {3 d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3} \]
9/4*b^3*d*n^3*(d+e*x^(1/3))^2/e^3-2/9*b^3*n^3*(d+e*x^(1/3))^3/e^3+18*a*b^2 *d^2*n^2*x^(1/3)/e^2-18*b^3*d^2*n^3*x^(1/3)/e^2+18*b^3*d^2*n^2*(d+e*x^(1/3 ))*ln(c*(d+e*x^(1/3))^n)/e^3-9/2*b^2*d*n^2*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e* x^(1/3))^n))/e^3+2/3*b^2*n^2*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))/e ^3-9*b*d^2*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^3+9/2*b*d*n*(d+ e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^3-b*n*(d+e*x^(1/3))^3*(a+b*ln (c*(d+e*x^(1/3))^n))^2/e^3+3*d^2*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n)) ^3/e^3-3*d*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^3+(d+e*x^(1/3)) ^3*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^3
Time = 0.15 (sec) , antiderivative size = 362, normalized size of antiderivative = 0.83 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \left (-510 d^2+57 d e \sqrt [3]{x}-8 e^2 x^{2/3}\right ) \sqrt [3]{x}-6 a b^2 n^2 \left (23 d^3-66 d^2 e \sqrt [3]{x}+15 d e^2 x^{2/3}-4 e^3 x\right )+36 a^3 \left (d^3+e^3 x\right )-18 a^2 b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )+6 b \left (18 a^2 \left (d^2-d e \sqrt [3]{x}+e^2 x^{2/3}\right )-6 a b n \left (11 d^2-5 d e \sqrt [3]{x}+2 e^2 x^{2/3}\right )+b^2 n^2 \left (85 d^2-19 d e \sqrt [3]{x}+4 e^2 x^{2/3}\right )\right ) \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+18 b^2 \left (6 a \left (d^3+e^3 x\right )-b n \left (11 d^3+6 d^2 e \sqrt [3]{x}-3 d e^2 x^{2/3}+2 e^3 x\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )+36 b^3 \left (d^3+e^3 x\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{36 e^3} \]
(b^3*e*n^3*(-510*d^2 + 57*d*e*x^(1/3) - 8*e^2*x^(2/3))*x^(1/3) - 6*a*b^2*n ^2*(23*d^3 - 66*d^2*e*x^(1/3) + 15*d*e^2*x^(2/3) - 4*e^3*x) + 36*a^3*(d^3 + e^3*x) - 18*a^2*b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/3) + 2*e^3* x) + 6*b*(18*a^2*(d^2 - d*e*x^(1/3) + e^2*x^(2/3)) - 6*a*b*n*(11*d^2 - 5*d *e*x^(1/3) + 2*e^2*x^(2/3)) + b^2*n^2*(85*d^2 - 19*d*e*x^(1/3) + 4*e^2*x^( 2/3)))*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n] + 18*b^2*(6*a*(d^3 + e^3*x ) - b*n*(11*d^3 + 6*d^2*e*x^(1/3) - 3*d*e^2*x^(2/3) + 2*e^3*x))*Log[c*(d + e*x^(1/3))^n]^2 + 36*b^3*(d^3 + e^3*x)*Log[c*(d + e*x^(1/3))^n]^3)/(36*e^ 3)
Time = 0.65 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2901, 2848, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2901 |
| \(\displaystyle 3 \int x^{2/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle 3 \int \left (\frac {\left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^2}-\frac {2 d \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^2}+\frac {d^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^2}\right )d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {2 b^2 n^2 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{9 e^3}-\frac {3 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{2 e^3}+\frac {6 a b^2 d^2 n^2 \sqrt [3]{x}}{e^2}-\frac {3 b d^2 n \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{e^3}+\frac {d^2 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}-\frac {b n \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 e^3}+\frac {3 b d n \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{2 e^3}+\frac {\left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{3 e^3}-\frac {d \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^3}+\frac {6 b^3 d^2 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^3}-\frac {6 b^3 d^2 n^3 \sqrt [3]{x}}{e^2}-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{27 e^3}+\frac {3 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^2}{4 e^3}\right )\) |
3*((3*b^3*d*n^3*(d + e*x^(1/3))^2)/(4*e^3) - (2*b^3*n^3*(d + e*x^(1/3))^3) /(27*e^3) + (6*a*b^2*d^2*n^2*x^(1/3))/e^2 - (6*b^3*d^2*n^3*x^(1/3))/e^2 + (6*b^3*d^2*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^3 - (3*b^2*d*n^ 2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(2*e^3) + (2*b^2*n^2 *(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(9*e^3) - (3*b*d^2*n* (d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^3 + (3*b*d*n*(d + e* x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^3) - (b*n*(d + e*x^(1/ 3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^3) + (d^2*(d + e*x^(1/3))*( a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^3 - (d*(d + e*x^(1/3))^2*(a + b*Log[c *(d + e*x^(1/3))^n])^3)/e^3 + ((d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/ 3))^n])^3)/(3*e^3))
3.5.59.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_), x_Symbo l] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*Log[c* (d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && FractionQ[n]
\[\int {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}d x\]
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}} \]
1/36*(36*b^3*e^3*x*log(c)^3 + 36*(b^3*e^3*n^3*x + b^3*d^3*n^3)*log(e*x^(1/ 3) + d)^3 - 36*(b^3*e^3*n - 3*a*b^2*e^3)*x*log(c)^2 + 18*(3*b^3*d*e^2*n^3* x^(2/3) - 6*b^3*d^2*e*n^3*x^(1/3) - 11*b^3*d^3*n^3 + 6*a*b^2*d^3*n^2 - 2*( b^3*e^3*n^3 - 3*a*b^2*e^3*n^2)*x + 6*(b^3*e^3*n^2*x + b^3*d^3*n^2)*log(c)) *log(e*x^(1/3) + d)^2 + 12*(2*b^3*e^3*n^2 - 6*a*b^2*e^3*n + 9*a^2*b*e^3)*x *log(c) - 4*(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)* x + 6*(85*b^3*d^3*n^3 - 66*a*b^2*d^3*n^2 + 18*a^2*b*d^3*n + 18*(b^3*e^3*n* x + b^3*d^3*n)*log(c)^2 + 2*(2*b^3*e^3*n^3 - 6*a*b^2*e^3*n^2 + 9*a^2*b*e^3 *n)*x - 6*(11*b^3*d^3*n^2 - 6*a*b^2*d^3*n + 2*(b^3*e^3*n^2 - 3*a*b^2*e^3*n )*x)*log(c) - 3*(5*b^3*d*e^2*n^3 - 6*b^3*d*e^2*n^2*log(c) - 6*a*b^2*d*e^2* n^2)*x^(2/3) + 6*(11*b^3*d^2*e*n^3 - 6*b^3*d^2*e*n^2*log(c) - 6*a*b^2*d^2* e*n^2)*x^(1/3))*log(e*x^(1/3) + d) + 3*(19*b^3*d*e^2*n^3 + 18*b^3*d*e^2*n* log(c)^2 - 30*a*b^2*d*e^2*n^2 + 18*a^2*b*d*e^2*n - 6*(5*b^3*d*e^2*n^2 - 6* a*b^2*d*e^2*n)*log(c))*x^(2/3) - 6*(85*b^3*d^2*e*n^3 + 18*b^3*d^2*e*n*log( c)^2 - 66*a*b^2*d^2*e*n^2 + 18*a^2*b*d^2*e*n - 6*(11*b^3*d^2*e*n^2 - 6*a*b ^2*d^2*e*n)*log(c))*x^(1/3))/e^3
\[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\int \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \]
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 \]
1/2*(e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2* x^(1/3))/e^3) + 6*x*log((e*x^(1/3) + d)^n*c))*a^2*b + 1/6*(6*e*n*(6*d^3*lo g(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*log( (e*x^(1/3) + d)^n*c) + 18*x*log((e*x^(1/3) + d)^n*c)^2 - (18*d^3*log(e*x^( 1/3) + d)^2 - 4*e^3*x + 66*d^3*log(e*x^(1/3) + d) + 15*d*e^2*x^(2/3) - 66* d^2*e*x^(1/3))*n^2/e^3)*a*b^2 + 1/36*(18*e*n*(6*d^3*log(e*x^(1/3) + d)/e^4 - (2*e^2*x - 3*d*e*x^(2/3) + 6*d^2*x^(1/3))/e^3)*log((e*x^(1/3) + d)^n*c) ^2 + 36*x*log((e*x^(1/3) + d)^n*c)^3 + e*n*((36*d^3*log(e*x^(1/3) + d)^3 + 198*d^3*log(e*x^(1/3) + d)^2 - 8*e^3*x + 510*d^3*log(e*x^(1/3) + d) + 57* d*e^2*x^(2/3) - 510*d^2*e*x^(1/3))*n^2/e^4 - 6*(18*d^3*log(e*x^(1/3) + d)^ 2 - 4*e^3*x + 66*d^3*log(e*x^(1/3) + d) + 15*d*e^2*x^(2/3) - 66*d^2*e*x^(1 /3))*n*log((e*x^(1/3) + d)^n*c)/e^4))*b^3 + a^3*x
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 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]
1/36*(36*b^3*e*x*log(c)^3 + (36*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)^3/e^2 - 108*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)^3/e^2 + 108*(e*x^(1/3) + d)* d^2*log(e*x^(1/3) + d)^3/e^2 - 36*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)^2/e ^2 + 162*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)^2/e^2 - 324*(e*x^(1/3) + d )*d^2*log(e*x^(1/3) + d)^2/e^2 + 24*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)/e ^2 - 162*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)/e^2 + 648*(e*x^(1/3) + d)* d^2*log(e*x^(1/3) + d)/e^2 - 8*(e*x^(1/3) + d)^3/e^2 + 81*(e*x^(1/3) + d)^ 2*d/e^2 - 648*(e*x^(1/3) + d)*d^2/e^2)*b^3*n^3 + 6*(18*(e*x^(1/3) + d)^3*l og(e*x^(1/3) + d)^2/e^2 - 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)^2/e^2 + 54*(e*x^(1/3) + d)*d^2*log(e*x^(1/3) + d)^2/e^2 - 12*(e*x^(1/3) + d)^3*l og(e*x^(1/3) + d)/e^2 + 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)/e^2 - 10 8*(e*x^(1/3) + d)*d^2*log(e*x^(1/3) + d)/e^2 + 4*(e*x^(1/3) + d)^3/e^2 - 2 7*(e*x^(1/3) + d)^2*d/e^2 + 108*(e*x^(1/3) + d)*d^2/e^2)*b^3*n^2*log(c) + 18*(6*(e*x^(1/3) + d)^3*log(e*x^(1/3) + d)/e^2 - 18*(e*x^(1/3) + d)^2*d*lo g(e*x^(1/3) + d)/e^2 + 18*(e*x^(1/3) + d)*d^2*log(e*x^(1/3) + d)/e^2 - 2*( e*x^(1/3) + d)^3/e^2 + 9*(e*x^(1/3) + d)^2*d/e^2 - 18*(e*x^(1/3) + d)*d^2/ e^2)*b^3*n*log(c)^2 + 108*a*b^2*e*x*log(c)^2 + 6*(18*(e*x^(1/3) + d)^3*log (e*x^(1/3) + d)^2/e^2 - 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)^2/e^2 + 54*(e*x^(1/3) + d)*d^2*log(e*x^(1/3) + d)^2/e^2 - 12*(e*x^(1/3) + d)^3*log (e*x^(1/3) + d)/e^2 + 54*(e*x^(1/3) + d)^2*d*log(e*x^(1/3) + d)/e^2 - 1...
Time = 1.88 (sec) , antiderivative size = 558, normalized size of antiderivative = 1.27 \[ \int \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=x\,\left (a^3-a^2\,b\,n+\frac {2\,a\,b^2\,n^2}{3}-\frac {2\,b^3\,n^3}{9}\right )-x^{2/3}\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{2\,e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{4\,e}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3\,\left (b^3\,x+\frac {b^3\,d^3}{e^3}\right )+{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2\,\left (\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{2\,e^3}-x^{2/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}\right )+b^2\,x\,\left (3\,a-b\,n\right )+\frac {d\,x^{1/3}\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {9\,a\,b^2\,d}{e}\right )}{e}\right )+x^{1/3}\,\left (\frac {d\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{e^2}\right )+\frac {\ln \left (d+e\,x^{1/3}\right )\,\left (18\,a^2\,b\,d^3\,n-66\,a\,b^2\,d^3\,n^2+85\,b^3\,d^3\,n^3\right )}{6\,e^3}+\frac {\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\,\left (\frac {x^{1/3}\,\left (\frac {d\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+6\,b^3\,d^2\,n^2\right )}{e}-\frac {x^{2/3}\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{2\,e}+\frac {b\,e\,x\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3}\right )}{e} \]
x*(a^3 - (2*b^3*n^3)/9 + (2*a*b^2*n^2)/3 - a^2*b*n) - x^(2/3)*((d*(3*a^3 - (2*b^3*n^3)/3 + 2*a*b^2*n^2 - 3*a^2*b*n))/(2*e) - (d*(6*a^3 + 5*b^3*n^3 - 6*a*b^2*n^2))/(4*e)) + log(c*(d + e*x^(1/3))^n)^3*(b^3*x + (b^3*d^3)/e^3) + log(c*(d + e*x^(1/3))^n)^2*((d*(6*a*b^2*d^2 - 11*b^3*d^2*n))/(2*e^3) - x^(2/3)*((3*b^2*d*(3*a - b*n))/(2*e) - (9*a*b^2*d)/(2*e)) + b^2*x*(3*a - b *n) + (d*x^(1/3)*((3*b^2*d*(3*a - b*n))/e - (9*a*b^2*d)/e))/e) + x^(1/3)*( (d*((d*(3*a^3 - (2*b^3*n^3)/3 + 2*a*b^2*n^2 - 3*a^2*b*n))/e - (d*(6*a^3 + 5*b^3*n^3 - 6*a*b^2*n^2))/(2*e)))/e + (b^2*d^2*n^2*(6*a - 11*b*n))/e^2) + (log(d + e*x^(1/3))*(85*b^3*d^3*n^3 - 66*a*b^2*d^3*n^2 + 18*a^2*b*d^3*n))/ (6*e^3) + (log(c*(d + e*x^(1/3))^n)*((x^(1/3)*((d*(b*d*e*(9*a^2 + 2*b^2*n^ 2 - 6*a*b*n) - 3*b*d*e*(3*a^2 - b^2*n^2)))/e + 6*b^3*d^2*n^2))/e - (x^(2/3 )*(b*d*e*(9*a^2 + 2*b^2*n^2 - 6*a*b*n) - 3*b*d*e*(3*a^2 - b^2*n^2)))/(2*e) + (b*e*x*(9*a^2 + 2*b^2*n^2 - 6*a*b*n))/3))/e