\(\int x^3 (a+b \log (c (d+e x^{2/3})^n))^3 \, dx\) [481]

Optimal result

Integrand size = 24, antiderivative size = 913 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:

-9*a*b^2*d^5*n^2*x^(2/3)/e^5-9*b^3*d^5*n^2*(d+e*x^(2/3))*ln(c*(d+e*x^(2/3) 
)^n)/e^6+45/8*b^2*d^4*n^2*(d+e*x^(2/3))^2*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6- 
10/3*b^2*d^3*n^2*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6+45/32*b^2 
*d^2*n^2*(d+e*x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-9/25*b^2*d*n^2*(d 
+e*x^(2/3))^5*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6+9/2*b*d^5*n*(d+e*x^(2/3))*(a 
+b*ln(c*(d+e*x^(2/3))^n))^2/e^6-45/8*b*d^4*n*(d+e*x^(2/3))^2*(a+b*ln(c*(d+ 
e*x^(2/3))^n))^2/e^6+5*b*d^3*n*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n)) 
^2/e^6-45/16*b*d^2*n*(d+e*x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6+9/1 
0*b*d*n*(d+e*x^(2/3))^5*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6+1/4*(d+e*x^(2/3) 
)^6*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6+9*b^3*d^5*n^3*x^(2/3)/e^5+1/24*b^2*n 
^2*(d+e*x^(2/3))^6*(a+b*ln(c*(d+e*x^(2/3))^n))/e^6-1/8*b*n*(d+e*x^(2/3))^6 
*(a+b*ln(c*(d+e*x^(2/3))^n))^2/e^6-45/16*b^3*d^4*n^3*(d+e*x^(2/3))^2/e^6+1 
0/9*b^3*d^3*n^3*(d+e*x^(2/3))^3/e^6-45/128*b^3*d^2*n^3*(d+e*x^(2/3))^4/e^6 
+9/125*b^3*d*n^3*(d+e*x^(2/3))^5/e^6-3/2*d^5*(d+e*x^(2/3))*(a+b*ln(c*(d+e* 
x^(2/3))^n))^3/e^6+15/4*d^4*(d+e*x^(2/3))^2*(a+b*ln(c*(d+e*x^(2/3))^n))^3/ 
e^6-5*d^3*(d+e*x^(2/3))^3*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6+15/4*d^2*(d+e* 
x^(2/3))^4*(a+b*ln(c*(d+e*x^(2/3))^n))^3/e^6-3/2*d*(d+e*x^(2/3))^5*(a+b*ln 
(c*(d+e*x^(2/3))^n))^3/e^6-1/144*b^3*n^3*(d+e*x^(2/3))^6/e^6
 

Mathematica [A] (verified)

Time = 0.96 (sec) , antiderivative size = 593, normalized size of antiderivative = 0.65 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\frac {e x^{2/3} \left (36000 a^3 e^5 x^{10/3}+b^3 n^3 \left (809340 d^5-140070 d^4 e x^{2/3}+41180 d^3 e^2 x^{4/3}-13785 d^2 e^3 x^2+4368 d e^4 x^{8/3}-1000 e^5 x^{10/3}\right )-60 a b^2 n^2 \left (8820 d^5-2610 d^4 e x^{2/3}+1140 d^3 e^2 x^{4/3}-555 d^2 e^3 x^2+264 d e^4 x^{8/3}-100 e^5 x^{10/3}\right )+1800 a^2 b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )\right )-60 b d^6 n \left (1800 a^2-8820 a b n+13489 b^2 n^2\right ) \log \left (d+e x^{2/3}\right )+60 b e x^{2/3} \left (1800 a^2 e^5 x^{10/3}+60 a b n \left (60 d^5-30 d^4 e x^{2/3}+20 d^3 e^2 x^{4/3}-15 d^2 e^3 x^2+12 d e^4 x^{8/3}-10 e^5 x^{10/3}\right )+b^2 n^2 \left (-8820 d^5+2610 d^4 e x^{2/3}-1140 d^3 e^2 x^{4/3}+555 d^2 e^3 x^2-264 d e^4 x^{8/3}+100 e^5 x^{10/3}\right )\right ) \log \left (c \left (d+e x^{2/3}\right )^n\right )+1800 b^2 \left (b n \left (147 d^6+60 d^5 e x^{2/3}-30 d^4 e^2 x^{4/3}+20 d^3 e^3 x^2-15 d^2 e^4 x^{8/3}+12 d e^5 x^{10/3}-10 e^6 x^4\right )-60 a \left (d^6-e^6 x^4\right )\right ) \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^4\right ) \log ^3\left (c \left (d+e x^{2/3}\right )^n\right )}{144000 e^6} \] Input:

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

Output:

(e*x^(2/3)*(36000*a^3*e^5*x^(10/3) + b^3*n^3*(809340*d^5 - 140070*d^4*e*x^ 
(2/3) + 41180*d^3*e^2*x^(4/3) - 13785*d^2*e^3*x^2 + 4368*d*e^4*x^(8/3) - 1 
000*e^5*x^(10/3)) - 60*a*b^2*n^2*(8820*d^5 - 2610*d^4*e*x^(2/3) + 1140*d^3 
*e^2*x^(4/3) - 555*d^2*e^3*x^2 + 264*d*e^4*x^(8/3) - 100*e^5*x^(10/3)) + 1 
800*a^2*b*n*(60*d^5 - 30*d^4*e*x^(2/3) + 20*d^3*e^2*x^(4/3) - 15*d^2*e^3*x 
^2 + 12*d*e^4*x^(8/3) - 10*e^5*x^(10/3))) - 60*b*d^6*n*(1800*a^2 - 8820*a* 
b*n + 13489*b^2*n^2)*Log[d + e*x^(2/3)] + 60*b*e*x^(2/3)*(1800*a^2*e^5*x^( 
10/3) + 60*a*b*n*(60*d^5 - 30*d^4*e*x^(2/3) + 20*d^3*e^2*x^(4/3) - 15*d^2* 
e^3*x^2 + 12*d*e^4*x^(8/3) - 10*e^5*x^(10/3)) + b^2*n^2*(-8820*d^5 + 2610* 
d^4*e*x^(2/3) - 1140*d^3*e^2*x^(4/3) + 555*d^2*e^3*x^2 - 264*d*e^4*x^(8/3) 
 + 100*e^5*x^(10/3)))*Log[c*(d + e*x^(2/3))^n] + 1800*b^2*(b*n*(147*d^6 + 
60*d^5*e*x^(2/3) - 30*d^4*e^2*x^(4/3) + 20*d^3*e^3*x^2 - 15*d^2*e^4*x^(8/3 
) + 12*d*e^5*x^(10/3) - 10*e^6*x^4) - 60*a*(d^6 - e^6*x^4))*Log[c*(d + e*x 
^(2/3))^n]^2 - 36000*b^3*(d^6 - e^6*x^4)*Log[c*(d + e*x^(2/3))^n]^3)/(1440 
00*e^6)
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 915, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2904, 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.

\(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle \frac {3}{2} \int x^{10/3} \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3dx^{2/3}\)

\(\Big \downarrow \) 2848

\(\displaystyle \frac {3}{2} \int \left (-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^5}{e^5}+\frac {5 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^4}{e^5}-\frac {10 \left (d+e x^{2/3}\right )^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^3}{e^5}+\frac {10 \left (d+e x^{2/3}\right )^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d^2}{e^5}-\frac {5 \left (d+e x^{2/3}\right )^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 d}{e^5}+\frac {\left (d+e x^{2/3}\right )^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{e^5}\right )dx^{2/3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{2} \left (-\frac {b^3 n^3 \left (d+e x^{2/3}\right )^6}{216 e^6}+\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^6}{6 e^6}-\frac {b n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^6}{12 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^6}{36 e^6}+\frac {6 b^3 d n^3 \left (d+e x^{2/3}\right )^5}{125 e^6}-\frac {d \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^5}{e^6}+\frac {3 b d n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^5}{5 e^6}-\frac {6 b^2 d n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^5}{25 e^6}-\frac {15 b^3 d^2 n^3 \left (d+e x^{2/3}\right )^4}{64 e^6}+\frac {5 d^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^4}{2 e^6}-\frac {15 b d^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^4}{8 e^6}+\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^4}{16 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e x^{2/3}\right )^3}{27 e^6}-\frac {10 d^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^3}{3 e^6}+\frac {10 b d^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^3}{3 e^6}-\frac {20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^3}{9 e^6}-\frac {15 b^3 d^4 n^3 \left (d+e x^{2/3}\right )^2}{8 e^6}+\frac {5 d^4 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )^2}{2 e^6}-\frac {15 b d^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )^2}{4 e^6}+\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (d+e x^{2/3}\right )^2}{4 e^6}-\frac {d^5 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \left (d+e x^{2/3}\right )}{e^6}+\frac {3 b d^5 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \left (d+e x^{2/3}\right )}{e^6}-\frac {6 b^3 d^5 n^2 \log \left (c \left (d+e x^{2/3}\right )^n\right ) \left (d+e x^{2/3}\right )}{e^6}+\frac {6 b^3 d^5 n^3 x^{2/3}}{e^5}-\frac {6 a b^2 d^5 n^2 x^{2/3}}{e^5}\right )\)

Input:

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

Output:

(3*((-15*b^3*d^4*n^3*(d + e*x^(2/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e*x 
^(2/3))^3)/(27*e^6) - (15*b^3*d^2*n^3*(d + e*x^(2/3))^4)/(64*e^6) + (6*b^3 
*d*n^3*(d + e*x^(2/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(2/3))^6)/(216*e^6 
) - (6*a*b^2*d^5*n^2*x^(2/3))/e^5 + (6*b^3*d^5*n^3*x^(2/3))/e^5 - (6*b^3*d 
^5*n^2*(d + e*x^(2/3))*Log[c*(d + e*x^(2/3))^n])/e^6 + (15*b^2*d^4*n^2*(d 
+ e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n]))/(4*e^6) - (20*b^2*d^3*n^2 
*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n]))/(9*e^6) + (15*b^2*d^2 
*n^2*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n]))/(16*e^6) - (6*b^2 
*d*n^2*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n]))/(25*e^6) + (b^2 
*n^2*(d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n]))/(36*e^6) + (3*b*d 
^5*n*(d + e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/e^6 - (15*b*d^4*n 
*(d + e*x^(2/3))^2*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(4*e^6) + (10*b*d^3 
*n*(d + e*x^(2/3))^3*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(3*e^6) - (15*b*d 
^2*n*(d + e*x^(2/3))^4*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(8*e^6) + (3*b* 
d*n*(d + e*x^(2/3))^5*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(5*e^6) - (b*n*( 
d + e*x^(2/3))^6*(a + b*Log[c*(d + e*x^(2/3))^n])^2)/(12*e^6) - (d^5*(d + 
e*x^(2/3))*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/e^6 + (5*d^4*(d + e*x^(2/3) 
)^2*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^6) - (10*d^3*(d + e*x^(2/3))^ 
3*(a + b*Log[c*(d + e*x^(2/3))^n])^3)/(3*e^6) + (5*d^2*(d + e*x^(2/3))^4*( 
a + b*Log[c*(d + e*x^(2/3))^n])^3)/(2*e^6) - (d*(d + e*x^(2/3))^5*(a + ...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2848

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]
 

rule 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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])
 

Maple [F]

\[\int x^{3} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1241, normalized size of antiderivative = 1.36 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/144000*(36000*b^3*e^6*x^4*log(c)^3 - 1000*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 
 + 18*a^2*b*e^6*n - 36*a^3*e^6)*x^4 + 36000*(b^3*e^6*n^3*x^4 - b^3*d^6*n^3 
)*log(e*x^(2/3) + d)^3 + 20*(2059*b^3*d^3*e^3*n^3 - 3420*a*b^2*d^3*e^3*n^2 
 + 1800*a^2*b*d^3*e^3*n)*x^2 + 1800*(20*b^3*d^3*e^3*n^3*x^2 + 147*b^3*d^6* 
n^3 - 60*a*b^2*d^6*n^2 - 10*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2)*x^4 + 60*(b^3* 
e^6*n^2*x^4 - b^3*d^6*n^2)*log(c) - 15*(b^3*d^2*e^4*n^3*x^2 - 4*b^3*d^5*e* 
n^3)*x^(2/3) + 6*(2*b^3*d*e^5*n^3*x^3 - 5*b^3*d^4*e^2*n^3*x)*x^(1/3))*log( 
e*x^(2/3) + d)^2 + 18000*(2*b^3*d^3*e^3*n*x^2 - (b^3*e^6*n - 6*a*b^2*e^6)* 
x^4)*log(c)^2 - 60*(13489*b^3*d^6*n^3 - 8820*a*b^2*d^6*n^2 + 1800*a^2*b*d^ 
6*n - 100*(b^3*e^6*n^3 - 6*a*b^2*e^6*n^2 + 18*a^2*b*e^6*n)*x^4 + 60*(19*b^ 
3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x^2 - 1800*(b^3*e^6*n*x^4 - b^3*d^6* 
n)*log(c)^2 - 60*(20*b^3*d^3*e^3*n^2*x^2 + 147*b^3*d^6*n^2 - 60*a*b^2*d^6* 
n - 10*(b^3*e^6*n^2 - 6*a*b^2*e^6*n)*x^4)*log(c) + 15*(588*b^3*d^5*e*n^3 - 
 240*a*b^2*d^5*e*n^2 - (37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2)*x^2 + 6 
0*(b^3*d^2*e^4*n^2*x^2 - 4*b^3*d^5*e*n^2)*log(c))*x^(2/3) + 6*(4*(11*b^3*d 
*e^5*n^3 - 30*a*b^2*d*e^5*n^2)*x^3 - 15*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4 
*e^2*n^2)*x - 60*(2*b^3*d*e^5*n^2*x^3 - 5*b^3*d^4*e^2*n^2*x)*log(c))*x^(1/ 
3))*log(e*x^(2/3) + d) + 1200*(5*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e 
^6)*x^4 - 3*(19*b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x^2)*log(c) + 15*(53 
956*b^3*d^5*e*n^3 - 35280*a*b^2*d^5*e*n^2 + 7200*a^2*b*d^5*e*n - (919*b...
 

Sympy [F(-1)]

Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 680, normalized size of antiderivative = 0.74 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*b^3*x^4*log((e*x^(2/3) + d)^n*c)^3 + 3/4*a*b^2*x^4*log((e*x^(2/3) + d) 
^n*c)^2 + 3/4*a^2*b*x^4*log((e*x^(2/3) + d)^n*c) + 1/4*a^3*x^4 - 1/80*a^2* 
b*e*n*(60*d^6*log(e*x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12*d*e^4*x^(10/3) + 1 
5*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*e*x^(4/3) - 60*d^5*x^(2/3))/e^ 
6) - 1/2400*(60*e*n*(60*d^6*log(e*x^(2/3) + d)/e^7 + (10*e^5*x^4 - 12*d*e^ 
4*x^(10/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30*d^4*e*x^(4/3) - 60*d 
^5*x^(2/3))/e^6)*log((e*x^(2/3) + d)^n*c) - (100*e^6*x^4 - 264*d*e^5*x^(10 
/3) + 555*d^2*e^4*x^(8/3) - 1140*d^3*e^3*x^2 + 1800*d^6*log(e*x^(2/3) + d) 
^2 + 2610*d^4*e^2*x^(4/3) + 8820*d^6*log(e*x^(2/3) + d) - 8820*d^5*e*x^(2/ 
3))*n^2/e^6)*a*b^2 - 1/144000*(1800*e*n*(60*d^6*log(e*x^(2/3) + d)/e^7 + ( 
10*e^5*x^4 - 12*d*e^4*x^(10/3) + 15*d^2*e^3*x^(8/3) - 20*d^3*e^2*x^2 + 30* 
d^4*e*x^(4/3) - 60*d^5*x^(2/3))/e^6)*log((e*x^(2/3) + d)^n*c)^2 + e*n*((10 
00*e^6*x^4 - 4368*d*e^5*x^(10/3) + 36000*d^6*log(e*x^(2/3) + d)^3 + 13785* 
d^2*e^4*x^(8/3) - 41180*d^3*e^3*x^2 + 264600*d^6*log(e*x^(2/3) + d)^2 + 14 
0070*d^4*e^2*x^(4/3) + 809340*d^6*log(e*x^(2/3) + d) - 809340*d^5*e*x^(2/3 
))*n^2/e^7 - 60*(100*e^6*x^4 - 264*d*e^5*x^(10/3) + 555*d^2*e^4*x^(8/3) - 
1140*d^3*e^3*x^2 + 1800*d^6*log(e*x^(2/3) + d)^2 + 2610*d^4*e^2*x^(4/3) + 
8820*d^6*log(e*x^(2/3) + d) - 8820*d^5*e*x^(2/3))*n*log((e*x^(2/3) + d)^n* 
c)/e^7))*b^3
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2104 vs. \(2 (787) = 1574\).

Time = 0.59 (sec) , antiderivative size = 2104, normalized size of antiderivative = 2.30 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*b^3*x^4*log(c)^3 + 3/4*a*b^2*x^4*log(c)^2 + 3/4*a^2*b*x^4*log(c) + 1/1 
44000*(36000*(e*x^(2/3) + d)^6*log(e*x^(2/3) + d)^3/e^6 - 216000*(e*x^(2/3 
) + d)^5*d*log(e*x^(2/3) + d)^3/e^6 + 540000*(e*x^(2/3) + d)^4*d^2*log(e*x 
^(2/3) + d)^3/e^6 - 720000*(e*x^(2/3) + d)^3*d^3*log(e*x^(2/3) + d)^3/e^6 
+ 540000*(e*x^(2/3) + d)^2*d^4*log(e*x^(2/3) + d)^3/e^6 - 18000*(e*x^(2/3) 
 + d)^6*log(e*x^(2/3) + d)^2/e^6 + 129600*(e*x^(2/3) + d)^5*d*log(e*x^(2/3 
) + d)^2/e^6 - 405000*(e*x^(2/3) + d)^4*d^2*log(e*x^(2/3) + d)^2/e^6 + 720 
000*(e*x^(2/3) + d)^3*d^3*log(e*x^(2/3) + d)^2/e^6 - 810000*(e*x^(2/3) + d 
)^2*d^4*log(e*x^(2/3) + d)^2/e^6 + 6000*(e*x^(2/3) + d)^6*log(e*x^(2/3) + 
d)/e^6 - 51840*(e*x^(2/3) + d)^5*d*log(e*x^(2/3) + d)/e^6 + 202500*(e*x^(2 
/3) + d)^4*d^2*log(e*x^(2/3) + d)/e^6 - 480000*(e*x^(2/3) + d)^3*d^3*log(e 
*x^(2/3) + d)/e^6 + 810000*(e*x^(2/3) + d)^2*d^4*log(e*x^(2/3) + d)/e^6 - 
1000*(e*x^(2/3) + d)^6/e^6 + 10368*(e*x^(2/3) + d)^5*d/e^6 - 50625*(e*x^(2 
/3) + d)^4*d^2/e^6 + 160000*(e*x^(2/3) + d)^3*d^3/e^6 - 405000*(e*x^(2/3) 
+ d)^2*d^4/e^6 - 216000*((e*x^(2/3) + d)*log(e*x^(2/3) + d)^3 - 3*(e*x^(2/ 
3) + d)*log(e*x^(2/3) + d)^2 + 6*(e*x^(2/3) + d)*log(e*x^(2/3) + d) - 6*e* 
x^(2/3) - 6*d)*d^5/e^6)*b^3*n^3 + 1/4*a^3*x^4 + 1/2400*(1800*(e*x^(2/3) + 
d)^6*log(e*x^(2/3) + d)^2/e^6 - 10800*(e*x^(2/3) + d)^5*d*log(e*x^(2/3) + 
d)^2/e^6 + 27000*(e*x^(2/3) + d)^4*d^2*log(e*x^(2/3) + d)^2/e^6 - 36000*(e 
*x^(2/3) + d)^3*d^3*log(e*x^(2/3) + d)^2/e^6 + 27000*(e*x^(2/3) + d)^2*...
 

Mupad [B] (verification not implemented)

Time = 35.14 (sec) , antiderivative size = 992, normalized size of antiderivative = 1.09 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(a^3*x^4)/4 + (b^3*x^4*log(c*(d + e*x^(2/3))^n)^3)/4 - (b^3*n^3*x^4)/144 + 
 (3*a*b^2*x^4*log(c*(d + e*x^(2/3))^n)^2)/4 - (b^3*n*x^4*log(c*(d + e*x^(2 
/3))^n)^2)/8 + (b^3*n^2*x^4*log(c*(d + e*x^(2/3))^n))/24 + (a*b^2*n^2*x^4) 
/24 - (b^3*d^6*log(c*(d + e*x^(2/3))^n)^3)/(4*e^6) + (3*a^2*b*x^4*log(c*(d 
 + e*x^(2/3))^n))/4 - (a^2*b*n*x^4)/8 - (a*b^2*n*x^4*log(c*(d + e*x^(2/3)) 
^n))/4 - (13489*b^3*d^6*n^3*log(d + e*x^(2/3)))/(2400*e^6) + (2059*b^3*d^3 
*n^3*x^2)/(7200*e^3) - (919*b^3*d^2*n^3*x^(8/3))/(9600*e^2) - (4669*b^3*d^ 
4*n^3*x^(4/3))/(4800*e^4) + (13489*b^3*d^5*n^3*x^(2/3))/(2400*e^5) - (3*a* 
b^2*d^6*log(c*(d + e*x^(2/3))^n)^2)/(4*e^6) + (147*b^3*d^6*n*log(c*(d + e* 
x^(2/3))^n)^2)/(80*e^6) + (91*b^3*d*n^3*x^(10/3))/(3000*e) - (3*a^2*b*d^6* 
n*log(d + e*x^(2/3)))/(4*e^6) + (3*b^3*d*n*x^(10/3)*log(c*(d + e*x^(2/3))^ 
n)^2)/(20*e) - (11*b^3*d*n^2*x^(10/3)*log(c*(d + e*x^(2/3))^n))/(100*e) + 
(a^2*b*d^3*n*x^2)/(4*e^3) - (3*a^2*b*d^2*n*x^(8/3))/(16*e^2) - (3*a^2*b*d^ 
4*n*x^(4/3))/(8*e^4) + (3*a^2*b*d^5*n*x^(2/3))/(4*e^5) - (11*a*b^2*d*n^2*x 
^(10/3))/(100*e) + (147*a*b^2*d^6*n^2*log(d + e*x^(2/3)))/(40*e^6) + (b^3* 
d^3*n*x^2*log(c*(d + e*x^(2/3))^n)^2)/(4*e^3) - (19*b^3*d^3*n^2*x^2*log(c* 
(d + e*x^(2/3))^n))/(40*e^3) - (3*b^3*d^2*n*x^(8/3)*log(c*(d + e*x^(2/3))^ 
n)^2)/(16*e^2) + (37*b^3*d^2*n^2*x^(8/3)*log(c*(d + e*x^(2/3))^n))/(160*e^ 
2) - (3*b^3*d^4*n*x^(4/3)*log(c*(d + e*x^(2/3))^n)^2)/(8*e^4) + (87*b^3*d^ 
4*n^2*x^(4/3)*log(c*(d + e*x^(2/3))^n))/(80*e^4) + (3*b^3*d^5*n*x^(2/3)...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1006, normalized size of antiderivative = 1.10 \[ \int x^3 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(108000*x**(2/3)*log((x**(2/3)*e + d)**n*c)**2*b**3*d**5*e*n - 27000*x**(2 
/3)*log((x**(2/3)*e + d)**n*c)**2*b**3*d**2*e**4*n*x**2 + 216000*x**(2/3)* 
log((x**(2/3)*e + d)**n*c)*a*b**2*d**5*e*n - 54000*x**(2/3)*log((x**(2/3)* 
e + d)**n*c)*a*b**2*d**2*e**4*n*x**2 - 529200*x**(2/3)*log((x**(2/3)*e + d 
)**n*c)*b**3*d**5*e*n**2 + 33300*x**(2/3)*log((x**(2/3)*e + d)**n*c)*b**3* 
d**2*e**4*n**2*x**2 + 108000*x**(2/3)*a**2*b*d**5*e*n - 27000*x**(2/3)*a** 
2*b*d**2*e**4*n*x**2 - 529200*x**(2/3)*a*b**2*d**5*e*n**2 + 33300*x**(2/3) 
*a*b**2*d**2*e**4*n**2*x**2 + 809340*x**(2/3)*b**3*d**5*e*n**3 - 13785*x** 
(2/3)*b**3*d**2*e**4*n**3*x**2 - 54000*x**(1/3)*log((x**(2/3)*e + d)**n*c) 
**2*b**3*d**4*e**2*n*x + 21600*x**(1/3)*log((x**(2/3)*e + d)**n*c)**2*b**3 
*d*e**5*n*x**3 - 108000*x**(1/3)*log((x**(2/3)*e + d)**n*c)*a*b**2*d**4*e* 
*2*n*x + 43200*x**(1/3)*log((x**(2/3)*e + d)**n*c)*a*b**2*d*e**5*n*x**3 + 
156600*x**(1/3)*log((x**(2/3)*e + d)**n*c)*b**3*d**4*e**2*n**2*x - 15840*x 
**(1/3)*log((x**(2/3)*e + d)**n*c)*b**3*d*e**5*n**2*x**3 - 54000*x**(1/3)* 
a**2*b*d**4*e**2*n*x + 21600*x**(1/3)*a**2*b*d*e**5*n*x**3 + 156600*x**(1/ 
3)*a*b**2*d**4*e**2*n**2*x - 15840*x**(1/3)*a*b**2*d*e**5*n**2*x**3 - 1400 
70*x**(1/3)*b**3*d**4*e**2*n**3*x + 4368*x**(1/3)*b**3*d*e**5*n**3*x**3 - 
36000*log((x**(2/3)*e + d)**n*c)**3*b**3*d**6 + 36000*log((x**(2/3)*e + d) 
**n*c)**3*b**3*e**6*x**4 - 108000*log((x**(2/3)*e + d)**n*c)**2*a*b**2*d** 
6 + 108000*log((x**(2/3)*e + d)**n*c)**2*a*b**2*e**6*x**4 + 264600*log(...