3.5.58 \(\int x (a+b \log (c (d+e \sqrt [3]{x})^n))^3 \, dx\) [458]

3.5.58.1 Optimal result

Integrand size = 22, antiderivative size = 907 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=-\frac {45 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^6}+\frac {20 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^6}-\frac {45 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^4}{64 e^6}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^6}-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^6}{72 e^6}-\frac {18 a b^2 d^5 n^2 \sqrt [3]{x}}{e^5}+\frac {18 b^3 d^5 n^3 \sqrt [3]{x}}{e^5}-\frac {18 b^3 d^5 n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^6}+\frac {45 b^2 d^4 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 )}{4 e^6}-\frac {20 b^2 d^3 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^6}+\frac {45 b^2 d^2 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 )}{16 e^6}-\frac {18 b^2 d 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^6}+\frac {b^2 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 )}{12 e^6}+\frac {9 b d^5 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^6}-\frac {45 b d^4 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}{4 e^6}+\frac {10 b d^3 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^6}-\frac {45 b d^2 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}{8 e^6}+\frac {9 b d 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^6}-\frac {b 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}{4 e^6}-\frac {3 d^5 \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^6}+\frac {15 d^4 \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}{2 e^6}-\frac {10 d^3 \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^6}+\frac {15 d^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 )^3}{2 e^6}-\frac {3 d \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^6}+\frac {\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}{2 e^6} \]

output

-1/72*b^3*n^3*(d+e*x^(1/3))^6/e^6-3*d^5*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/ 
3))^n))^3/e^6+15/2*d^4*(d+e*x^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^6-1 
0*d^3*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^6+15/2*d^2*(d+e*x^(1 
/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^6-3*d*(d+e*x^(1/3))^5*(a+b*ln(c*(d+ 
e*x^(1/3))^n))^3/e^6-18*a*b^2*d^5*n^2*x^(1/3)/e^5-18*b^3*d^5*n^2*(d+e*x^(1 
/3))*ln(c*(d+e*x^(1/3))^n)/e^6+45/4*b^2*d^4*n^2*(d+e*x^(1/3))^2*(a+b*ln(c* 
(d+e*x^(1/3))^n))/e^6-20/3*b^2*d^3*n^2*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1 
/3))^n))/e^6+45/16*b^2*d^2*n^2*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n)) 
/e^6-18/25*b^2*d*n^2*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))/e^6+9*b*d 
^5*n*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^6-45/4*b*d^4*n*(d+e*x^( 
1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^6+10*b*d^3*n*(d+e*x^(1/3))^3*(a+b* 
ln(c*(d+e*x^(1/3))^n))^2/e^6-45/8*b*d^2*n*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x 
^(1/3))^n))^2/e^6+9/5*b*d*n*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))^2/ 
e^6-45/8*b^3*d^4*n^3*(d+e*x^(1/3))^2/e^6+20/9*b^3*d^3*n^3*(d+e*x^(1/3))^3/ 
e^6-45/64*b^3*d^2*n^3*(d+e*x^(1/3))^4/e^6+18/125*b^3*d*n^3*(d+e*x^(1/3))^5 
/e^6+18*b^3*d^5*n^3*x^(1/3)/e^5+1/12*b^2*n^2*(d+e*x^(1/3))^6*(a+b*ln(c*(d+ 
e*x^(1/3))^n))/e^6-1/4*b*n*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e 
^6+1/2*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^6
 

3.5.58.2 Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.65 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt [3]{x} \left (809340 d^5-140070 d^4 e \sqrt [3]{x}+41180 d^3 e^2 x^{2/3}-13785 d^2 e^3 x+4368 d e^4 x^{4/3}-1000 e^5 x^{5/3}\right )+1800 a^2 b n \left (147 d^6+60 d^5 e \sqrt [3]{x}-30 d^4 e^2 x^{2/3}+20 d^3 e^3 x-15 d^2 e^4 x^{4/3}+12 d e^5 x^{5/3}-10 e^6 x^2\right )-36000 a^3 \left (d^6-e^6 x^2\right )+60 a b^2 n^2 \left (8111 d^6-8820 d^5 e \sqrt [3]{x}+2610 d^4 e^2 x^{2/3}-1140 d^3 e^3 x+555 d^2 e^4 x^{4/3}-264 d e^5 x^{5/3}+100 e^6 x^2\right )-60 b \left (b^2 n^2 \left (13489 d^6+8820 d^5 e \sqrt [3]{x}-2610 d^4 e^2 x^{2/3}+1140 d^3 e^3 x-555 d^2 e^4 x^{4/3}+264 d e^5 x^{5/3}-100 e^6 x^2\right )-60 a b n \left (147 d^6+60 d^5 e \sqrt [3]{x}-30 d^4 e^2 x^{2/3}+20 d^3 e^3 x-15 d^2 e^4 x^{4/3}+12 d e^5 x^{5/3}-10 e^6 x^2\right )+1800 a^2 \left (d^6-e^6 x^2\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )-1800 b^2 \left (60 a \left (d^6-e^6 x^2\right )+b n \left (-147 d^6-60 d^5 e \sqrt [3]{x}+30 d^4 e^2 x^{2/3}-20 d^3 e^3 x+15 d^2 e^4 x^{4/3}-12 d e^5 x^{5/3}+10 e^6 x^2\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-36000 b^3 \left (d^6-e^6 x^2\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{72000 e^6} \]

input

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

output

(b^3*e*n^3*x^(1/3)*(809340*d^5 - 140070*d^4*e*x^(1/3) + 41180*d^3*e^2*x^(2 
/3) - 13785*d^2*e^3*x + 4368*d*e^4*x^(4/3) - 1000*e^5*x^(5/3)) + 1800*a^2* 
b*n*(147*d^6 + 60*d^5*e*x^(1/3) - 30*d^4*e^2*x^(2/3) + 20*d^3*e^3*x - 15*d 
^2*e^4*x^(4/3) + 12*d*e^5*x^(5/3) - 10*e^6*x^2) - 36000*a^3*(d^6 - e^6*x^2 
) + 60*a*b^2*n^2*(8111*d^6 - 8820*d^5*e*x^(1/3) + 2610*d^4*e^2*x^(2/3) - 1 
140*d^3*e^3*x + 555*d^2*e^4*x^(4/3) - 264*d*e^5*x^(5/3) + 100*e^6*x^2) - 6 
0*b*(b^2*n^2*(13489*d^6 + 8820*d^5*e*x^(1/3) - 2610*d^4*e^2*x^(2/3) + 1140 
*d^3*e^3*x - 555*d^2*e^4*x^(4/3) + 264*d*e^5*x^(5/3) - 100*e^6*x^2) - 60*a 
*b*n*(147*d^6 + 60*d^5*e*x^(1/3) - 30*d^4*e^2*x^(2/3) + 20*d^3*e^3*x - 15* 
d^2*e^4*x^(4/3) + 12*d*e^5*x^(5/3) - 10*e^6*x^2) + 1800*a^2*(d^6 - e^6*x^2 
))*Log[c*(d + e*x^(1/3))^n] - 1800*b^2*(60*a*(d^6 - e^6*x^2) + b*n*(-147*d 
^6 - 60*d^5*e*x^(1/3) + 30*d^4*e^2*x^(2/3) - 20*d^3*e^3*x + 15*d^2*e^4*x^( 
4/3) - 12*d*e^5*x^(5/3) + 10*e^6*x^2))*Log[c*(d + e*x^(1/3))^n]^2 - 36000* 
b^3*(d^6 - e^6*x^2)*Log[c*(d + e*x^(1/3))^n]^3)/(72000*e^6)
 

3.5.58.3 Rubi [A] (verified)

Time = 1.21 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int x^{5/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3d\sqrt [3]{x}\)

\(\Big \downarrow \) 2848

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

\(\Big \downarrow \) 2009

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

input

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

output

3*((-15*b^3*d^4*n^3*(d + e*x^(1/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e*x^ 
(1/3))^3)/(27*e^6) - (15*b^3*d^2*n^3*(d + e*x^(1/3))^4)/(64*e^6) + (6*b^3* 
d*n^3*(d + e*x^(1/3))^5)/(125*e^6) - (b^3*n^3*(d + e*x^(1/3))^6)/(216*e^6) 
 - (6*a*b^2*d^5*n^2*x^(1/3))/e^5 + (6*b^3*d^5*n^3*x^(1/3))/e^5 - (6*b^3*d^ 
5*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^6 + (15*b^2*d^4*n^2*(d + 
 e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^6) - (20*b^2*d^3*n^2* 
(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(9*e^6) + (15*b^2*d^2* 
n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n]))/(16*e^6) - (6*b^2* 
d*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n]))/(25*e^6) + (b^2* 
n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n]))/(36*e^6) + (3*b*d^ 
5*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^6 - (15*b*d^4*n* 
(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^6) + (10*b*d^3* 
n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^6) - (15*b*d^ 
2*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(8*e^6) + (3*b*d 
*n*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^6) - (b*n*(d 
 + e*x^(1/3))^6*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(12*e^6) - (d^5*(d + e 
*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^6 + (5*d^4*(d + e*x^(1/3)) 
^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (10*d^3*(d + e*x^(1/3))^3 
*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(3*e^6) + (5*d^2*(d + e*x^(1/3))^4*(a 
 + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^6) - (d*(d + e*x^(1/3))^5*(a + b...
 

3.5.58.3.1 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])
 

3.5.58.4 Maple [F]

\[\int x {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}d x\]

input

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

output

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

3.5.58.5 Fricas [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 1190, normalized size of antiderivative = 1.31 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

input

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

output

1/72000*(36000*b^3*e^6*x^2*log(c)^3 + 36000*(b^3*e^6*n^3*x^2 - b^3*d^6*n^3 
)*log(e*x^(1/3) + d)^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^2 + 1800*(20*b^3*d^3*e^3*n^3*x + 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^2 + 60*(b^3*e^6*n^2*x 
^2 - b^3*d^6*n^2)*log(c) + 6*(2*b^3*d*e^5*n^3*x - 5*b^3*d^4*e^2*n^3)*x^(2/ 
3) - 15*(b^3*d^2*e^4*n^3*x - 4*b^3*d^5*e*n^3)*x^(1/3))*log(e*x^(1/3) + d)^ 
2 + 18000*(2*b^3*d^3*e^3*n*x - (b^3*e^6*n - 6*a*b^2*e^6)*x^2)*log(c)^2 + 2 
0*(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 
 - 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^2 - 1800*(b^3*e^6*n*x^2 - 
b^3*d^6*n)*log(c)^2 + 60*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x - 6 
0*(20*b^3*d^3*e^3*n^2*x + 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^2)*log(c) - 6*(435*b^3*d^4*e^2*n^3 - 300*a*b^2*d^4*e 
^2*n^2 - 4*(11*b^3*d*e^5*n^3 - 30*a*b^2*d*e^5*n^2)*x + 60*(2*b^3*d*e^5*n^2 
*x - 5*b^3*d^4*e^2*n^2)*log(c))*x^(2/3) + 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 + 60*(b^3*d^2* 
e^4*n^2*x - 4*b^3*d^5*e*n^2)*log(c))*x^(1/3))*log(e*x^(1/3) + d) + 1200*(5 
*(b^3*e^6*n^2 - 6*a*b^2*e^6*n + 18*a^2*b*e^6)*x^2 - 3*(19*b^3*d^3*e^3*n^2 
- 20*a*b^2*d^3*e^3*n)*x)*log(c) - 6*(23345*b^3*d^4*e^2*n^3 - 26100*a*b^2*d 
^4*e^2*n^2 + 9000*a^2*b*d^4*e^2*n - 1800*(2*b^3*d*e^5*n*x - 5*b^3*d^4*e...
 

3.5.58.6 Sympy [F]

\[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\int x \left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}\, dx \]

input

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

output

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

3.5.58.7 Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 668, normalized size of antiderivative = 0.74 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {1}{2} \, b^{3} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{3} + \frac {3}{2} \, a b^{2} x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} - \frac {1}{40} \, a^{2} b e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} + \frac {3}{2} \, a^{2} b x^{2} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + \frac {1}{2} \, a^{3} x^{2} - \frac {1}{1200} \, {\left (60 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) - \frac {{\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{6}}\right )} a b^{2} - \frac {1}{72000} \, {\left (1800 \, e n {\left (\frac {60 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )}{e^{7}} + \frac {10 \, e^{5} x^{2} - 12 \, d e^{4} x^{\frac {5}{3}} + 15 \, d^{2} e^{3} x^{\frac {4}{3}} - 20 \, d^{3} e^{2} x + 30 \, d^{4} e x^{\frac {2}{3}} - 60 \, d^{5} x^{\frac {1}{3}}}{e^{6}}\right )} \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )^{2} + e n {\left (\frac {{\left (36000 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{3} + 1000 \, e^{6} x^{2} + 264600 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 4368 \, d e^{5} x^{\frac {5}{3}} + 13785 \, d^{2} e^{4} x^{\frac {4}{3}} - 41180 \, d^{3} e^{3} x + 809340 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 140070 \, d^{4} e^{2} x^{\frac {2}{3}} - 809340 \, d^{5} e x^{\frac {1}{3}}\right )} n^{2}}{e^{7}} - \frac {60 \, {\left (100 \, e^{6} x^{2} + 1800 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right )^{2} - 264 \, d e^{5} x^{\frac {5}{3}} + 555 \, d^{2} e^{4} x^{\frac {4}{3}} - 1140 \, d^{3} e^{3} x + 8820 \, d^{6} \log \left (e x^{\frac {1}{3}} + d\right ) + 2610 \, d^{4} e^{2} x^{\frac {2}{3}} - 8820 \, d^{5} e x^{\frac {1}{3}}\right )} n \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right )}{e^{7}}\right )}\right )} b^{3} \]

input

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

output

1/2*b^3*x^2*log((e*x^(1/3) + d)^n*c)^3 + 3/2*a*b^2*x^2*log((e*x^(1/3) + d) 
^n*c)^2 - 1/40*a^2*b*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12 
*d*e^4*x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60 
*d^5*x^(1/3))/e^6) + 3/2*a^2*b*x^2*log((e*x^(1/3) + d)^n*c) + 1/2*a^3*x^2 
- 1/1200*(60*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 - 12*d*e^4*x 
^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) - 60*d^5*x^( 
1/3))/e^6)*log((e*x^(1/3) + d)^n*c) - (100*e^6*x^2 + 1800*d^6*log(e*x^(1/3 
) + d)^2 - 264*d*e^5*x^(5/3) + 555*d^2*e^4*x^(4/3) - 1140*d^3*e^3*x + 8820 
*d^6*log(e*x^(1/3) + d) + 2610*d^4*e^2*x^(2/3) - 8820*d^5*e*x^(1/3))*n^2/e 
^6)*a*b^2 - 1/72000*(1800*e*n*(60*d^6*log(e*x^(1/3) + d)/e^7 + (10*e^5*x^2 
 - 12*d*e^4*x^(5/3) + 15*d^2*e^3*x^(4/3) - 20*d^3*e^2*x + 30*d^4*e*x^(2/3) 
 - 60*d^5*x^(1/3))/e^6)*log((e*x^(1/3) + d)^n*c)^2 + e*n*((36000*d^6*log(e 
*x^(1/3) + d)^3 + 1000*e^6*x^2 + 264600*d^6*log(e*x^(1/3) + d)^2 - 4368*d* 
e^5*x^(5/3) + 13785*d^2*e^4*x^(4/3) - 41180*d^3*e^3*x + 809340*d^6*log(e*x 
^(1/3) + d) + 140070*d^4*e^2*x^(2/3) - 809340*d^5*e*x^(1/3))*n^2/e^7 - 60* 
(100*e^6*x^2 + 1800*d^6*log(e*x^(1/3) + d)^2 - 264*d*e^5*x^(5/3) + 555*d^2 
*e^4*x^(4/3) - 1140*d^3*e^3*x + 8820*d^6*log(e*x^(1/3) + d) + 2610*d^4*e^2 
*x^(2/3) - 8820*d^5*e*x^(1/3))*n*log((e*x^(1/3) + d)^n*c)/e^7))*b^3
 

3.5.58.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2160 vs. \(2 (787) = 1574\).

Time = 0.33 (sec) , antiderivative size = 2160, normalized size of antiderivative = 2.38 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

input

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

output

1/72000*(36000*b^3*e*x^2*log(c)^3 + 108000*a*b^2*e*x^2*log(c)^2 + (36000*( 
e*x^(1/3) + d)^6*log(e*x^(1/3) + d)^3/e^5 - 216000*(e*x^(1/3) + d)^5*d*log 
(e*x^(1/3) + d)^3/e^5 + 540000*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)^3/ 
e^5 - 720000*(e*x^(1/3) + d)^3*d^3*log(e*x^(1/3) + d)^3/e^5 + 540000*(e*x^ 
(1/3) + d)^2*d^4*log(e*x^(1/3) + d)^3/e^5 - 216000*(e*x^(1/3) + d)*d^5*log 
(e*x^(1/3) + d)^3/e^5 - 18000*(e*x^(1/3) + d)^6*log(e*x^(1/3) + d)^2/e^5 + 
 129600*(e*x^(1/3) + d)^5*d*log(e*x^(1/3) + d)^2/e^5 - 405000*(e*x^(1/3) + 
 d)^4*d^2*log(e*x^(1/3) + d)^2/e^5 + 720000*(e*x^(1/3) + d)^3*d^3*log(e*x^ 
(1/3) + d)^2/e^5 - 810000*(e*x^(1/3) + d)^2*d^4*log(e*x^(1/3) + d)^2/e^5 + 
 648000*(e*x^(1/3) + d)*d^5*log(e*x^(1/3) + d)^2/e^5 + 6000*(e*x^(1/3) + d 
)^6*log(e*x^(1/3) + d)/e^5 - 51840*(e*x^(1/3) + d)^5*d*log(e*x^(1/3) + d)/ 
e^5 + 202500*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)/e^5 - 480000*(e*x^(1 
/3) + d)^3*d^3*log(e*x^(1/3) + d)/e^5 + 810000*(e*x^(1/3) + d)^2*d^4*log(e 
*x^(1/3) + d)/e^5 - 1296000*(e*x^(1/3) + d)*d^5*log(e*x^(1/3) + d)/e^5 - 1 
000*(e*x^(1/3) + d)^6/e^5 + 10368*(e*x^(1/3) + d)^5*d/e^5 - 50625*(e*x^(1/ 
3) + d)^4*d^2/e^5 + 160000*(e*x^(1/3) + d)^3*d^3/e^5 - 405000*(e*x^(1/3) + 
 d)^2*d^4/e^5 + 1296000*(e*x^(1/3) + d)*d^5/e^5)*b^3*n^3 + 60*(1800*(e*x^( 
1/3) + d)^6*log(e*x^(1/3) + d)^2/e^5 - 10800*(e*x^(1/3) + d)^5*d*log(e*x^( 
1/3) + d)^2/e^5 + 27000*(e*x^(1/3) + d)^4*d^2*log(e*x^(1/3) + d)^2/e^5 - 3 
6000*(e*x^(1/3) + d)^3*d^3*log(e*x^(1/3) + d)^2/e^5 + 27000*(e*x^(1/3) ...
 

3.5.58.9 Mupad [B] (verification not implemented)

Time = 9.19 (sec) , antiderivative size = 979, normalized size of antiderivative = 1.08 \[ \int x \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {a^3\,x^2}{2}+\frac {b^3\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{2}-\frac {b^3\,n^3\,x^2}{72}+\frac {3\,a\,b^2\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2}-\frac {b^3\,n\,x^2\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{4}+\frac {b^3\,n^2\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{12}+\frac {a\,b^2\,n^2\,x^2}{12}-\frac {b^3\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^3}{2\,e^6}+\frac {3\,a^2\,b\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}-\frac {a^2\,b\,n\,x^2}{4}-\frac {a\,b^2\,n\,x^2\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2}-\frac {13489\,b^3\,d^6\,n^3\,\ln \left (d+e\,x^{1/3}\right )}{1200\,e^6}-\frac {919\,b^3\,d^2\,n^3\,x^{4/3}}{4800\,e^2}-\frac {4669\,b^3\,d^4\,n^3\,x^{2/3}}{2400\,e^4}+\frac {13489\,b^3\,d^5\,n^3\,x^{1/3}}{1200\,e^5}-\frac {3\,a\,b^2\,d^6\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^6}+\frac {147\,b^3\,d^6\,n\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{40\,e^6}+\frac {2059\,b^3\,d^3\,n^3\,x}{3600\,e^3}+\frac {91\,b^3\,d\,n^3\,x^{5/3}}{1500\,e}-\frac {3\,a^2\,b\,d^6\,n\,\ln \left (d+e\,x^{1/3}\right )}{2\,e^6}+\frac {b^3\,d^3\,n\,x\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^3}-\frac {19\,b^3\,d^3\,n^2\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{20\,e^3}+\frac {3\,b^3\,d\,n\,x^{5/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{10\,e}-\frac {11\,b^3\,d\,n^2\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{50\,e}-\frac {19\,a\,b^2\,d^3\,n^2\,x}{20\,e^3}-\frac {11\,a\,b^2\,d\,n^2\,x^{5/3}}{50\,e}-\frac {3\,a^2\,b\,d^2\,n\,x^{4/3}}{8\,e^2}-\frac {3\,a^2\,b\,d^4\,n\,x^{2/3}}{4\,e^4}+\frac {3\,a^2\,b\,d^5\,n\,x^{1/3}}{2\,e^5}+\frac {147\,a\,b^2\,d^6\,n^2\,\ln \left (d+e\,x^{1/3}\right )}{20\,e^6}-\frac {3\,b^3\,d^2\,n\,x^{4/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{8\,e^2}+\frac {37\,b^3\,d^2\,n^2\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{80\,e^2}-\frac {3\,b^3\,d^4\,n\,x^{2/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{4\,e^4}+\frac {87\,b^3\,d^4\,n^2\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{40\,e^4}+\frac {3\,b^3\,d^5\,n\,x^{1/3}\,{\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}^2}{2\,e^5}-\frac {147\,b^3\,d^5\,n^2\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{20\,e^5}+\frac {37\,a\,b^2\,d^2\,n^2\,x^{4/3}}{80\,e^2}+\frac {87\,a\,b^2\,d^4\,n^2\,x^{2/3}}{40\,e^4}-\frac {147\,a\,b^2\,d^5\,n^2\,x^{1/3}}{20\,e^5}+\frac {a^2\,b\,d^3\,n\,x}{2\,e^3}+\frac {3\,a^2\,b\,d\,n\,x^{5/3}}{10\,e}+\frac {a\,b^2\,d^3\,n\,x\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^3}+\frac {3\,a\,b^2\,d\,n\,x^{5/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{5\,e}-\frac {3\,a\,b^2\,d^2\,n\,x^{4/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{4\,e^2}-\frac {3\,a\,b^2\,d^4\,n\,x^{2/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{2\,e^4}+\frac {3\,a\,b^2\,d^5\,n\,x^{1/3}\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )}{e^5} \]

input

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

output

(a^3*x^2)/2 + (b^3*x^2*log(c*(d + e*x^(1/3))^n)^3)/2 - (b^3*n^3*x^2)/72 + 
(3*a*b^2*x^2*log(c*(d + e*x^(1/3))^n)^2)/2 - (b^3*n*x^2*log(c*(d + e*x^(1/ 
3))^n)^2)/4 + (b^3*n^2*x^2*log(c*(d + e*x^(1/3))^n))/12 + (a*b^2*n^2*x^2)/ 
12 - (b^3*d^6*log(c*(d + e*x^(1/3))^n)^3)/(2*e^6) + (3*a^2*b*x^2*log(c*(d 
+ e*x^(1/3))^n))/2 - (a^2*b*n*x^2)/4 - (a*b^2*n*x^2*log(c*(d + e*x^(1/3))^ 
n))/2 - (13489*b^3*d^6*n^3*log(d + e*x^(1/3)))/(1200*e^6) - (919*b^3*d^2*n 
^3*x^(4/3))/(4800*e^2) - (4669*b^3*d^4*n^3*x^(2/3))/(2400*e^4) + (13489*b^ 
3*d^5*n^3*x^(1/3))/(1200*e^5) - (3*a*b^2*d^6*log(c*(d + e*x^(1/3))^n)^2)/( 
2*e^6) + (147*b^3*d^6*n*log(c*(d + e*x^(1/3))^n)^2)/(40*e^6) + (2059*b^3*d 
^3*n^3*x)/(3600*e^3) + (91*b^3*d*n^3*x^(5/3))/(1500*e) - (3*a^2*b*d^6*n*lo 
g(d + e*x^(1/3)))/(2*e^6) + (b^3*d^3*n*x*log(c*(d + e*x^(1/3))^n)^2)/(2*e^ 
3) - (19*b^3*d^3*n^2*x*log(c*(d + e*x^(1/3))^n))/(20*e^3) + (3*b^3*d*n*x^( 
5/3)*log(c*(d + e*x^(1/3))^n)^2)/(10*e) - (11*b^3*d*n^2*x^(5/3)*log(c*(d + 
 e*x^(1/3))^n))/(50*e) - (19*a*b^2*d^3*n^2*x)/(20*e^3) - (11*a*b^2*d*n^2*x 
^(5/3))/(50*e) - (3*a^2*b*d^2*n*x^(4/3))/(8*e^2) - (3*a^2*b*d^4*n*x^(2/3)) 
/(4*e^4) + (3*a^2*b*d^5*n*x^(1/3))/(2*e^5) + (147*a*b^2*d^6*n^2*log(d + e* 
x^(1/3)))/(20*e^6) - (3*b^3*d^2*n*x^(4/3)*log(c*(d + e*x^(1/3))^n)^2)/(8*e 
^2) + (37*b^3*d^2*n^2*x^(4/3)*log(c*(d + e*x^(1/3))^n))/(80*e^2) - (3*b^3* 
d^4*n*x^(2/3)*log(c*(d + e*x^(1/3))^n)^2)/(4*e^4) + (87*b^3*d^4*n^2*x^(2/3 
)*log(c*(d + e*x^(1/3))^n))/(40*e^4) + (3*b^3*d^5*n*x^(1/3)*log(c*(d + ...