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

Optimal result

Integrand size = 24, antiderivative size = 1357 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:

18*a*b^2*d^8*n^2*x^(1/3)/e^8+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+5 
6/3*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^9-9*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+1/3*(d+ 
e*x^(1/3))^9*(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^9-252/125*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+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)...
 

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 808, normalized size of antiderivative = 0.60 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\frac {b^3 e n^3 \sqrt [3]{x} \left (-76356985320 d^8+15542491860 d^7 e \sqrt [3]{x}-5483495640 d^6 e^2 x^{2/3}[phone] d^5 e^3 x-1075607064 d^4 e^4 x^{4/3}[phone] d^3 e^5 x^{5/3}-219465000 d^2 e^6 x^2+83734875 d e^7 x^{7/3}-21952000 e^8 x^{8/3}\right )-2520 a b^2 n^2 \left (26853209 d^9-17965080 d^8 e \sqrt [3]{x}+5807340 d^7 e^2 x^{2/3}-2813160 d^6 e^3 x+1580670 d^5 e^4 x^{4/3}-947016 d^4 e^5 x^{5/3}+577500 d^3 e^6 x^2-343800 d^2 e^7 x^{7/3}+187425 d e^8 x^{8/3}-78400 e^9 x^3\right )[phone] a^3 \left (d^9+e^9 x^3\right )-3175200 a^2 b n \left (7129 d^9+2520 d^8 e \sqrt [3]{x}-1260 d^7 e^2 x^{2/3}+840 d^6 e^3 x-630 d^5 e^4 x^{4/3}+504 d^4 e^5 x^{5/3}-420 d^3 e^6 x^2+360 d^2 e^7 x^{7/3}-315 d e^8 x^{8/3}+280 e^9 x^3\right )+2520 b \left (3175200 a^2 \left (d^9+e^9 x^3\right )-2520 a b n \left (7129 d^9+2520 d^8 e \sqrt [3]{x}-1260 d^7 e^2 x^{2/3}+840 d^6 e^3 x-630 d^5 e^4 x^{4/3}+504 d^4 e^5 x^{5/3}-420 d^3 e^6 x^2+360 d^2 e^7 x^{7/3}-315 d e^8 x^{8/3}+280 e^9 x^3\right )+b^2 n^2 \left (30300391 d^9+17965080 d^8 e \sqrt [3]{x}-5807340 d^7 e^2 x^{2/3}+2813160 d^6 e^3 x-1580670 d^5 e^4 x^{4/3}+947016 d^4 e^5 x^{5/3}-577500 d^3 e^6 x^2+343800 d^2 e^7 x^{7/3}-187425 d e^8 x^{8/3}+78400 e^9 x^3\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )+3175200 b^2 \left (2520 a \left (d^9+e^9 x^3\right )-b n \left (7129 d^9+2520 d^8 e \sqrt [3]{x}-1260 d^7 e^2 x^{2/3}+840 d^6 e^3 x-630 d^5 e^4 x^{4/3}+504 d^4 e^5 x^{5/3}-420 d^3 e^6 x^2+360 d^2 e^7 x^{7/3}-315 d e^8 x^{8/3}+280 e^9 x^3\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )[phone] b^3 \left (d^9+e^9 x^3\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{8001504000 e^9} \] Input:

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

Output:

(b^3*e*n^3*x^(1/3)*(-76356985320*d^8 + 15542491860*d^7*e*x^(1/3) - 5483495 
640*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)) - 2520*a*b^2*n^2*(26853209*d^9 - 17965080*d^8*e*x 
^(1/3) + 5807340*d^7*e^2*x^(2/3) - 2813160*d^6*e^3*x + 1580670*d^5*e^4*x^( 
4/3) - 947016*d^4*e^5*x^(5/3) + 577500*d^3*e^6*x^2 - 343800*d^2*e^7*x^(7/3 
) + 187425*d*e^8*x^(8/3) - 78400*e^9*x^3) + 2667168000*a^3*(d^9 + e^9*x^3) 
 - 3175200*a^2*b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3) + 
 840*d^6*e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e^5*x^(5/3) - 420*d^3*e^6*x 
^2 + 360*d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x^3) + 2520*b*(3175 
200*a^2*(d^9 + e^9*x^3) - 2520*a*b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260 
*d^7*e^2*x^(2/3) + 840*d^6*e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e^5*x^(5/ 
3) - 420*d^3*e^6*x^2 + 360*d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x 
^3) + b^2*n^2*(30300391*d^9 + 17965080*d^8*e*x^(1/3) - 5807340*d^7*e^2*x^( 
2/3) + 2813160*d^6*e^3*x - 1580670*d^5*e^4*x^(4/3) + 947016*d^4*e^5*x^(5/3 
) - 577500*d^3*e^6*x^2 + 343800*d^2*e^7*x^(7/3) - 187425*d*e^8*x^(8/3) + 7 
8400*e^9*x^3))*Log[c*(d + e*x^(1/3))^n] + 3175200*b^2*(2520*a*(d^9 + e^9*x 
^3) - b*n*(7129*d^9 + 2520*d^8*e*x^(1/3) - 1260*d^7*e^2*x^(2/3) + 840*d^6* 
e^3*x - 630*d^5*e^4*x^(4/3) + 504*d^4*e^5*x^(5/3) - 420*d^3*e^6*x^2 + 360* 
d^2*e^7*x^(7/3) - 315*d*e^8*x^(8/3) + 280*e^9*x^3))*Log[c*(d + e*x^(1/3...
 

Rubi [A] (verified)

Time = 3.20 (sec) , antiderivative size = 1366, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int x^{8/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3d\sqrt [3]{x}\)

\(\Big \downarrow \) 2848

\(\displaystyle 3 \int \left (\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^8}{e^8}-\frac {8 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^7}{e^8}+\frac {28 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^6}{e^8}-\frac {56 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^5}{e^8}+\frac {70 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^4}{e^8}-\frac {56 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^3}{e^8}+\frac {28 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^2}{e^8}-\frac {8 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d}{e^8}+\frac {\left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^8}\right )d\sqrt [3]{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (-\frac {2 b^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{2187 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}{9 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}{27 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}{243 e^9}+\frac {3 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^8}{256 e^9}-\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 )^8}{e^9}+\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 )^8}{8 e^9}-\frac {3 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 {24 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^9}+\frac {4 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 {12 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 {24 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}{27 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}{3 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}{3 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}{9 e^9}-\frac {84 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^9}+\frac {14 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 {42 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 {84 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 {21 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^4}{16 e^9}-\frac {14 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 {21 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 {21 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}{27 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}{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}{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}{9 e^9}+\frac {3 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^2}{e^9}-\frac {4 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 {6 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 {6 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 {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 {3 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 {6 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 {6 b^3 d^8 n^3 \sqrt [3]{x}}{e^8}+\frac {6 a b^2 d^8 n^2 \sqrt [3]{x}}{e^8}\right )\)

Input:

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

Output:

3*((3*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)/(27*e^9) + (21*b^3*d^5*n^3*(d + e*x^(1/3))^4)/(16*e^9) - (84*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)/(27*e^ 
9) - (24*b^3*d^2*n^3*(d + e*x^(1/3))^7)/(343*e^9) + (3*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)/(2187*e^9) + (6*a*b^2* 
d^8*n^2*x^(1/3))/e^8 - (6*b^3*d^8*n^3*x^(1/3))/e^8 + (6*b^3*d^8*n^2*(d + e 
*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^9 - (6*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]))/(9*e^9) - (21*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) + (84*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]))/(9*e^9) + (24*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) - (3*b^2*d*n^ 
2*(d + e*x^(1/3))^8*(a + b*Log[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]))/(243*e^9) - (3*b*d^8 
*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^9 + (6*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)/(3*e^9) + (21*b*d^5*n*(d 
 + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(2*e^9) - (42*b*d^4*n* 
(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^9) + (14*b*d...
 

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]

\[\int x^{2} {\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}d x\]

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1688, normalized size of antiderivative = 1.24 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/8001504000*(2667168000*b^3*e^9*x^3*log(c)^3 - 10976000*(2*b^3*e^9*n^3 - 
18*a*b^2*e^9*n^2 + 81*a^2*b*e^9*n - 243*a^3*e^9)*x^3 + 2667168000*(b^3*e^9 
*n^3*x^3 + b^3*d^9*n^3)*log(e*x^(1/3) + d)^3 + 10500*(47485*b^3*d^3*e^6*n^ 
3 - 138600*a*b^2*d^3*e^6*n^2 + 127008*a^2*b*d^3*e^6*n)*x^2 + 3175200*(420* 
b^3*d^3*e^6*n^3*x^2 - 840*b^3*d^6*e^3*n^3*x - 7129*b^3*d^9*n^3 + 2520*a*b^ 
2*d^9*n^2 - 280*(b^3*e^9*n^3 - 9*a*b^2*e^9*n^2)*x^3 + 2520*(b^3*e^9*n^2*x^ 
3 + b^3*d^9*n^2)*log(c) + 63*(5*b^3*d*e^8*n^3*x^2 - 8*b^3*d^4*e^5*n^3*x + 
20*b^3*d^7*e^2*n^3)*x^(2/3) - 90*(4*b^3*d^2*e^7*n^3*x^2 - 7*b^3*d^5*e^4*n^ 
3*x + 28*b^3*d^8*e*n^3)*x^(1/3))*log(e*x^(1/3) + d)^2 + 444528000*(3*b^3*d 
^3*e^6*n*x^2 - 6*b^3*d^6*e^3*n*x - 2*(b^3*e^9*n - 9*a*b^2*e^9)*x^3)*log(c) 
^2 - 840*(6527971*b^3*d^6*e^3*n^3 - 8439480*a*b^2*d^6*e^3*n^2 + 3175200*a^ 
2*b*d^6*e^3*n)*x + 2520*(30300391*b^3*d^9*n^3 - 17965080*a*b^2*d^9*n^2 + 3 
175200*a^2*b*d^9*n + 39200*(2*b^3*e^9*n^3 - 18*a*b^2*e^9*n^2 + 81*a^2*b*e^ 
9*n)*x^3 - 2100*(275*b^3*d^3*e^6*n^3 - 504*a*b^2*d^3*e^6*n^2)*x^2 + 317520 
0*(b^3*e^9*n*x^3 + b^3*d^9*n)*log(c)^2 + 840*(3349*b^3*d^6*e^3*n^3 - 2520* 
a*b^2*d^6*e^3*n^2)*x + 2520*(420*b^3*d^3*e^6*n^2*x^2 - 840*b^3*d^6*e^3*n^2 
*x - 7129*b^3*d^9*n^2 + 2520*a*b^2*d^9*n - 280*(b^3*e^9*n^2 - 9*a*b^2*e^9* 
n)*x^3)*log(c) - 63*(92180*b^3*d^7*e^2*n^3 - 50400*a*b^2*d^7*e^2*n^2 + 175 
*(17*b^3*d*e^8*n^3 - 72*a*b^2*d*e^8*n^2)*x^2 - 8*(1879*b^3*d^4*e^5*n^3 - 2 
520*a*b^2*d^4*e^5*n^2)*x - 2520*(5*b^3*d*e^8*n^2*x^2 - 8*b^3*d^4*e^5*n^...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 867, normalized size of antiderivative = 0.64 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*b^3*x^3*log((e*x^(1/3) + d)^n*c)^3 + a*b^2*x^3*log((e*x^(1/3) + d)^n*c 
)^2 + a^2*b*x^3*log((e*x^(1/3) + d)^n*c) + 1/3*a^3*x^3 + 1/2520*a^2*b*e*n* 
(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 360 
*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^( 
4/3) + 840*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9) + 1/317 
5200*(2520*e*n*(2520*d^9*log(e*x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^ 
7*x^(8/3) + 360*d^2*e^6*x^(7/3) - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 
630*d^5*e^3*x^(4/3) + 840*d^6*e^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3 
))/e^9)*log((e*x^(1/3) + d)^n*c) + (78400*e^9*x^3 - 187425*d*e^8*x^(8/3) + 
 343800*d^2*e^7*x^(7/3) - 577500*d^3*e^6*x^2 - 3175200*d^9*log(e*x^(1/3) + 
 d)^2 + 947016*d^4*e^5*x^(5/3) - 1580670*d^5*e^4*x^(4/3) + 2813160*d^6*e^3 
*x - 17965080*d^9*log(e*x^(1/3) + d) - 5807340*d^7*e^2*x^(2/3) + 17965080* 
d^8*e*x^(1/3))*n^2/e^9)*a*b^2 + 1/8001504000*(3175200*e*n*(2520*d^9*log(e* 
x^(1/3) + d)/e^10 - (280*e^8*x^3 - 315*d*e^7*x^(8/3) + 360*d^2*e^6*x^(7/3) 
 - 420*d^3*e^5*x^2 + 504*d^4*e^4*x^(5/3) - 630*d^5*e^3*x^(4/3) + 840*d^6*e 
^2*x - 1260*d^7*e*x^(2/3) + 2520*d^8*x^(1/3))/e^9)*log((e*x^(1/3) + d)^n*c 
)^2 - e*n*((21952000*e^9*x^3 - 2667168000*d^9*log(e*x^(1/3) + d)^3 - 83734 
875*d*e^8*x^(8/3) + 219465000*d^2*e^7*x^(7/3) - 498592500*d^3*e^6*x^2 - 22 
636000800*d^9*log(e*x^(1/3) + d)^2 + 1075607064*d^4*e^5*x^(5/3) - 23403309 
30*d^5*e^4*x^(4/3) + 5483495640*d^6*e^3*x - 76356985320*d^9*log(e*x^(1/...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3240 vs. \(2 (1189) = 2378\).

Time = 0.17 (sec) , antiderivative size = 3240, normalized size of antiderivative = 2.39 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/8001504000*(2667168000*b^3*e*x^3*log(c)^3 + 8001504000*a*b^2*e*x^3*log(c 
)^2 + 8001504000*a^2*b*e*x^3*log(c) + (2667168000*(e*x^(1/3) + d)^9*log(e* 
x^(1/3) + d)^3/e^8 - 24004512000*(e*x^(1/3) + d)^8*d*log(e*x^(1/3) + d)^3/ 
e^8 + 96018048000*(e*x^(1/3) + d)^7*d^2*log(e*x^(1/3) + d)^3/e^8 - 2240421 
12000*(e*x^(1/3) + d)^6*d^3*log(e*x^(1/3) + d)^3/e^8 + 336063168000*(e*x^( 
1/3) + d)^5*d^4*log(e*x^(1/3) + d)^3/e^8 - 336063168000*(e*x^(1/3) + d)^4* 
d^5*log(e*x^(1/3) + d)^3/e^8 + 224042112000*(e*x^(1/3) + d)^3*d^6*log(e*x^ 
(1/3) + d)^3/e^8 - 96018048000*(e*x^(1/3) + d)^2*d^7*log(e*x^(1/3) + d)^3/ 
e^8 + 24004512000*(e*x^(1/3) + d)*d^8*log(e*x^(1/3) + d)^3/e^8 - 889056000 
*(e*x^(1/3) + d)^9*log(e*x^(1/3) + d)^2/e^8 + 9001692000*(e*x^(1/3) + d)^8 
*d*log(e*x^(1/3) + d)^2/e^8 - 41150592000*(e*x^(1/3) + d)^7*d^2*log(e*x^(1 
/3) + d)^2/e^8 + 112021056000*(e*x^(1/3) + d)^6*d^3*log(e*x^(1/3) + d)^2/e 
^8 - 201637900800*(e*x^(1/3) + d)^5*d^4*log(e*x^(1/3) + d)^2/e^8 + 2520473 
76000*(e*x^(1/3) + d)^4*d^5*log(e*x^(1/3) + d)^2/e^8 - 224042112000*(e*x^( 
1/3) + d)^3*d^6*log(e*x^(1/3) + d)^2/e^8 + 144027072000*(e*x^(1/3) + d)^2* 
d^7*log(e*x^(1/3) + d)^2/e^8 - 72013536000*(e*x^(1/3) + d)*d^8*log(e*x^(1/ 
3) + d)^2/e^8 + 197568000*(e*x^(1/3) + d)^9*log(e*x^(1/3) + d)/e^8 - 22504 
23000*(e*x^(1/3) + d)^8*d*log(e*x^(1/3) + d)/e^8 + 11757312000*(e*x^(1/3) 
+ d)^7*d^2*log(e*x^(1/3) + d)/e^8 - 37340352000*(e*x^(1/3) + d)^6*d^3*log( 
e*x^(1/3) + d)/e^8 + 80655160320*(e*x^(1/3) + d)^5*d^4*log(e*x^(1/3) + ...
 

Mupad [B] (verification not implemented)

Time = 21.04 (sec) , antiderivative size = 1386, normalized size of antiderivative = 1.02 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(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) - (21 
34141*b^3*d^4*n^3*x^(5/3))/(15876000*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) - (30300391* 
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*...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1405, normalized size of antiderivative = 1.04 \[ \int x^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(4000752000*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**7*e**2*n - 1600 
300800*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**4*e**5*n*x + 1000188 
000*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d*e**8*n*x**2 + 8001504000 
*x**(2/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**7*e**2*n - 3200601600*x**(2 
/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**4*e**5*n*x + 2000376000*x**(2/3)* 
log((x**(1/3)*e + d)**n*c)*a*b**2*d*e**8*n*x**2 - 14634496800*x**(2/3)*log 
((x**(1/3)*e + d)**n*c)*b**3*d**7*e**2*n**2 + 2386480320*x**(2/3)*log((x** 
(1/3)*e + d)**n*c)*b**3*d**4*e**5*n**2*x - 472311000*x**(2/3)*log((x**(1/3 
)*e + d)**n*c)*b**3*d*e**8*n**2*x**2 + 4000752000*x**(2/3)*a**2*b*d**7*e** 
2*n - 1600300800*x**(2/3)*a**2*b*d**4*e**5*n*x + 1000188000*x**(2/3)*a**2* 
b*d*e**8*n*x**2 - 14634496800*x**(2/3)*a*b**2*d**7*e**2*n**2 + 2386480320* 
x**(2/3)*a*b**2*d**4*e**5*n**2*x - 472311000*x**(2/3)*a*b**2*d*e**8*n**2*x 
**2 + 15542491860*x**(2/3)*b**3*d**7*e**2*n**3 - 1075607064*x**(2/3)*b**3* 
d**4*e**5*n**3*x + 83734875*x**(2/3)*b**3*d*e**8*n**3*x**2 - 8001504000*x* 
*(1/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**8*e*n + 2000376000*x**(1/3)*l 
og((x**(1/3)*e + d)**n*c)**2*b**3*d**5*e**4*n*x - 1143072000*x**(1/3)*log( 
(x**(1/3)*e + d)**n*c)**2*b**3*d**2*e**7*n*x**2 - 16003008000*x**(1/3)*log 
((x**(1/3)*e + d)**n*c)*a*b**2*d**8*e*n + 4000752000*x**(1/3)*log((x**(1/3 
)*e + d)**n*c)*a*b**2*d**5*e**4*n*x - 2286144000*x**(1/3)*log((x**(1/3)*e 
+ d)**n*c)*a*b**2*d**2*e**7*n*x**2 + 45272001600*x**(1/3)*log((x**(1/3)...