\(\int x^3 (a+b \log (c (d+e \sqrt [3]{x})^n))^3 \, dx\) [456]

Optimal result

Integrand size = 24, antiderivative size = 1835 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:

-1188/49*b^2*d^5*n^2*(d+e*x^(1/3))^7*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12+1485 
/128*b^2*d^4*n^2*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12-110/27*b 
^2*d^3*n^2*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12+99/100*b^2*d^2 
*n^2*(d+e*x^(1/3))^10*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12-18/121*b^2*d*n^2*(d 
+e*x^(1/3))^11*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12+9*b*d^11*n*(d+e*x^(1/3))*( 
a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12-99/4*b*d^10*n*(d+e*x^(1/3))^2*(a+b*ln(c* 
(d+e*x^(1/3))^n))^2/e^12+55*b*d^9*n*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3) 
)^n))^2/e^12-1485/16*b*d^8*n*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))^2 
/e^12+594/5*b*d^7*n*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12-231 
/2*b*d^6*n*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12+594/7*b*d^5* 
n*(d+e*x^(1/3))^7*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12-1485/32*b*d^4*n*(d+e* 
x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12+55/3*b*d^3*n*(d+e*x^(1/3))^9 
*(a+b*ln(c*(d+e*x^(1/3))^n))^2/e^12-99/20*b*d^2*n*(d+e*x^(1/3))^10*(a+b*ln 
(c*(d+e*x^(1/3))^n))^2/e^12+9/11*b*d*n*(d+e*x^(1/3))^11*(a+b*ln(c*(d+e*x^( 
1/3))^n))^2/e^12-18*b^3*d^11*n^2*(d+e*x^(1/3))*ln(c*(d+e*x^(1/3))^n)/e^12- 
18*a*b^2*d^11*n^2*x^(1/3)/e^11+99/4*b^2*d^10*n^2*(d+e*x^(1/3))^2*(a+b*ln(c 
*(d+e*x^(1/3))^n))/e^12-110/3*b^2*d^9*n^2*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x 
^(1/3))^n))/e^12+1485/32*b^2*d^8*n^2*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3 
))^n))/e^12-1188/25*b^2*d^7*n^2*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n) 
)/e^12+77/2*b^2*d^6*n^2*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))/e^1...
 

Mathematica [A] (verified)

Time = 1.45 (sec) , antiderivative size = 1025, normalized size of antiderivative = 0.56 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

(e*x^(1/3)*(3550000608000*a^3*e^11*x^(11/3) + b^3*n^3*(119225632485960*d^1 
1 - 26563616859780*d^10*e*x^(1/3) + 10242678720120*d^9*e^2*x^(2/3) - 48363 
09598890*d^8*e^3*x + 2516628075192*d^7*e^4*x^(4/3) - 1373077023780*d^6*e^5 
*x^(5/3) + 761128152840*d^5*e^6*x^2 - 417533743935*d^4*e^7*x^(7/3) + 22016 
1492320*d^3*e^8*x^(8/3) - 106944990768*d^2*e^9*x^3 + 44119404000*d*e^10*x^ 
(10/3) - 12326391000*e^11*x^(11/3)) - 27720*a*b^2*n^2*(2384502120*d^11 - 8 
08051860*d^10*e*x^(1/3) + 410634840*d^9*e^2*x^(2/3) - 243942930*d^8*e^3*x 
+ 156734424*d^7*e^4*x^(4/3) - 104998740*d^6*e^5*x^(5/3) + 71703720*d^5*e^6 
*x^2 - 49019355*d^4*e^7*x^(7/3) + 32900560*d^3*e^8*x^(8/3) - 21072744*d^2* 
e^9*x^3 + 12171600*d*e^10*x^(10/3) - 5336100*e^11*x^(11/3)) + 384199200*a^ 
2*b*n*(27720*d^11 - 13860*d^10*e*x^(1/3) + 9240*d^9*e^2*x^(2/3) - 6930*d^8 
*e^3*x + 5544*d^7*e^4*x^(4/3) - 4620*d^6*e^5*x^(5/3) + 3960*d^5*e^6*x^2 - 
3465*d^4*e^7*x^(7/3) + 3080*d^3*e^8*x^(8/3) - 2772*d^2*e^9*x^3 + 2520*d*e^ 
10*x^(10/3) - 2310*e^11*x^(11/3))) - 27720*b*d^12*n*(384199200*a^2 - 23845 
02120*a*b*n + 4301068993*b^2*n^2)*Log[d + e*x^(1/3)] + 27720*b*e*x^(1/3)*( 
384199200*a^2*e^11*x^(11/3) + 27720*a*b*n*(27720*d^11 - 13860*d^10*e*x^(1/ 
3) + 9240*d^9*e^2*x^(2/3) - 6930*d^8*e^3*x + 5544*d^7*e^4*x^(4/3) - 4620*d 
^6*e^5*x^(5/3) + 3960*d^5*e^6*x^2 - 3465*d^4*e^7*x^(7/3) + 3080*d^3*e^8*x^ 
(8/3) - 2772*d^2*e^9*x^3 + 2520*d*e^10*x^(10/3) - 2310*e^11*x^(11/3)) + b^ 
2*n^2*(-2384502120*d^11 + 808051860*d^10*e*x^(1/3) - 410634840*d^9*e^2*...
 

Rubi [A] (verified)

Time = 4.64 (sec) , antiderivative size = 1843, 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.

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

\(\Big \downarrow \) 2904

\(\displaystyle 3 \int x^{11/3} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3d\sqrt [3]{x}\)

\(\Big \downarrow \) 2848

\(\displaystyle 3 \int \left (-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^{11}}{e^{11}}+\frac {11 \left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^{10}}{e^{11}}-\frac {55 \left (d+e \sqrt [3]{x}\right )^2 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^9}{e^{11}}+\frac {165 \left (d+e \sqrt [3]{x}\right )^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^8}{e^{11}}-\frac {330 \left (d+e \sqrt [3]{x}\right )^4 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^7}{e^{11}}+\frac {462 \left (d+e \sqrt [3]{x}\right )^5 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^6}{e^{11}}-\frac {462 \left (d+e \sqrt [3]{x}\right )^6 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^5}{e^{11}}+\frac {330 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^4}{e^{11}}-\frac {165 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^3}{e^{11}}+\frac {55 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d^2}{e^{11}}-\frac {11 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 d}{e^{11}}+\frac {\left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{e^{11}}\right )d\sqrt [3]{x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \left (-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^{12}}{3456 e^{12}}+\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 )^{12}}{12 e^{12}}-\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 )^{12}}{48 e^{12}}+\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 )^{12}}{288 e^{12}}+\frac {6 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^{11}}{1331 e^{12}}-\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 )^{11}}{e^{12}}+\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 )^{11}}{11 e^{12}}-\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 )^{11}}{121 e^{12}}-\frac {33 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^{10}}{1000 e^{12}}+\frac {11 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 )^{10}}{2 e^{12}}-\frac {33 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 )^{10}}{20 e^{12}}+\frac {33 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 )^{10}}{100 e^{12}}+\frac {110 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{729 e^{12}}-\frac {55 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 )^9}{3 e^{12}}+\frac {55 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 )^9}{9 e^{12}}-\frac {110 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 )^9}{81 e^{12}}-\frac {495 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^8}{1024 e^{12}}+\frac {165 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 )^8}{4 e^{12}}-\frac {495 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 )^8}{32 e^{12}}+\frac {495 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 )^8}{128 e^{12}}+\frac {396 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^{12}}-\frac {66 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 )^7}{e^{12}}+\frac {198 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 )^7}{7 e^{12}}-\frac {396 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 )^7}{49 e^{12}}-\frac {77 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^6}{36 e^{12}}+\frac {77 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 )^6}{e^{12}}-\frac {77 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 )^6}{2 e^{12}}+\frac {77 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 )^6}{6 e^{12}}+\frac {396 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^{12}}-\frac {66 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 )^5}{e^{12}}+\frac {198 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 )^5}{5 e^{12}}-\frac {396 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 )^5}{25 e^{12}}-\frac {495 b^3 d^8 n^3 \left (d+e \sqrt [3]{x}\right )^4}{128 e^{12}}+\frac {165 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 )^4}{4 e^{12}}-\frac {495 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 )^4}{16 e^{12}}+\frac {495 b^2 d^8 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}{32 e^{12}}+\frac {110 b^3 d^9 n^3 \left (d+e \sqrt [3]{x}\right )^3}{27 e^{12}}-\frac {55 d^9 \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^{12}}+\frac {55 b d^9 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^{12}}-\frac {110 b^2 d^9 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^{12}}-\frac {33 b^3 d^{10} n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^{12}}+\frac {11 d^{10} \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^{12}}-\frac {33 b d^{10} 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^{12}}+\frac {33 b^2 d^{10} 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^{12}}-\frac {d^{11} \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^{12}}+\frac {3 b d^{11} 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^{12}}-\frac {6 b^3 d^{11} n^2 \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right ) \left (d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {6 b^3 d^{11} n^3 \sqrt [3]{x}}{e^{11}}-\frac {6 a b^2 d^{11} n^2 \sqrt [3]{x}}{e^{11}}\right )\)

Input:

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

Output:

3*((-33*b^3*d^10*n^3*(d + e*x^(1/3))^2)/(8*e^12) + (110*b^3*d^9*n^3*(d + e 
*x^(1/3))^3)/(27*e^12) - (495*b^3*d^8*n^3*(d + e*x^(1/3))^4)/(128*e^12) + 
(396*b^3*d^7*n^3*(d + e*x^(1/3))^5)/(125*e^12) - (77*b^3*d^6*n^3*(d + e*x^ 
(1/3))^6)/(36*e^12) + (396*b^3*d^5*n^3*(d + e*x^(1/3))^7)/(343*e^12) - (49 
5*b^3*d^4*n^3*(d + e*x^(1/3))^8)/(1024*e^12) + (110*b^3*d^3*n^3*(d + e*x^( 
1/3))^9)/(729*e^12) - (33*b^3*d^2*n^3*(d + e*x^(1/3))^10)/(1000*e^12) + (6 
*b^3*d*n^3*(d + e*x^(1/3))^11)/(1331*e^12) - (b^3*n^3*(d + e*x^(1/3))^12)/ 
(3456*e^12) - (6*a*b^2*d^11*n^2*x^(1/3))/e^11 + (6*b^3*d^11*n^3*x^(1/3))/e 
^11 - (6*b^3*d^11*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/e^12 + (33 
*b^2*d^10*n^2*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n]))/(4*e^12) 
 - (110*b^2*d^9*n^2*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^n]))/(9 
*e^12) + (495*b^2*d^8*n^2*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n 
]))/(32*e^12) - (396*b^2*d^7*n^2*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^( 
1/3))^n]))/(25*e^12) + (77*b^2*d^6*n^2*(d + e*x^(1/3))^6*(a + b*Log[c*(d + 
 e*x^(1/3))^n]))/(6*e^12) - (396*b^2*d^5*n^2*(d + e*x^(1/3))^7*(a + b*Log[ 
c*(d + e*x^(1/3))^n]))/(49*e^12) + (495*b^2*d^4*n^2*(d + e*x^(1/3))^8*(a + 
 b*Log[c*(d + e*x^(1/3))^n]))/(128*e^12) - (110*b^2*d^3*n^2*(d + e*x^(1/3) 
)^9*(a + b*Log[c*(d + e*x^(1/3))^n]))/(81*e^12) + (33*b^2*d^2*n^2*(d + e*x 
^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n]))/(100*e^12) - (6*b^2*d*n^2*(d 
+ e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n]))/(121*e^12) + (b^2*n^2...
 

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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 2183, normalized size of antiderivative = 1.19 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/14200002432000*(3550000608000*b^3*e^12*x^4*log(c)^3 - 12326391000*(b^3*e 
^12*n^3 - 12*a*b^2*e^12*n^2 + 72*a^2*b*e^12*n - 288*a^3*e^12)*x^4 + 603680 
*(364699*b^3*d^3*e^9*n^3 - 1510740*a*b^2*d^3*e^9*n^2 + 1960200*a^2*b*d^3*e 
^9*n)*x^3 + 3550000608000*(b^3*e^12*n^3*x^4 - b^3*d^12*n^3)*log(e*x^(1/3) 
+ d)^3 - 4620*(297202819*b^3*d^6*e^6*n^3 - 629992440*a*b^2*d^6*e^6*n^2 + 3 
84199200*a^2*b*d^6*e^6*n)*x^2 + 384199200*(3080*b^3*d^3*e^9*n^3*x^3 - 4620 
*b^3*d^6*e^6*n^3*x^2 + 9240*b^3*d^9*e^3*n^3*x + 86021*b^3*d^12*n^3 - 27720 
*a*b^2*d^12*n^2 - 2310*(b^3*e^12*n^3 - 12*a*b^2*e^12*n^2)*x^4 + 27720*(b^3 
*e^12*n^2*x^4 - b^3*d^12*n^2)*log(c) + 63*(40*b^3*d*e^11*n^3*x^3 - 55*b^3* 
d^4*e^8*n^3*x^2 + 88*b^3*d^7*e^5*n^3*x - 220*b^3*d^10*e^2*n^3)*x^(2/3) - 1 
98*(14*b^3*d^2*e^10*n^3*x^3 - 20*b^3*d^5*e^7*n^3*x^2 + 35*b^3*d^8*e^4*n^3* 
x - 140*b^3*d^11*e*n^3)*x^(1/3))*log(e*x^(1/3) + d)^2 + 295833384000*(4*b^ 
3*d^3*e^9*n*x^3 - 6*b^3*d^6*e^6*n*x^2 + 12*b^3*d^9*e^3*n*x - 3*(b^3*e^12*n 
 - 12*a*b^2*e^12)*x^4)*log(c)^2 + 9240*(1108515013*b^3*d^9*e^3*n^3 - 12319 
04520*a*b^2*d^9*e^3*n^2 + 384199200*a^2*b*d^9*e^3*n)*x - 27720*(4301068993 
*b^3*d^12*n^3 - 2384502120*a*b^2*d^12*n^2 + 384199200*a^2*b*d^12*n - 53361 
00*(b^3*e^12*n^3 - 12*a*b^2*e^12*n^2 + 72*a^2*b*e^12*n)*x^4 + 43120*(763*b 
^3*d^3*e^9*n^3 - 1980*a*b^2*d^3*e^9*n^2)*x^3 - 4620*(22727*b^3*d^6*e^6*n^3 
 - 27720*a*b^2*d^6*e^6*n^2)*x^2 - 384199200*(b^3*e^12*n*x^4 - b^3*d^12*n)* 
log(c)^2 + 9240*(44441*b^3*d^9*e^3*n^3 - 27720*a*b^2*d^9*e^3*n^2)*x - 2...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1064, normalized size of antiderivative = 0.58 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*b^3*x^4*log((e*x^(1/3) + d)^n*c)^3 + 3/4*a*b^2*x^4*log((e*x^(1/3) + d) 
^n*c)^2 + 3/4*a^2*b*x^4*log((e*x^(1/3) + d)^n*c) + 1/4*a^3*x^4 - 1/36960*a 
^2*b*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + (2310*e^11*x^4 - 2520*d*e^1 
0*x^(11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/ 
3) - 3960*d^5*e^6*x^(7/3) + 4620*d^6*e^5*x^2 - 5544*d^7*e^4*x^(5/3) + 6930 
*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10*e*x^(2/3) - 27720*d^11*x^(1 
/3))/e^12) - 1/512265600*(27720*e*n*(27720*d^12*log(e*x^(1/3) + d)/e^13 + 
(2310*e^11*x^4 - 2520*d*e^10*x^(11/3) + 2772*d^2*e^9*x^(10/3) - 3080*d^3*e 
^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6*x^(7/3) + 4620*d^6*e^5*x^2 - 
5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240*d^9*e^2*x + 13860*d^10* 
e*x^(2/3) - 27720*d^11*x^(1/3))/e^12)*log((e*x^(1/3) + d)^n*c) - (5336100* 
e^12*x^4 - 12171600*d*e^11*x^(11/3) + 21072744*d^2*e^10*x^(10/3) - 3290056 
0*d^3*e^9*x^3 + 49019355*d^4*e^8*x^(8/3) - 71703720*d^5*e^7*x^(7/3) + 1049 
98740*d^6*e^6*x^2 + 384199200*d^12*log(e*x^(1/3) + d)^2 - 156734424*d^7*e^ 
5*x^(5/3) + 243942930*d^8*e^4*x^(4/3) - 410634840*d^9*e^3*x + 2384502120*d 
^12*log(e*x^(1/3) + d) + 808051860*d^10*e^2*x^(2/3) - 2384502120*d^11*e*x^ 
(1/3))*n^2/e^12)*a*b^2 - 1/14200002432000*(384199200*e*n*(27720*d^12*log(e 
*x^(1/3) + d)/e^13 + (2310*e^11*x^4 - 2520*d*e^10*x^(11/3) + 2772*d^2*e^9* 
x^(10/3) - 3080*d^3*e^8*x^3 + 3465*d^4*e^7*x^(8/3) - 3960*d^5*e^6*x^(7/3) 
+ 4620*d^6*e^5*x^2 - 5544*d^7*e^4*x^(5/3) + 6930*d^8*e^3*x^(4/3) - 9240...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4320 vs. \(2 (1591) = 3182\).

Time = 0.19 (sec) , antiderivative size = 4320, normalized size of antiderivative = 2.35 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/14200002432000*(3550000608000*b^3*e*x^4*log(c)^3 + 10650001824000*a*b^2* 
e*x^4*log(c)^2 + 10650001824000*a^2*b*e*x^4*log(c) + 3550000608000*a^3*e*x 
^4 + (3550000608000*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)^3/e^11 - 4260000 
7296000*(e*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)^3/e^11 + 234300040128000*( 
e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)^3/e^11 - 781000133760000*(e*x^(1/ 
3) + d)^9*d^3*log(e*x^(1/3) + d)^3/e^11 + 1757250300960000*(e*x^(1/3) + d) 
^8*d^4*log(e*x^(1/3) + d)^3/e^11 - 2811600481536000*(e*x^(1/3) + d)^7*d^5* 
log(e*x^(1/3) + d)^3/e^11 + 3280200561792000*(e*x^(1/3) + d)^6*d^6*log(e*x 
^(1/3) + d)^3/e^11 - 2811600481536000*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) 
+ d)^3/e^11 + 1757250300960000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)^3/ 
e^11 - 781000133760000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)^3/e^11 + 2 
34300040128000*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)^3/e^11 - 42600007 
296000*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)^3/e^11 - 887500152000*(e*x^ 
(1/3) + d)^12*log(e*x^(1/3) + d)^2/e^11 + 11618183808000*(e*x^(1/3) + d)^1 
1*d*log(e*x^(1/3) + d)^2/e^11 - 70290012038400*(e*x^(1/3) + d)^10*d^2*log( 
e*x^(1/3) + d)^2/e^11 + 260333377920000*(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3 
) + d)^2/e^11 - 658968862860000*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)^2 
/e^11 + 1204971634944000*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)^2/e^11 - 
 1640100280896000*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)^2/e^11 + 168696 
0288921600*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)^2/e^11 - 1317937725...
 

Mupad [B] (verification not implemented)

Time = 21.28 (sec) , antiderivative size = 1802, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \] Input:

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

Output:

(a^3*x^4)/4 + (b^3*x^4*log(c*(d + e*x^(1/3))^n)^3)/4 - (b^3*n^3*x^4)/1152 
+ (3*a*b^2*x^4*log(c*(d + e*x^(1/3))^n)^2)/4 - (b^3*n*x^4*log(c*(d + e*x^( 
1/3))^n)^2)/16 + (b^3*n^2*x^4*log(c*(d + e*x^(1/3))^n))/96 + (a*b^2*n^2*x^ 
4)/96 - (b^3*d^12*log(c*(d + e*x^(1/3))^n)^3)/(4*e^12) + (3*a^2*b*x^4*log( 
c*(d + e*x^(1/3))^n))/4 - (a^2*b*n*x^4)/16 - (a*b^2*n*x^4*log(c*(d + e*x^( 
1/3))^n))/8 - (4301068993*b^3*d^12*n^3*log(d + e*x^(1/3)))/(512265600*e^12 
) + (364699*b^3*d^3*n^3*x^3)/(23522400*e^3) - (297202819*b^3*d^6*n^3*x^2)/ 
(3073593600*e^6) - (21871*b^3*d^2*n^3*x^(10/3))/(2904000*e^2) - (2459191*b 
^3*d^4*n^3*x^(8/3))/(83635200*e^4) + (192204079*b^3*d^5*n^3*x^(7/3))/(3585 
859200*e^5) + (453937243*b^3*d^7*n^3*x^(5/3))/(2561328000*e^7) - (69788017 
3*b^3*d^8*n^3*x^(4/3))/(2049062400*e^8) - (1916566873*b^3*d^10*n^3*x^(2/3) 
)/(1024531200*e^10) + (4301068993*b^3*d^11*n^3*x^(1/3))/(512265600*e^11) - 
 (3*a*b^2*d^12*log(c*(d + e*x^(1/3))^n)^2)/(4*e^12) + (86021*b^3*d^12*n*lo 
g(c*(d + e*x^(1/3))^n)^2)/(36960*e^12) + (397*b^3*d*n^3*x^(11/3))/(127776* 
e) + (1108515013*b^3*d^9*n^3*x)/(1536796800*e^9) - (3*a^2*b*d^12*n*log(d + 
 e*x^(1/3)))/(4*e^12) + (3*b^3*d*n*x^(11/3)*log(c*(d + e*x^(1/3))^n)^2)/(4 
4*e) - (23*b^3*d*n^2*x^(11/3)*log(c*(d + e*x^(1/3))^n))/(968*e) + (b^3*d^9 
*n*x*log(c*(d + e*x^(1/3))^n)^2)/(4*e^9) - (44441*b^3*d^9*n^2*x*log(c*(d + 
 e*x^(1/3))^n))/(55440*e^9) + (a^2*b*d^3*n*x^3)/(12*e^3) - (a^2*b*d^6*n*x^ 
2)/(8*e^6) - (23*a*b^2*d*n^2*x^(11/3))/(968*e) - (3*a^2*b*d^2*n*x^(10/3...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1816, normalized size of antiderivative = 0.99 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:

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

Output:

( - 5325000912000*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**10*e**2*n 
 + 2130000364800*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**7*e**5*n*x 
 - 1331250228000*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**4*e**8*n*x 
**2 + 968181984000*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d*e**11*n*x 
**3 - 10650001824000*x**(2/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**10*e**2 
*n + 4260000729600*x**(2/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**7*e**5*n* 
x - 2662500456000*x**(2/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**4*e**8*n*x 
**2 + 1936363968000*x**(2/3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d*e**11*n*x 
**3 + 22399197559200*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d**10*e**2*n 
**2 - 4344678233280*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d**7*e**5*n** 
2*x + 1358816520600*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d**4*e**8*n** 
2*x**2 - 337396752000*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d*e**11*n** 
2*x**3 - 5325000912000*x**(2/3)*a**2*b*d**10*e**2*n + 2130000364800*x**(2/ 
3)*a**2*b*d**7*e**5*n*x - 1331250228000*x**(2/3)*a**2*b*d**4*e**8*n*x**2 + 
 968181984000*x**(2/3)*a**2*b*d*e**11*n*x**3 + 22399197559200*x**(2/3)*a*b 
**2*d**10*e**2*n**2 - 4344678233280*x**(2/3)*a*b**2*d**7*e**5*n**2*x + 135 
8816520600*x**(2/3)*a*b**2*d**4*e**8*n**2*x**2 - 337396752000*x**(2/3)*a*b 
**2*d*e**11*n**2*x**3 - 26563616859780*x**(2/3)*b**3*d**10*e**2*n**3 + 251 
6628075192*x**(2/3)*b**3*d**7*e**5*n**3*x - 417533743935*x**(2/3)*b**3*d** 
4*e**8*n**3*x**2 + 44119404000*x**(2/3)*b**3*d*e**11*n**3*x**3 + 106500...