\(\int x^3 (a+b \log (c (d+e \sqrt [3]{x})^n))^3 \, dx\) [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=-\frac {99 b^3 d^{10} n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^{12}}+\frac {110 b^3 d^9 n^3 \left (d+e \sqrt [3]{x}\right )^3}{9 e^{12}}-\frac {1485 b^3 d^8 n^3 \left (d+e \sqrt [3]{x}\right )^4}{128 e^{12}}+\frac {1188 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^{12}}-\frac {77 b^3 d^6 n^3 \left (d+e \sqrt [3]{x}\right )^6}{12 e^{12}}+\frac {1188 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^{12}}-\frac {1485 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^8}{1024 e^{12}}+\frac {110 b^3 d^3 n^3 \left (d+e \sqrt [3]{x}\right )^9}{243 e^{12}}-\frac {99 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^{10}}{1000 e^{12}}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^{11}}{1331 e^{12}}-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^{12}}{1152 e^{12}}-\frac {18 a b^2 d^{11} n^2 \sqrt [3]{x}}{e^{11}}+\frac {18 b^3 d^{11} n^3 \sqrt [3]{x}}{e^{11}}-\frac {18 b^3 d^{11} n^2 \left (d+e \sqrt [3]{x}\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{e^{12}}+\frac {99 b^2 d^{10} 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^{12}}-\frac {110 b^2 d^9 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^{12}}+\frac {1485 b^2 d^8 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 )}{32 e^{12}}-\frac {1188 b^2 d^7 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^{12}}+\frac {77 b^2 d^6 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 )}{2 e^{12}}-\frac {1188 b^2 d^5 n^2 \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{49 e^{12}}+\frac {1485 b^2 d^4 n^2 \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{128 e^{12}}-\frac {110 b^2 d^3 n^2 \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{27 e^{12}}+\frac {99 b^2 d^2 n^2 \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{100 e^{12}}-\frac {18 b^2 d n^2 \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{121 e^{12}}+\frac {b^2 n^2 \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )}{96 e^{12}}+\frac {9 b d^{11} 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^{12}}-\frac {99 b d^{10} 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^{12}}+\frac {55 b d^9 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^{12}}-\frac {1485 b d^8 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}{16 e^{12}}+\frac {594 b d^7 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^{12}}-\frac {231 b d^6 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}{2 e^{12}}+\frac {594 b d^5 n \left (d+e \sqrt [3]{x}\right )^7 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{7 e^{12}}-\frac {1485 b d^4 n \left (d+e \sqrt [3]{x}\right )^8 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{32 e^{12}}+\frac {55 b d^3 n \left (d+e \sqrt [3]{x}\right )^9 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{3 e^{12}}-\frac {99 b d^2 n \left (d+e \sqrt [3]{x}\right )^{10} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{20 e^{12}}+\frac {9 b d n \left (d+e \sqrt [3]{x}\right )^{11} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{11 e^{12}}-\frac {b n \left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{16 e^{12}}-\frac {3 d^{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}{e^{12}}+\frac {33 d^{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}{2 e^{12}}-\frac {55 d^9 \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^{12}}+\frac {495 d^8 \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}{4 e^{12}}-\frac {198 d^7 \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^{12}}+\frac {231 d^6 \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}{e^{12}}-\frac {198 d^5 \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}{e^{12}}+\frac {495 d^4 \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}{4 e^{12}}-\frac {55 d^3 \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}{e^{12}}+\frac {33 d^2 \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}{2 e^{12}}-\frac {3 d \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^{12}}+\frac {\left (d+e \sqrt [3]{x}\right )^{12} \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{4 e^{12}} \]

[Out]

-1/1152*b^3*n^3*(d+e*x^(1/3))^12/e^12-3*d^11*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12+33/2*d^10*(d+e*x
^(1/3))^2*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-55*d^9*(d+e*x^(1/3))^3*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12+495/4*d
^8*(d+e*x^(1/3))^4*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-198*d^7*(d+e*x^(1/3))^5*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^
12+231*d^6*(d+e*x^(1/3))^6*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-198*d^5*(d+e*x^(1/3))^7*(a+b*ln(c*(d+e*x^(1/3))^
n))^3/e^12+495/4*d^4*(d+e*x^(1/3))^8*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-55*d^3*(d+e*x^(1/3))^9*(a+b*ln(c*(d+e*
x^(1/3))^n))^3/e^12+33/2*d^2*(d+e*x^(1/3))^10*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-3*d*(d+e*x^(1/3))^11*(a+b*ln(
c*(d+e*x^(1/3))^n))^3/e^12+1/4*(d+e*x^(1/3))^12*(a+b*ln(c*(d+e*x^(1/3))^n))^3/e^12-18*a*b^2*d^11*n^2*x^(1/3)/e
^11-18*b^3*d^11*n^2*(d+e*x^(1/3))*ln(c*(d+e*x^(1/3))^n)/e^12+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^12-1188/49*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^1
2-99/20*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^12-99/8*b^3*d^10*n^3*(d+e*x^(1/3))^2/e^12+110/9*b^3*d^9*n^3*(d+e*x^(1/3))^3/e^12-1485/128*b^
3*d^8*n^3*(d+e*x^(1/3))^4/e^12+1188/125*b^3*d^7*n^3*(d+e*x^(1/3))^5/e^12-77/12*b^3*d^6*n^3*(d+e*x^(1/3))^6/e^1
2+1188/343*b^3*d^5*n^3*(d+e*x^(1/3))^7/e^12-1485/1024*b^3*d^4*n^3*(d+e*x^(1/3))^8/e^12+110/243*b^3*d^3*n^3*(d+
e*x^(1/3))^9/e^12-99/1000*b^3*d^2*n^3*(d+e*x^(1/3))^10/e^12+18/1331*b^3*d*n^3*(d+e*x^(1/3))^11/e^12+18*b^3*d^1
1*n^3*x^(1/3)/e^11+1/96*b^2*n^2*(d+e*x^(1/3))^12*(a+b*ln(c*(d+e*x^(1/3))^n))/e^12-1/16*b*n*(d+e*x^(1/3))^12*(a
+b*ln(c*(d+e*x^(1/3))^n))^2/e^12

Rubi [A] (verified)

Time = 1.55 (sec) , antiderivative size = 1835, normalized size of antiderivative = 1.00, number of steps used = 52, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2504, 2448, 2436, 2333, 2332, 2437, 2342, 2341} \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=-\frac {b^3 n^3 \left (d+e \sqrt [3]{x}\right )^{12}}{1152 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}}{4 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}}{16 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}}{96 e^{12}}+\frac {18 b^3 d n^3 \left (d+e \sqrt [3]{x}\right )^{11}}{1331 e^{12}}-\frac {3 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 {9 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 {18 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 {99 b^3 d^2 n^3 \left (d+e \sqrt [3]{x}\right )^{10}}{1000 e^{12}}+\frac {33 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 {99 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 {99 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}{243 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}{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}{3 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}{27 e^{12}}-\frac {1485 b^3 d^4 n^3 \left (d+e \sqrt [3]{x}\right )^8}{1024 e^{12}}+\frac {495 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 {1485 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 {1485 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 {1188 b^3 d^5 n^3 \left (d+e \sqrt [3]{x}\right )^7}{343 e^{12}}-\frac {198 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 {594 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 {1188 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}{12 e^{12}}+\frac {231 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 {231 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}{2 e^{12}}+\frac {1188 b^3 d^7 n^3 \left (d+e \sqrt [3]{x}\right )^5}{125 e^{12}}-\frac {198 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 {594 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 {1188 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 {1485 b^3 d^8 n^3 \left (d+e \sqrt [3]{x}\right )^4}{128 e^{12}}+\frac {495 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 {1485 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 {1485 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}{9 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}{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}{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}{3 e^{12}}-\frac {99 b^3 d^{10} n^3 \left (d+e \sqrt [3]{x}\right )^2}{8 e^{12}}+\frac {33 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 {99 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 {99 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 {3 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 {9 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 {18 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 {18 b^3 d^{11} n^3 \sqrt [3]{x}}{e^{11}}-\frac {18 a b^2 d^{11} n^2 \sqrt [3]{x}}{e^{11}} \]

[In]

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

[Out]

(-99*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)/(9*e^12) - (1485*b^3*d^8*n
^3*(d + e*x^(1/3))^4)/(128*e^12) + (1188*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)/(12*e^12) + (1188*b^3*d^5*n^3*(d + e*x^(1/3))^7)/(343*e^12) - (1485*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)/(243*e^12) - (99*b^3*d^2*n^3*(d + e*x^(1/3))^10)/(1000*e^12)
+ (18*b^3*d*n^3*(d + e*x^(1/3))^11)/(1331*e^12) - (b^3*n^3*(d + e*x^(1/3))^12)/(1152*e^12) - (18*a*b^2*d^11*n^
2*x^(1/3))/e^11 + (18*b^3*d^11*n^3*x^(1/3))/e^11 - (18*b^3*d^11*n^2*(d + e*x^(1/3))*Log[c*(d + e*x^(1/3))^n])/
e^12 + (99*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]))/(3*e^12) + (1485*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) - (1188*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]))/(2*e^12) - (1188*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) + (1485*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]))/(27*e^12) + (99*
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) - (18*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*(d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n]))/
(96*e^12) + (9*b*d^11*n*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/e^12 - (99*b*d^10*n*(d + e*x^(1/3)
)^2*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(4*e^12) + (55*b*d^9*n*(d + e*x^(1/3))^3*(a + b*Log[c*(d + e*x^(1/3))^
n])^2)/e^12 - (1485*b*d^8*n*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) + (594*b*d^7*n*(d
+ e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(5*e^12) - (231*b*d^6*n*(d + e*x^(1/3))^6*(a + b*Log[c*(d +
e*x^(1/3))^n])^2)/(2*e^12) + (594*b*d^5*n*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(7*e^12) - (1
485*b*d^4*n*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(32*e^12) + (55*b*d^3*n*(d + e*x^(1/3))^9*(a
+ b*Log[c*(d + e*x^(1/3))^n])^2)/(3*e^12) - (99*b*d^2*n*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^2
)/(20*e^12) + (9*b*d*n*(d + e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(11*e^12) - (b*n*(d + e*x^(1/3))
^12*(a + b*Log[c*(d + e*x^(1/3))^n])^2)/(16*e^12) - (3*d^11*(d + e*x^(1/3))*(a + b*Log[c*(d + e*x^(1/3))^n])^3
)/e^12 + (33*d^10*(d + e*x^(1/3))^2*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (55*d^9*(d + e*x^(1/3))^3*(
a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495*d^8*(d + e*x^(1/3))^4*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^
12) - (198*d^7*(d + e*x^(1/3))^5*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (231*d^6*(d + e*x^(1/3))^6*(a + b*
Log[c*(d + e*x^(1/3))^n])^3)/e^12 - (198*d^5*(d + e*x^(1/3))^7*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + (495
*d^4*(d + e*x^(1/3))^8*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(4*e^12) - (55*d^3*(d + e*x^(1/3))^9*(a + b*Log[c*(
d + e*x^(1/3))^n])^3)/e^12 + (33*d^2*(d + e*x^(1/3))^10*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/(2*e^12) - (3*d*(d
+ e*x^(1/3))^11*(a + b*Log[c*(d + e*x^(1/3))^n])^3)/e^12 + ((d + e*x^(1/3))^12*(a + b*Log[c*(d + e*x^(1/3))^n
])^3)/(4*e^12)

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2437

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[(f*(x/d))^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2448

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(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 2504

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

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int x^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right ) \\ & = 3 \text {Subst}\left (\int \left (-\frac {d^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {11 d^{10} (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac {55 d^9 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {165 d^8 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac {330 d^7 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {462 d^6 (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac {462 d^5 (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {330 d^4 (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac {165 d^3 (d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {55 d^2 (d+e x)^9 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}-\frac {11 d (d+e x)^{10} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}+\frac {(d+e x)^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^{11}}\right ) \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3 \text {Subst}\left (\int (d+e x)^{11} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {(33 d) \text {Subst}\left (\int (d+e x)^{10} \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int (d+e x)^9 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int (d+e x)^8 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int (d+e x)^7 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int (d+e x)^6 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\sqrt [3]{x}\right )}{e^{11}} \\ & = \frac {3 \text {Subst}\left (\int x^{11} \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {(33 d) \text {Subst}\left (\int x^{10} \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (165 d^2\right ) \text {Subst}\left (\int x^9 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (495 d^3\right ) \text {Subst}\left (\int x^8 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (990 d^4\right ) \text {Subst}\left (\int x^7 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (1386 d^5\right ) \text {Subst}\left (\int x^6 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (1386 d^6\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (990 d^7\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (495 d^8\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (165 d^9\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}+\frac {\left (33 d^{10}\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}}-\frac {\left (3 d^{11}\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+e \sqrt [3]{x}\right )}{e^{12}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (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=\frac {e \sqrt [3]{x} \left (3550000608000 a^3 e^{11} x^{11/3}+b^3 n^3 \left (119225632485960 d^{11}-26563616859780 d^{10} e \sqrt [3]{x}[phone] d^9 e^2 x^{2/3}-4836309598890 d^8 e^3 x+2516628075192 d^7 e^4 x^{4/3}-1373077023780 d^6 e^5 x^{5/3}[phone] d^5 e^6 x^2-417533743935 d^4 e^7 x^{7/3}[phone] 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}\right )-27720 a b^2 n^2 \left (2384502120 d^{11}-808051860 d^{10} e \sqrt [3]{x}[phone] 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}[phone] d^5 e^6 x^2-49019355 d^4 e^7 x^{7/3}[phone] 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}\right )[phone] a^2 b n \left (27720 d^{11}-13860 d^{10} e \sqrt [3]{x}+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}\right )\right )-27720 b d^{12} n \left (384199200 a^2-2384502120 a b n+4301068993 b^2 n^2\right ) \log \left (d+e \sqrt [3]{x}\right )+27720 b e \sqrt [3]{x} \left (384199200 a^2 e^{11} x^{11/3}+27720 a b n \left (27720 d^{11}-13860 d^{10} e \sqrt [3]{x}+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}\right )+b^2 n^2 \left (-2384502120 d^{11}[phone] d^{10} e \sqrt [3]{x}-410634840 d^9 e^2 x^{2/3}[phone] d^8 e^3 x-156734424 d^7 e^4 x^{4/3}[phone] 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}[phone] d^2 e^9 x^3-12171600 d e^{10} x^{10/3}+5336100 e^{11} x^{11/3}\right )\right ) \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )[phone] b^2 \left (b n \left (86021 d^{12}+27720 d^{11} e \sqrt [3]{x}-13860 d^{10} e^2 x^{2/3}+9240 d^9 e^3 x-6930 d^8 e^4 x^{4/3}+5544 d^7 e^5 x^{5/3}-4620 d^6 e^6 x^2+3960 d^5 e^7 x^{7/3}-3465 d^4 e^8 x^{8/3}+3080 d^3 e^9 x^3-2772 d^2 e^{10} x^{10/3}+2520 d e^{11} x^{11/3}-2310 e^{12} x^4\right )-27720 a \left (d^{12}-e^{12} x^4\right )\right ) \log ^2\left (c \left (d+e \sqrt [3]{x}\right )^n\right )-3550000608000 b^3 \left (d^{12}-e^{12} x^4\right ) \log ^3\left (c \left (d+e \sqrt [3]{x}\right )^n\right )}{14200002432000 e^{12}} \]

[In]

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

[Out]

(e*x^(1/3)*(3550000608000*a^3*e^11*x^(11/3) + b^3*n^3*(119225632485960*d^11 - 26563616859780*d^10*e*x^(1/3) +
10242678720120*d^9*e^2*x^(2/3) - 4836309598890*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) + 220161492320*d^3*e^8*x^(8/3) - 10694499
0768*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
- 808051860*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) - 10
4998740*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) - 2107274
4*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^(1
1/3))) - 27720*b*d^12*n*(384199200*a^2 - 2384502120*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) + 308
0*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*x^(2/3) + 243942930*d^8*e^3*x - 156734424*d^7*e^4*x^(4/3) + 104
998740*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)))*Log[c*(d + e*x^(1/3))^n] + 384199200*b^2*(b*
n*(86021*d^12 + 27720*d^11*e*x^(1/3) - 13860*d^10*e^2*x^(2/3) + 9240*d^9*e^3*x - 6930*d^8*e^4*x^(4/3) + 5544*d
^7*e^5*x^(5/3) - 4620*d^6*e^6*x^2 + 3960*d^5*e^7*x^(7/3) - 3465*d^4*e^8*x^(8/3) + 3080*d^3*e^9*x^3 - 2772*d^2*
e^10*x^(10/3) + 2520*d*e^11*x^(11/3) - 2310*e^12*x^4) - 27720*a*(d^12 - e^12*x^4))*Log[c*(d + e*x^(1/3))^n]^2
- 3550000608000*b^3*(d^12 - e^12*x^4)*Log[c*(d + e*x^(1/3))^n]^3)/(14200002432000*e^12)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.52 (sec) , antiderivative size = 2183, normalized size of antiderivative = 1.19 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 + 384199200*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 - 5
5*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) - 198*(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 + 2958333840
00*(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 - 1231904520*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 - 5336100*(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 - 27720*(3080*b^3*d^3*e^9*n^2*x^3 - 4620*b^3*d^6*e^6*n^2*x^2 + 924
0*b^3*d^9*e^3*n^2*x + 86021*b^3*d^12*n^2 - 27720*a*b^2*d^12*n - 2310*(b^3*e^12*n^2 - 12*a*b^2*e^12*n)*x^4)*log
(c) - 63*(12826220*b^3*d^10*e^2*n^3 - 6098400*a*b^2*d^10*e^2*n^2 - 8400*(23*b^3*d*e^11*n^3 - 132*a*b^2*d*e^11*
n^2)*x^3 + 385*(2021*b^3*d^4*e^8*n^3 - 3960*a*b^2*d^4*e^8*n^2)*x^2 - 88*(28271*b^3*d^7*e^5*n^3 - 27720*a*b^2*d
^7*e^5*n^2)*x + 27720*(40*b^3*d*e^11*n^2*x^3 - 55*b^3*d^4*e^8*n^2*x^2 + 88*b^3*d^7*e^5*n^2*x - 220*b^3*d^10*e^
2*n^2)*log(c))*x^(2/3) + 198*(12042940*b^3*d^11*e*n^3 - 3880800*a*b^2*d^11*e*n^2 - 588*(181*b^3*d^2*e^10*n^3 -
660*a*b^2*d^2*e^10*n^2)*x^3 + 20*(18107*b^3*d^5*e^7*n^3 - 27720*a*b^2*d^5*e^7*n^2)*x^2 - 35*(35201*b^3*d^8*e^
4*n^3 - 27720*a*b^2*d^8*e^4*n^2)*x + 27720*(14*b^3*d^2*e^10*n^2*x^3 - 20*b^3*d^5*e^7*n^2*x^2 + 35*b^3*d^8*e^4*
n^2*x - 140*b^3*d^11*e*n^2)*log(c))*x^(1/3))*log(e*x^(1/3) + d) + 42688800*(3465*(b^3*e^12*n^2 - 12*a*b^2*e^12
*n + 72*a^2*b*e^12)*x^4 - 28*(763*b^3*d^3*e^9*n^2 - 1980*a*b^2*d^3*e^9*n)*x^3 + 3*(22727*b^3*d^6*e^6*n^2 - 277
20*a*b^2*d^6*e^6*n)*x^2 - 6*(44441*b^3*d^9*e^3*n^2 - 27720*a*b^2*d^9*e^3*n)*x)*log(c) - 63*(421644712060*b^3*d
^10*e^2*n^3 - 355542818400*a*b^2*d^10*e^2*n^2 + 84523824000*a^2*b*d^10*e^2*n - 1764000*(397*b^3*d*e^11*n^3 - 3
036*a*b^2*d*e^11*n^2 + 8712*a^2*b*d*e^11*n)*x^3 + 2695*(2459191*b^3*d^4*e^8*n^3 - 8003160*a*b^2*d^4*e^8*n^2 +
7840800*a^2*b*d^4*e^8*n)*x^2 - 384199200*(40*b^3*d*e^11*n*x^3 - 55*b^3*d^4*e^8*n*x^2 + 88*b^3*d^7*e^5*n*x - 22
0*b^3*d^10*e^2*n)*log(c)^2 - 88*(453937243*b^3*d^7*e^5*n^3 - 783672120*a*b^2*d^7*e^5*n^2 + 384199200*a^2*b*d^7
*e^5*n)*x - 27720*(12826220*b^3*d^10*e^2*n^2 - 6098400*a*b^2*d^10*e^2*n - 8400*(23*b^3*d*e^11*n^2 - 132*a*b^2*
d*e^11*n)*x^3 + 385*(2021*b^3*d^4*e^8*n^2 - 3960*a*b^2*d^4*e^8*n)*x^2 - 88*(28271*b^3*d^7*e^5*n^2 - 27720*a*b^
2*d^7*e^5*n)*x)*log(c))*x^(2/3) + 198*(602149659020*b^3*d^11*e*n^3 - 333830296800*a*b^2*d^11*e*n^2 + 537878880
00*a^2*b*d^11*e*n - 24696*(21871*b^3*d^2*e^10*n^3 - 119460*a*b^2*d^2*e^10*n^2 + 217800*a^2*b*d^2*e^10*n)*x^3 +
20*(192204079*b^3*d^5*e^7*n^3 - 501926040*a*b^2*d^5*e^7*n^2 + 384199200*a^2*b*d^5*e^7*n)*x^2 - 384199200*(14*
b^3*d^2*e^10*n*x^3 - 20*b^3*d^5*e^7*n*x^2 + 35*b^3*d^8*e^4*n*x - 140*b^3*d^11*e*n)*log(c)^2 - 35*(697880173*b^
3*d^8*e^4*n^3 - 975771720*a*b^2*d^8*e^4*n^2 + 384199200*a^2*b*d^8*e^4*n)*x - 27720*(12042940*b^3*d^11*e*n^2 -
3880800*a*b^2*d^11*e*n - 588*(181*b^3*d^2*e^10*n^2 - 660*a*b^2*d^2*e^10*n)*x^3 + 20*(18107*b^3*d^5*e^7*n^2 - 2
7720*a*b^2*d^5*e^7*n)*x^2 - 35*(35201*b^3*d^8*e^4*n^2 - 27720*a*b^2*d^8*e^4*n)*x)*log(c))*x^(1/3))/e^12

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 1064, normalized size of antiderivative = 0.58 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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
^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) - 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) -
32900560*d^3*e^9*x^3 + 49019355*d^4*e^8*x^(8/3) - 71703720*d^5*e^7*x^(7/3) + 104998740*d^6*e^6*x^2 + 38419920
0*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 + 23
84502120*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) + 2
772*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 - 554
4*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)*l
og((e*x^(1/3) + d)^n*c)^2 + e*n*((12326391000*e^12*x^4 - 44119404000*d*e^11*x^(11/3) + 106944990768*d^2*e^10*x
^(10/3) - 220161492320*d^3*e^9*x^3 + 3550000608000*d^12*log(e*x^(1/3) + d)^3 + 417533743935*d^4*e^8*x^(8/3) -
761128152840*d^5*e^7*x^(7/3) + 1373077023780*d^6*e^6*x^2 + 33049199383200*d^12*log(e*x^(1/3) + d)^2 - 25166280
75192*d^7*e^5*x^(5/3) + 4836309598890*d^8*e^4*x^(4/3) - 10242678720120*d^9*e^3*x + 119225632485960*d^12*log(e*
x^(1/3) + d) + 26563616859780*d^10*e^2*x^(2/3) - 119225632485960*d^11*e*x^(1/3))*n^2/e^13 - 27720*(5336100*e^1
2*x^4 - 12171600*d*e^11*x^(11/3) + 21072744*d^2*e^10*x^(10/3) - 32900560*d^3*e^9*x^3 + 49019355*d^4*e^8*x^(8/3
) - 71703720*d^5*e^7*x^(7/3) + 104998740*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*log((e*x^(1/3) + d)^n*c)/e^13))*b^3

Giac [B] (verification not implemented)

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

Time = 0.56 (sec) , antiderivative size = 4320, normalized size of antiderivative = 2.35 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 - 42600
007296000*(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 + 3280
200561792000*(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 + 234300040128000*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)^3/e^11 -
42600007296000*(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)^11*d*log(e*x^(1/3) + d)^2/e^11 - 70290012038400*(e*x^(1/3) + d)^1
0*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 - 6589688628
60000*(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 + 1686960288921600*(e*x^(1/3) +
d)^5*d^7*log(e*x^(1/3) + d)^2/e^11 - 1317937725720000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)^2/e^11 + 781000
133760000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)^2/e^11 - 351450060192000*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/
3) + d)^2/e^11 + 127800021888000*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)^2/e^11 + 147916692000*(e*x^(1/3) + d)
^12*log(e*x^(1/3) + d)/e^11 - 2112397056000*(e*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)/e^11 + 14058002407680*(e*x
^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 - 57851861760000*(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)/e^11 + 16
4742215715000*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 - 344277609984000*(e*x^(1/3) + d)^7*d^5*log(e*x^(1
/3) + d)/e^11 + 546700093632000*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e^11 - 674784115568640*(e*x^(1/3) + d
)^5*d^7*log(e*x^(1/3) + d)/e^11 + 658968862860000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)/e^11 - 520666755840
000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 + 351450060192000*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)/
e^11 - 255600043776000*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 - 12326391000*(e*x^(1/3) + d)^12/e^11 + 19
2036096000*(e*x^(1/3) + d)^11*d/e^11 - 1405800240768*(e*x^(1/3) + d)^10*d^2/e^11 + 6427984640000*(e*x^(1/3) +
d)^9*d^3/e^11 - 20592776964375*(e*x^(1/3) + d)^8*d^4/e^11 + 49182515712000*(e*x^(1/3) + d)^7*d^5/e^11 - 911166
82272000*(e*x^(1/3) + d)^6*d^6/e^11 + 134956823113728*(e*x^(1/3) + d)^5*d^7/e^11 - 164742215715000*(e*x^(1/3)
+ d)^4*d^8/e^11 + 173555585280000*(e*x^(1/3) + d)^3*d^9/e^11 - 175725030096000*(e*x^(1/3) + d)^2*d^10/e^11 + 2
55600043776000*(e*x^(1/3) + d)*d^11/e^11)*b^3*n^3 + 27720*(384199200*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)^2/e
^11 - 4610390400*(e*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)^2/e^11 + 25357147200*(e*x^(1/3) + d)^10*d^2*log(e*x^(
1/3) + d)^2/e^11 - 84523824000*(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)^2/e^11 + 190178604000*(e*x^(1/3) + d)^
8*d^4*log(e*x^(1/3) + d)^2/e^11 - 304285766400*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)^2/e^11 + 355000060800*
(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)^2/e^11 - 304285766400*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)^2/e^11
+ 190178604000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)^2/e^11 - 84523824000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1
/3) + d)^2/e^11 + 25357147200*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)^2/e^11 - 4610390400*(e*x^(1/3) + d)*d^
11*log(e*x^(1/3) + d)^2/e^11 - 64033200*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^11 + 838252800*(e*x^(1/3) + d)
^11*d*log(e*x^(1/3) + d)/e^11 - 5071429440*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 + 18783072000*(e*x^(
1/3) + d)^9*d^3*log(e*x^(1/3) + d)/e^11 - 47544651000*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 + 86938790
400*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)/e^11 - 118333353600*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e^11
+ 121714306560*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 - 95089302000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3
) + d)/e^11 + 56349216000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 - 25357147200*(e*x^(1/3) + d)^2*d^10*l
og(e*x^(1/3) + d)/e^11 + 9220780800*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 + 5336100*(e*x^(1/3) + d)^12/
e^11 - 76204800*(e*x^(1/3) + d)^11*d/e^11 + 507142944*(e*x^(1/3) + d)^10*d^2/e^11 - 2087008000*(e*x^(1/3) + d)
^9*d^3/e^11 + 5943081375*(e*x^(1/3) + d)^8*d^4/e^11 - 12419827200*(e*x^(1/3) + d)^7*d^5/e^11 + 19722225600*(e*
x^(1/3) + d)^6*d^6/e^11 - 24342861312*(e*x^(1/3) + d)^5*d^7/e^11 + 23772325500*(e*x^(1/3) + d)^4*d^8/e^11 - 18
783072000*(e*x^(1/3) + d)^3*d^9/e^11 + 12678573600*(e*x^(1/3) + d)^2*d^10/e^11 - 9220780800*(e*x^(1/3) + d)*d^
11/e^11)*b^3*n^2*log(c) + 384199200*(27720*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^11 - 332640*(e*x^(1/3) + d)
^11*d*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 - 6098400*(e*x^(1/3) +
d)^9*d^3*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 - 21954240*(e*x^(1/3
) + d)^7*d^5*log(e*x^(1/3) + d)/e^11 + 25613280*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e^11 - 21954240*(e*x^
(1/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)/e^11 - 6098400*(e
*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)/e^11 - 332640*
(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 - 2310*(e*x^(1/3) + d)^12/e^11 + 30240*(e*x^(1/3) + d)^11*d/e^11
- 182952*(e*x^(1/3) + d)^10*d^2/e^11 + 677600*(e*x^(1/3) + d)^9*d^3/e^11 - 1715175*(e*x^(1/3) + d)^8*d^4/e^11
+ 3136320*(e*x^(1/3) + d)^7*d^5/e^11 - 4268880*(e*x^(1/3) + d)^6*d^6/e^11 + 4390848*(e*x^(1/3) + d)^5*d^7/e^11
- 3430350*(e*x^(1/3) + d)^4*d^8/e^11 + 2032800*(e*x^(1/3) + d)^3*d^9/e^11 - 914760*(e*x^(1/3) + d)^2*d^10/e^1
1 + 332640*(e*x^(1/3) + d)*d^11/e^11)*b^3*n*log(c)^2 + 27720*(384199200*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)^
2/e^11 - 4610390400*(e*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)^2/e^11 + 25357147200*(e*x^(1/3) + d)^10*d^2*log(e*
x^(1/3) + d)^2/e^11 - 84523824000*(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)^2/e^11 + 190178604000*(e*x^(1/3) +
d)^8*d^4*log(e*x^(1/3) + d)^2/e^11 - 304285766400*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)^2/e^11 + 3550000608
00*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)^2/e^11 - 304285766400*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)^2/e
^11 + 190178604000*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)^2/e^11 - 84523824000*(e*x^(1/3) + d)^3*d^9*log(e*x
^(1/3) + d)^2/e^11 + 25357147200*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)^2/e^11 - 4610390400*(e*x^(1/3) + d)
*d^11*log(e*x^(1/3) + d)^2/e^11 - 64033200*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^11 + 838252800*(e*x^(1/3) +
d)^11*d*log(e*x^(1/3) + d)/e^11 - 5071429440*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 + 18783072000*(e*
x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)/e^11 - 47544651000*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 + 86938
790400*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)/e^11 - 118333353600*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e
^11 + 121714306560*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 - 95089302000*(e*x^(1/3) + d)^4*d^8*log(e*x^(
1/3) + d)/e^11 + 56349216000*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 - 25357147200*(e*x^(1/3) + d)^2*d^1
0*log(e*x^(1/3) + d)/e^11 + 9220780800*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 + 5336100*(e*x^(1/3) + d)^
12/e^11 - 76204800*(e*x^(1/3) + d)^11*d/e^11 + 507142944*(e*x^(1/3) + d)^10*d^2/e^11 - 2087008000*(e*x^(1/3) +
d)^9*d^3/e^11 + 5943081375*(e*x^(1/3) + d)^8*d^4/e^11 - 12419827200*(e*x^(1/3) + d)^7*d^5/e^11 + 19722225600*
(e*x^(1/3) + d)^6*d^6/e^11 - 24342861312*(e*x^(1/3) + d)^5*d^7/e^11 + 23772325500*(e*x^(1/3) + d)^4*d^8/e^11 -
18783072000*(e*x^(1/3) + d)^3*d^9/e^11 + 12678573600*(e*x^(1/3) + d)^2*d^10/e^11 - 9220780800*(e*x^(1/3) + d)
*d^11/e^11)*a*b^2*n^2 + 768398400*(27720*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^11 - 332640*(e*x^(1/3) + d)^1
1*d*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e^11 - 6098400*(e*x^(1/3) + d)
^9*d^3*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d)/e^11 - 21954240*(e*x^(1/3)
+ d)^7*d^5*log(e*x^(1/3) + d)/e^11 + 25613280*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3) + d)/e^11 - 21954240*(e*x^(1
/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^4*d^8*log(e*x^(1/3) + d)/e^11 - 6098400*(e*x
^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^2*d^10*log(e*x^(1/3) + d)/e^11 - 332640*(e
*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 - 2310*(e*x^(1/3) + d)^12/e^11 + 30240*(e*x^(1/3) + d)^11*d/e^11 -
182952*(e*x^(1/3) + d)^10*d^2/e^11 + 677600*(e*x^(1/3) + d)^9*d^3/e^11 - 1715175*(e*x^(1/3) + d)^8*d^4/e^11 +
3136320*(e*x^(1/3) + d)^7*d^5/e^11 - 4268880*(e*x^(1/3) + d)^6*d^6/e^11 + 4390848*(e*x^(1/3) + d)^5*d^7/e^11 -
3430350*(e*x^(1/3) + d)^4*d^8/e^11 + 2032800*(e*x^(1/3) + d)^3*d^9/e^11 - 914760*(e*x^(1/3) + d)^2*d^10/e^11
+ 332640*(e*x^(1/3) + d)*d^11/e^11)*a*b^2*n*log(c) + 384199200*(27720*(e*x^(1/3) + d)^12*log(e*x^(1/3) + d)/e^
11 - 332640*(e*x^(1/3) + d)^11*d*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^10*d^2*log(e*x^(1/3) + d)/e
^11 - 6098400*(e*x^(1/3) + d)^9*d^3*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^8*d^4*log(e*x^(1/3) + d
)/e^11 - 21954240*(e*x^(1/3) + d)^7*d^5*log(e*x^(1/3) + d)/e^11 + 25613280*(e*x^(1/3) + d)^6*d^6*log(e*x^(1/3)
+ d)/e^11 - 21954240*(e*x^(1/3) + d)^5*d^7*log(e*x^(1/3) + d)/e^11 + 13721400*(e*x^(1/3) + d)^4*d^8*log(e*x^(
1/3) + d)/e^11 - 6098400*(e*x^(1/3) + d)^3*d^9*log(e*x^(1/3) + d)/e^11 + 1829520*(e*x^(1/3) + d)^2*d^10*log(e*
x^(1/3) + d)/e^11 - 332640*(e*x^(1/3) + d)*d^11*log(e*x^(1/3) + d)/e^11 - 2310*(e*x^(1/3) + d)^12/e^11 + 30240
*(e*x^(1/3) + d)^11*d/e^11 - 182952*(e*x^(1/3) + d)^10*d^2/e^11 + 677600*(e*x^(1/3) + d)^9*d^3/e^11 - 1715175*
(e*x^(1/3) + d)^8*d^4/e^11 + 3136320*(e*x^(1/3) + d)^7*d^5/e^11 - 4268880*(e*x^(1/3) + d)^6*d^6/e^11 + 4390848
*(e*x^(1/3) + d)^5*d^7/e^11 - 3430350*(e*x^(1/3) + d)^4*d^8/e^11 + 2032800*(e*x^(1/3) + d)^3*d^9/e^11 - 914760
*(e*x^(1/3) + d)^2*d^10/e^11 + 332640*(e*x^(1/3) + d)*d^11/e^11)*a^2*b*n)/e

Mupad [B] (verification not implemented)

Time = 9.93 (sec) , antiderivative size = 1802, normalized size of antiderivative = 0.98 \[ \int x^3 \left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

(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))
/(3585859200*e^5) + (453937243*b^3*d^7*n^3*x^(5/3))/(2561328000*e^7) - (697880173*b^3*d^8*n^3*x^(4/3))/(204906
2400*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*log(c*(d + e*x^(1/3))^n)^2)/(3
6960*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)/(44*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))/(40*e^2) - (3*a^2*b*d^4*n*x^(8/3))/(32*e^4) - (44441*a*b^2*d
^9*n^2*x)/(55440*e^9) + (3*a^2*b*d^5*n*x^(7/3))/(28*e^5) + (3*a^2*b*d^7*n*x^(5/3))/(20*e^7) - (3*a^2*b*d^8*n*x
^(4/3))/(16*e^8) - (3*a^2*b*d^10*n*x^(2/3))/(8*e^10) + (3*a^2*b*d^11*n*x^(1/3))/(4*e^11) + (86021*a*b^2*d^12*n
^2*log(d + e*x^(1/3)))/(18480*e^12) + (b^3*d^3*n*x^3*log(c*(d + e*x^(1/3))^n)^2)/(12*e^3) - (763*b^3*d^3*n^2*x
^3*log(c*(d + e*x^(1/3))^n))/(11880*e^3) - (b^3*d^6*n*x^2*log(c*(d + e*x^(1/3))^n)^2)/(8*e^6) + (22727*b^3*d^6
*n^2*x^2*log(c*(d + e*x^(1/3))^n))/(110880*e^6) - (3*b^3*d^2*n*x^(10/3)*log(c*(d + e*x^(1/3))^n)^2)/(40*e^2) +
(181*b^3*d^2*n^2*x^(10/3)*log(c*(d + e*x^(1/3))^n))/(4400*e^2) - (3*b^3*d^4*n*x^(8/3)*log(c*(d + e*x^(1/3))^n
)^2)/(32*e^4) + (2021*b^3*d^4*n^2*x^(8/3)*log(c*(d + e*x^(1/3))^n))/(21120*e^4) + (3*b^3*d^5*n*x^(7/3)*log(c*(
d + e*x^(1/3))^n)^2)/(28*e^5) - (18107*b^3*d^5*n^2*x^(7/3)*log(c*(d + e*x^(1/3))^n))/(129360*e^5) + (3*b^3*d^7
*n*x^(5/3)*log(c*(d + e*x^(1/3))^n)^2)/(20*e^7) - (28271*b^3*d^7*n^2*x^(5/3)*log(c*(d + e*x^(1/3))^n))/(92400*
e^7) - (3*b^3*d^8*n*x^(4/3)*log(c*(d + e*x^(1/3))^n)^2)/(16*e^8) + (35201*b^3*d^8*n^2*x^(4/3)*log(c*(d + e*x^(
1/3))^n))/(73920*e^8) - (3*b^3*d^10*n*x^(2/3)*log(c*(d + e*x^(1/3))^n)^2)/(8*e^10) + (58301*b^3*d^10*n^2*x^(2/
3)*log(c*(d + e*x^(1/3))^n))/(36960*e^10) + (3*b^3*d^11*n*x^(1/3)*log(c*(d + e*x^(1/3))^n)^2)/(4*e^11) - (8602
1*b^3*d^11*n^2*x^(1/3)*log(c*(d + e*x^(1/3))^n))/(18480*e^11) - (763*a*b^2*d^3*n^2*x^3)/(11880*e^3) + (22727*a
*b^2*d^6*n^2*x^2)/(110880*e^6) + (181*a*b^2*d^2*n^2*x^(10/3))/(4400*e^2) + (2021*a*b^2*d^4*n^2*x^(8/3))/(21120
*e^4) - (18107*a*b^2*d^5*n^2*x^(7/3))/(129360*e^5) - (28271*a*b^2*d^7*n^2*x^(5/3))/(92400*e^7) + (35201*a*b^2*
d^8*n^2*x^(4/3))/(73920*e^8) + (58301*a*b^2*d^10*n^2*x^(2/3))/(36960*e^10) - (86021*a*b^2*d^11*n^2*x^(1/3))/(1
8480*e^11) + (3*a^2*b*d*n*x^(11/3))/(44*e) + (a^2*b*d^9*n*x)/(4*e^9) + (3*a*b^2*d*n*x^(11/3)*log(c*(d + e*x^(1
/3))^n))/(22*e) + (a*b^2*d^9*n*x*log(c*(d + e*x^(1/3))^n))/(2*e^9) + (a*b^2*d^3*n*x^3*log(c*(d + e*x^(1/3))^n)
)/(6*e^3) - (a*b^2*d^6*n*x^2*log(c*(d + e*x^(1/3))^n))/(4*e^6) - (3*a*b^2*d^2*n*x^(10/3)*log(c*(d + e*x^(1/3))
^n))/(20*e^2) - (3*a*b^2*d^4*n*x^(8/3)*log(c*(d + e*x^(1/3))^n))/(16*e^4) + (3*a*b^2*d^5*n*x^(7/3)*log(c*(d +
e*x^(1/3))^n))/(14*e^5) + (3*a*b^2*d^7*n*x^(5/3)*log(c*(d + e*x^(1/3))^n))/(10*e^7) - (3*a*b^2*d^8*n*x^(4/3)*l
og(c*(d + e*x^(1/3))^n))/(8*e^8) - (3*a*b^2*d^10*n*x^(2/3)*log(c*(d + e*x^(1/3))^n))/(4*e^10) + (3*a*b^2*d^11*
n*x^(1/3)*log(c*(d + e*x^(1/3))^n))/(2*e^11)