Integrand size = 24, antiderivative size = 907 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}+\frac {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{72 e^6}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}}-\frac {18 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}+\frac {18 b^3 d^5 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^6}-\frac {45 b^2 d^4 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 e^6}+\frac {20 b^2 d^3 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^6}-\frac {45 b^2 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{16 e^6}+\frac {18 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{25 e^6}-\frac {b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 e^6}-\frac {9 b d^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^4 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}-\frac {10 b d^3 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^6}+\frac {45 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{8 e^6}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{5 e^6}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{4 e^6}+\frac {3 d^5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^4 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {10 d^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {15 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^6}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^6 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{2 e^6} \]
45/8*b^3*d^4*n^3*(d+e/x^(1/3))^2/e^6-20/9*b^3*d^3*n^3*(d+e/x^(1/3))^3/e^6+ 45/64*b^3*d^2*n^3*(d+e/x^(1/3))^4/e^6-18/125*b^3*d*n^3*(d+e/x^(1/3))^5/e^6 -18*b^3*d^5*n^3/e^5/x^(1/3)-1/12*b^2*n^2*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^ (1/3))^n))/e^6+1/4*b*n*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+1 /72*b^3*n^3*(d+e/x^(1/3))^6/e^6+3*d^5*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3) )^n))^3/e^6-15/2*d^4*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6+10* d^3*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6-15/2*d^2*(d+e/x^(1/3 ))^4*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6+3*d*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/ x^(1/3))^n))^3/e^6-1/2*(d+e/x^(1/3))^6*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^6-4 5/16*b^2*d^2*n^2*(d+e/x^(1/3))^4*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+18/25*b^2 *d*n^2*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6-9*b*d^5*n*(d+e/x^(1 /3))*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+45/4*b*d^4*n*(d+e/x^(1/3))^2*(a+b*l n(c*(d+e/x^(1/3))^n))^2/e^6-10*b*d^3*n*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1 /3))^n))^2/e^6+45/8*b*d^2*n*(d+e/x^(1/3))^4*(a+b*ln(c*(d+e/x^(1/3))^n))^2/ e^6-9/5*b*d*n*(d+e/x^(1/3))^5*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^6+18*b^3*d^5 *n^2*(d+e/x^(1/3))*ln(c*(d+e/x^(1/3))^n)/e^6-45/4*b^2*d^4*n^2*(d+e/x^(1/3) )^2*(a+b*ln(c*(d+e/x^(1/3))^n))/e^6+20/3*b^2*d^3*n^2*(d+e/x^(1/3))^3*(a+b* ln(c*(d+e/x^(1/3))^n))/e^6+18*a*b^2*d^5*n^2/e^5/x^(1/3)
Time = 0.87 (sec) , antiderivative size = 962, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {-36000 a^3 e^6+18000 a^2 b e^6 n-6000 a b^2 e^6 n^2+1000 b^3 e^6 n^3-21600 a^2 b d e^5 n \sqrt [3]{x}+15840 a b^2 d e^5 n^2 \sqrt [3]{x}-4368 b^3 d e^5 n^3 \sqrt [3]{x}+27000 a^2 b d^2 e^4 n x^{2/3}-33300 a b^2 d^2 e^4 n^2 x^{2/3}+13785 b^3 d^2 e^4 n^3 x^{2/3}-36000 a^2 b d^3 e^3 n x+68400 a b^2 d^3 e^3 n^2 x-41180 b^3 d^3 e^3 n^3 x+54000 a^2 b d^4 e^2 n x^{4/3}-156600 a b^2 d^4 e^2 n^2 x^{4/3}+140070 b^3 d^4 e^2 n^3 x^{4/3}-108000 a^2 b d^5 e n x^{5/3}+529200 a b^2 d^5 e n^2 x^{5/3}-809340 b^3 d^5 e n^3 x^{5/3}-72000 b^3 d^6 n^3 x^2 \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )-36000 b^3 e^6 \log ^3\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+108000 a^2 b d^6 n x^2 \log \left (e+d \sqrt [3]{x}\right )-529200 a b^2 d^6 n^2 x^2 \log \left (e+d \sqrt [3]{x}\right )+809340 b^3 d^6 n^3 x^2 \log \left (e+d \sqrt [3]{x}\right )+3600 b^2 d^6 n^2 x^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (-20 a+49 b n-20 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (3 \log \left (e+d \sqrt [3]{x}\right )-\log (x)\right )-36000 a^2 b d^6 n x^2 \log (x)+176400 a b^2 d^6 n^2 x^2 \log (x)-269780 b^3 d^6 n^3 x^2 \log (x)+1800 b^2 d^6 n^2 x^2 \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (60 a-147 b n+60 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+60 b n \log \left (e+d \sqrt [3]{x}\right )-20 b n \log (x)\right )+1800 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (e \left (-60 a e^5+10 b e^5 n-12 b d e^4 n \sqrt [3]{x}+15 b d^2 e^3 n x^{2/3}-20 b d^3 e^2 n x+30 b d^4 e n x^{4/3}-60 b d^5 n x^{5/3}\right )+60 b d^6 n x^2 \log \left (e+d \sqrt [3]{x}\right )-20 b d^6 n x^2 \log (x)\right )-60 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (1800 a^2 e^6+b^2 e n^2 \left (100 e^5-264 d e^4 \sqrt [3]{x}+555 d^2 e^3 x^{2/3}-1140 d^3 e^2 x+2610 d^4 e x^{4/3}-8820 d^5 x^{5/3}\right )-60 a b e n \left (10 e^5-12 d e^4 \sqrt [3]{x}+15 d^2 e^3 x^{2/3}-20 d^3 e^2 x+30 d^4 e x^{4/3}-60 d^5 x^{5/3}\right )+180 b d^6 n (-20 a+49 b n) x^2 \log \left (e+d \sqrt [3]{x}\right )+60 b d^6 n (20 a-49 b n) x^2 \log (x)\right )}{72000 e^6 x^2} \]
(-36000*a^3*e^6 + 18000*a^2*b*e^6*n - 6000*a*b^2*e^6*n^2 + 1000*b^3*e^6*n^ 3 - 21600*a^2*b*d*e^5*n*x^(1/3) + 15840*a*b^2*d*e^5*n^2*x^(1/3) - 4368*b^3 *d*e^5*n^3*x^(1/3) + 27000*a^2*b*d^2*e^4*n*x^(2/3) - 33300*a*b^2*d^2*e^4*n ^2*x^(2/3) + 13785*b^3*d^2*e^4*n^3*x^(2/3) - 36000*a^2*b*d^3*e^3*n*x + 684 00*a*b^2*d^3*e^3*n^2*x - 41180*b^3*d^3*e^3*n^3*x + 54000*a^2*b*d^4*e^2*n*x ^(4/3) - 156600*a*b^2*d^4*e^2*n^2*x^(4/3) + 140070*b^3*d^4*e^2*n^3*x^(4/3) - 108000*a^2*b*d^5*e*n*x^(5/3) + 529200*a*b^2*d^5*e*n^2*x^(5/3) - 809340* b^3*d^5*e*n^3*x^(5/3) - 72000*b^3*d^6*n^3*x^2*Log[d + e/x^(1/3)]^3 - 36000 *b^3*e^6*Log[c*(d + e/x^(1/3))^n]^3 + 108000*a^2*b*d^6*n*x^2*Log[e + d*x^( 1/3)] - 529200*a*b^2*d^6*n^2*x^2*Log[e + d*x^(1/3)] + 809340*b^3*d^6*n^3*x ^2*Log[e + d*x^(1/3)] + 3600*b^2*d^6*n^2*x^2*Log[d + e/x^(1/3)]*(-20*a + 4 9*b*n - 20*b*Log[c*(d + e/x^(1/3))^n])*(3*Log[e + d*x^(1/3)] - Log[x]) - 3 6000*a^2*b*d^6*n*x^2*Log[x] + 176400*a*b^2*d^6*n^2*x^2*Log[x] - 269780*b^3 *d^6*n^3*x^2*Log[x] + 1800*b^2*d^6*n^2*x^2*Log[d + e/x^(1/3)]^2*(60*a - 14 7*b*n + 60*b*Log[c*(d + e/x^(1/3))^n] + 60*b*n*Log[e + d*x^(1/3)] - 20*b*n *Log[x]) + 1800*b^2*Log[c*(d + e/x^(1/3))^n]^2*(e*(-60*a*e^5 + 10*b*e^5*n - 12*b*d*e^4*n*x^(1/3) + 15*b*d^2*e^3*n*x^(2/3) - 20*b*d^3*e^2*n*x + 30*b* d^4*e*n*x^(4/3) - 60*b*d^5*n*x^(5/3)) + 60*b*d^6*n*x^2*Log[e + d*x^(1/3)] - 20*b*d^6*n*x^2*Log[x]) - 60*b*Log[c*(d + e/x^(1/3))^n]*(1800*a^2*e^6 + b ^2*e*n^2*(100*e^5 - 264*d*e^4*x^(1/3) + 555*d^2*e^3*x^(2/3) - 1140*d^3*...
Time = 1.23 (sec) , antiderivative size = 913, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^{5/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -3 \int \left (-\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 d^5}{e^5}+\frac {5 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 d^4}{e^5}-\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 d^3}{e^5}+\frac {10 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 d^2}{e^5}-\frac {5 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 d}{e^5}+\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^5}\right )d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{216 e^6}+\frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{6 e^6}-\frac {b n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{12 e^6}+\frac {b^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{36 e^6}+\frac {6 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}-\frac {d \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{e^6}+\frac {3 b d n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{5 e^6}-\frac {6 b^2 d n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{25 e^6}-\frac {15 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}+\frac {5 d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{2 e^6}-\frac {15 b d^2 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{8 e^6}+\frac {15 b^2 d^2 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{16 e^6}+\frac {20 b^3 d^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{27 e^6}-\frac {10 d^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{3 e^6}+\frac {10 b d^3 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{3 e^6}-\frac {20 b^2 d^3 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^6}-\frac {15 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}+\frac {5 d^4 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{2 e^6}-\frac {15 b d^4 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}+\frac {15 b^2 d^4 n^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^6}-\frac {d^5 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {3 b d^5 n \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {6 b^3 d^5 n^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {6 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}-\frac {6 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}}\right )\) |
-3*((-15*b^3*d^4*n^3*(d + e/x^(1/3))^2)/(8*e^6) + (20*b^3*d^3*n^3*(d + e/x ^(1/3))^3)/(27*e^6) - (15*b^3*d^2*n^3*(d + e/x^(1/3))^4)/(64*e^6) + (6*b^3 *d*n^3*(d + e/x^(1/3))^5)/(125*e^6) - (b^3*n^3*(d + e/x^(1/3))^6)/(216*e^6 ) - (6*a*b^2*d^5*n^2)/(e^5*x^(1/3)) + (6*b^3*d^5*n^3)/(e^5*x^(1/3)) - (6*b ^3*d^5*n^2*(d + e/x^(1/3))*Log[c*(d + e/x^(1/3))^n])/e^6 + (15*b^2*d^4*n^2 *(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(4*e^6) - (20*b^2*d^3 *n^2*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*e^6) + (15*b^2 *d^2*n^2*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n]))/(16*e^6) - (6 *b^2*d*n^2*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n]))/(25*e^6) + (b^2*n^2*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n]))/(36*e^6) + (3 *b*d^5*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^6 - (15*b*d ^4*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(4*e^6) + (10*b *d^3*n*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(3*e^6) - (15 *b*d^2*n*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(8*e^6) + ( 3*b*d*n*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(5*e^6) - (b *n*(d + e/x^(1/3))^6*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(12*e^6) - (d^5*( d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 + (5*d^4*(d + e/x^( 1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6) - (10*d^3*(d + e/x^(1/ 3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(3*e^6) + (5*d^2*(d + e/x^(1/3)) ^4*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/(2*e^6) - (d*(d + e/x^(1/3))^5*(...
3.6.7.3.1 Defintions of rubi rules used
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]
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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]
Time = 0.45 (sec) , antiderivative size = 1404, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]
1/72000*(1000*b^3*e^6*n^3 - 6000*a*b^2*e^6*n^2 + 18000*a^2*b*e^6*n - 36000 *a^3*e^6 + 36000*(b^3*e^6*x^2 - b^3*e^6)*log(c)^3 + 36000*(b^3*d^6*n^3*x^2 - b^3*e^6*n^3)*log((d*x + e*x^(2/3))/x)^3 + 20*(1800*a^3*e^6 + (2059*b^3* d^3*e^3 - 50*b^3*e^6)*n^3 - 60*(57*a*b^2*d^3*e^3 - 5*a*b^2*e^6)*n^2 + 900* (2*a^2*b*d^3*e^3 - a^2*b*e^6)*n)*x^2 - 18000*(2*b^3*d^3*e^3*n*x - b^3*e^6* n + 6*a*b^2*e^6 - (6*a*b^2*e^6 + (2*b^3*d^3*e^3 - b^3*e^6)*n)*x^2)*log(c)^ 2 - 1800*(20*b^3*d^3*e^3*n^3*x - 10*b^3*e^6*n^3 + 60*a*b^2*e^6*n^2 + 3*(49 *b^3*d^6*n^3 - 20*a*b^2*d^6*n^2)*x^2 - 60*(b^3*d^6*n^2*x^2 - b^3*e^6*n^2)* log(c) + 15*(4*b^3*d^5*e*n^3*x - b^3*d^2*e^4*n^3)*x^(2/3) - 6*(5*b^3*d^4*e ^2*n^3*x - 2*b^3*d*e^5*n^3)*x^(1/3))*log((d*x + e*x^(2/3))/x)^2 - 20*(2059 *b^3*d^3*e^3*n^3 - 3420*a*b^2*d^3*e^3*n^2 + 1800*a^2*b*d^3*e^3*n)*x - 1200 *(5*b^3*e^6*n^2 - 30*a*b^2*e^6*n + 90*a^2*b*e^6 - (90*a^2*b*e^6 - (57*b^3* d^3*e^3 - 5*b^3*e^6)*n^2 + 30*(2*a*b^2*d^3*e^3 - a*b^2*e^6)*n)*x^2 - 3*(19 *b^3*d^3*e^3*n^2 - 20*a*b^2*d^3*e^3*n)*x)*log(c) - 60*(100*b^3*e^6*n^3 - 6 00*a*b^2*e^6*n^2 + 1800*a^2*b*e^6*n - (13489*b^3*d^6*n^3 - 8820*a*b^2*d^6* n^2 + 1800*a^2*b*d^6*n)*x^2 - 1800*(b^3*d^6*n*x^2 - b^3*e^6*n)*log(c)^2 - 60*(19*b^3*d^3*e^3*n^3 - 20*a*b^2*d^3*e^3*n^2)*x + 60*(20*b^3*d^3*e^3*n^2* x - 10*b^3*e^6*n^2 + 60*a*b^2*e^6*n + 3*(49*b^3*d^6*n^2 - 20*a*b^2*d^6*n)* x^2)*log(c) + 15*(37*b^3*d^2*e^4*n^3 - 60*a*b^2*d^2*e^4*n^2 - 12*(49*b^3*d ^5*e*n^3 - 20*a*b^2*d^5*e*n^2)*x + 60*(4*b^3*d^5*e*n^2*x - b^3*d^2*e^4*...
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 864, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]
1/40*a^2*b*e*n*(60*d^6*log(d*x^(1/3) + e)/e^7 - 20*d^6*log(x)/e^7 - (60*d^ 5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*e^2*x - 15*d^2*e^3*x^(2/3) + 12*d*e^ 4*x^(1/3) - 10*e^5)/(e^6*x^2)) + 1/1200*(60*e*n*(60*d^6*log(d*x^(1/3) + e) /e^7 - 20*d^6*log(x)/e^7 - (60*d^5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*e^2 *x - 15*d^2*e^3*x^(2/3) + 12*d*e^4*x^(1/3) - 10*e^5)/(e^6*x^2))*log(c*(d + e/x^(1/3))^n) - (1800*d^6*x^2*log(d*x^(1/3) + e)^2 + 200*d^6*x^2*log(x)^2 - 2940*d^6*x^2*log(x) - 8820*d^5*e*x^(5/3) + 2610*d^4*e^2*x^(4/3) - 1140* d^3*e^3*x + 555*d^2*e^4*x^(2/3) - 264*d*e^5*x^(1/3) + 100*e^6 - 60*(20*d^6 *x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e))*n^2/(e^6*x^2))*a*b^2 + 1/21 6000*(5400*e*n*(60*d^6*log(d*x^(1/3) + e)/e^7 - 20*d^6*log(x)/e^7 - (60*d^ 5*x^(5/3) - 30*d^4*e*x^(4/3) + 20*d^3*e^2*x - 15*d^2*e^3*x^(2/3) + 12*d*e^ 4*x^(1/3) - 10*e^5)/(e^6*x^2))*log(c*(d + e/x^(1/3))^n)^2 + e*n*((108000*d ^6*x^2*log(d*x^(1/3) + e)^3 - 4000*d^6*x^2*log(x)^3 + 88200*d^6*x^2*log(x) ^2 - 809340*d^6*x^2*log(x) - 2428020*d^5*e*x^(5/3) + 420210*d^4*e^2*x^(4/3 ) - 123540*d^3*e^3*x + 41355*d^2*e^4*x^(2/3) - 13104*d*e^5*x^(1/3) + 3000* e^6 - 5400*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e)^2 + 180*(2 00*d^6*x^2*log(x)^2 - 2940*d^6*x^2*log(x) + 13489*d^6*x^2)*log(d*x^(1/3) + e))*n^2/(e^7*x^2) - 180*(1800*d^6*x^2*log(d*x^(1/3) + e)^2 + 200*d^6*x^2* log(x)^2 - 2940*d^6*x^2*log(x) - 8820*d^5*e*x^(5/3) + 2610*d^4*e^2*x^(4/3) - 1140*d^3*e^3*x + 555*d^2*e^4*x^(2/3) - 264*d*e^5*x^(1/3) + 100*e^6 -...
Leaf count of result is larger than twice the leaf count of optimal. 1747 vs. \(2 (787) = 1574\).
Time = 0.43 (sec) , antiderivative size = 1747, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]
1/72000*(36000*(6*(d*x^(1/3) + e)*b^3*d^5*n^3/(e^5*x^(1/3)) - 15*(d*x^(1/3 ) + e)^2*b^3*d^4*n^3/(e^5*x^(2/3)) + 20*(d*x^(1/3) + e)^3*b^3*d^3*n^3/(e^5 *x) - 15*(d*x^(1/3) + e)^4*b^3*d^2*n^3/(e^5*x^(4/3)) + 6*(d*x^(1/3) + e)^5 *b^3*d*n^3/(e^5*x^(5/3)) - (d*x^(1/3) + e)^6*b^3*n^3/(e^5*x^2))*log((d*x^( 1/3) + e)/x^(1/3))^3 + 1800*(10*(b^3*n^3 - 6*b^3*n^2*log(c) - 6*a*b^2*n^2) *(d*x^(1/3) + e)^6/(e^5*x^2) - 72*(b^3*d*n^3 - 5*b^3*d*n^2*log(c) - 5*a*b^ 2*d*n^2)*(d*x^(1/3) + e)^5/(e^5*x^(5/3)) + 225*(b^3*d^2*n^3 - 4*b^3*d^2*n^ 2*log(c) - 4*a*b^2*d^2*n^2)*(d*x^(1/3) + e)^4/(e^5*x^(4/3)) - 400*(b^3*d^3 *n^3 - 3*b^3*d^3*n^2*log(c) - 3*a*b^2*d^3*n^2)*(d*x^(1/3) + e)^3/(e^5*x) + 450*(b^3*d^4*n^3 - 2*b^3*d^4*n^2*log(c) - 2*a*b^2*d^4*n^2)*(d*x^(1/3) + e )^2/(e^5*x^(2/3)) - 360*(b^3*d^5*n^3 - b^3*d^5*n^2*log(c) - a*b^2*d^5*n^2) *(d*x^(1/3) + e)/(e^5*x^(1/3)))*log((d*x^(1/3) + e)/x^(1/3))^2 - 60*(100*( b^3*n^3 - 6*b^3*n^2*log(c) + 18*b^3*n*log(c)^2 - 6*a*b^2*n^2 + 36*a*b^2*n* log(c) + 18*a^2*b*n)*(d*x^(1/3) + e)^6/(e^5*x^2) - 432*(2*b^3*d*n^3 - 10*b ^3*d*n^2*log(c) + 25*b^3*d*n*log(c)^2 - 10*a*b^2*d*n^2 + 50*a*b^2*d*n*log( c) + 25*a^2*b*d*n)*(d*x^(1/3) + e)^5/(e^5*x^(5/3)) + 3375*(b^3*d^2*n^3 - 4 *b^3*d^2*n^2*log(c) + 8*b^3*d^2*n*log(c)^2 - 4*a*b^2*d^2*n^2 + 16*a*b^2*d^ 2*n*log(c) + 8*a^2*b*d^2*n)*(d*x^(1/3) + e)^4/(e^5*x^(4/3)) - 4000*(2*b^3* d^3*n^3 - 6*b^3*d^3*n^2*log(c) + 9*b^3*d^3*n*log(c)^2 - 6*a*b^2*d^3*n^2 + 18*a*b^2*d^3*n*log(c) + 9*a^2*b*d^3*n)*(d*x^(1/3) + e)^3/(e^5*x) + 1350...
Time = 9.28 (sec) , antiderivative size = 992, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]
(b^3*n^3)/(72*x^2) - (b^3*log(c*(d + e/x^(1/3))^n)^3)/(2*x^2) - a^3/(2*x^2 ) - (3*a*b^2*log(c*(d + e/x^(1/3))^n)^2)/(2*x^2) + (b^3*n*log(c*(d + e/x^( 1/3))^n)^2)/(4*x^2) - (b^3*n^2*log(c*(d + e/x^(1/3))^n))/(12*x^2) - (a*b^2 *n^2)/(12*x^2) + (b^3*d^6*log(c*(d + e/x^(1/3))^n)^3)/(2*e^6) - (3*a^2*b*l og(c*(d + e/x^(1/3))^n))/(2*x^2) + (a^2*b*n)/(4*x^2) + (a*b^2*n*log(c*(d + e/x^(1/3))^n))/(2*x^2) + (13489*b^3*d^6*n^3*log(d + e/x^(1/3)))/(1200*e^6 ) - (2059*b^3*d^3*n^3)/(3600*e^3*x) + (919*b^3*d^2*n^3)/(4800*e^2*x^(4/3)) + (4669*b^3*d^4*n^3)/(2400*e^4*x^(2/3)) - (13489*b^3*d^5*n^3)/(1200*e^5*x ^(1/3)) + (3*a*b^2*d^6*log(c*(d + e/x^(1/3))^n)^2)/(2*e^6) - (147*b^3*d^6* n*log(c*(d + e/x^(1/3))^n)^2)/(40*e^6) - (91*b^3*d*n^3)/(1500*e*x^(5/3)) + (3*a^2*b*d^6*n*log(d + e/x^(1/3)))/(2*e^6) - (3*b^3*d*n*log(c*(d + e/x^(1 /3))^n)^2)/(10*e*x^(5/3)) + (11*b^3*d*n^2*log(c*(d + e/x^(1/3))^n))/(50*e* x^(5/3)) - (a^2*b*d^3*n)/(2*e^3*x) + (11*a*b^2*d*n^2)/(50*e*x^(5/3)) + (3* a^2*b*d^2*n)/(8*e^2*x^(4/3)) + (3*a^2*b*d^4*n)/(4*e^4*x^(2/3)) - (3*a^2*b* d^5*n)/(2*e^5*x^(1/3)) - (147*a*b^2*d^6*n^2*log(d + e/x^(1/3)))/(20*e^6) - (b^3*d^3*n*log(c*(d + e/x^(1/3))^n)^2)/(2*e^3*x) + (19*b^3*d^3*n^2*log(c* (d + e/x^(1/3))^n))/(20*e^3*x) + (3*b^3*d^2*n*log(c*(d + e/x^(1/3))^n)^2)/ (8*e^2*x^(4/3)) - (37*b^3*d^2*n^2*log(c*(d + e/x^(1/3))^n))/(80*e^2*x^(4/3 )) + (3*b^3*d^4*n*log(c*(d + e/x^(1/3))^n)^2)/(4*e^4*x^(2/3)) - (87*b^3*d^ 4*n^2*log(c*(d + e/x^(1/3))^n))/(40*e^4*x^(2/3)) - (3*b^3*d^5*n*log(c*(...