Integrand size = 24, antiderivative size = 438 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx=-\frac {9 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3}+\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac {18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac {18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}-\frac {18 b^3 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac {9 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^3}-\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}+\frac {9 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac {9 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 e^3}+\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac {3 d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac {3 d \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3} \] Output:
-9/4*b^3*d*n^3*(d+e/x^(1/3))^2/e^3+2/9*b^3*n^3*(d+e/x^(1/3))^3/e^3-18*a*b^ 2*d^2*n^2/e^2/x^(1/3)+18*b^3*d^2*n^3/e^2/x^(1/3)-18*b^3*d^2*n^2*(d+e/x^(1/ 3))*ln(c*(d+e/x^(1/3))^n)/e^3+9/2*b^2*d*n^2*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e /x^(1/3))^n))/e^3-2/3*b^2*n^2*(d+e/x^(1/3))^3*(a+b*ln(c*(d+e/x^(1/3))^n))/ e^3+9*b*d^2*n*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^3-9/2*b*d*n*(d +e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3))^n))^2/e^3+b*n*(d+e/x^(1/3))^3*(a+b*l n(c*(d+e/x^(1/3))^n))^2/e^3-3*d^2*(d+e/x^(1/3))*(a+b*ln(c*(d+e/x^(1/3))^n) )^3/e^3+3*d*(d+e/x^(1/3))^2*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^3-(d+e/x^(1/3) )^3*(a+b*ln(c*(d+e/x^(1/3))^n))^3/e^3
Time = 0.84 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.52 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {-36 a^3 e^3+36 a^2 b e^3 n-24 a b^2 e^3 n^2+8 b^3 e^3 n^3-54 a^2 b d e^2 n \sqrt [3]{x}+90 a b^2 d e^2 n^2 \sqrt [3]{x}-57 b^3 d e^2 n^3 \sqrt [3]{x}+108 a^2 b d^2 e n x^{2/3}-396 a b^2 d^2 e n^2 x^{2/3}+510 b^3 d^2 e n^3 x^{2/3}+72 b^3 d^3 n^3 x \log ^3\left (d+\frac {e}{\sqrt [3]{x}}\right )-36 b^3 e^3 \log ^3\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-108 a^2 b d^3 n x \log \left (e+d \sqrt [3]{x}\right )+396 a b^2 d^3 n^2 x \log \left (e+d \sqrt [3]{x}\right )-510 b^3 d^3 n^3 x \log \left (e+d \sqrt [3]{x}\right )+12 b^2 d^3 n^2 x \log \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right ) \left (3 \log \left (e+d \sqrt [3]{x}\right )-\log (x)\right )+36 a^2 b d^3 n x \log (x)-132 a b^2 d^3 n^2 x \log (x)+170 b^3 d^3 n^3 x \log (x)-18 b^2 d^3 n^2 x \log ^2\left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (6 a-11 b n+6 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+6 b n \log \left (e+d \sqrt [3]{x}\right )-2 b n \log (x)\right )+18 b^2 \log ^2\left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (e \left (-6 a e^2+2 b e^2 n-3 b d e n \sqrt [3]{x}+6 b d^2 n x^{2/3}\right )-6 b d^3 n x \log \left (e+d \sqrt [3]{x}\right )+2 b d^3 n x \log (x)\right )-6 b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \left (18 a^2 e^3-6 a b e n \left (2 e^2-3 d e \sqrt [3]{x}+6 d^2 x^{2/3}\right )+b^2 e n^2 \left (4 e^2-15 d e \sqrt [3]{x}+66 d^2 x^{2/3}\right )+6 b d^3 n (6 a-11 b n) x \log \left (e+d \sqrt [3]{x}\right )+2 b d^3 n (-6 a+11 b n) x \log (x)\right )}{36 e^3 x} \] Input:
Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,x]
Output:
(-36*a^3*e^3 + 36*a^2*b*e^3*n - 24*a*b^2*e^3*n^2 + 8*b^3*e^3*n^3 - 54*a^2* b*d*e^2*n*x^(1/3) + 90*a*b^2*d*e^2*n^2*x^(1/3) - 57*b^3*d*e^2*n^3*x^(1/3) + 108*a^2*b*d^2*e*n*x^(2/3) - 396*a*b^2*d^2*e*n^2*x^(2/3) + 510*b^3*d^2*e* n^3*x^(2/3) + 72*b^3*d^3*n^3*x*Log[d + e/x^(1/3)]^3 - 36*b^3*e^3*Log[c*(d + e/x^(1/3))^n]^3 - 108*a^2*b*d^3*n*x*Log[e + d*x^(1/3)] + 396*a*b^2*d^3*n ^2*x*Log[e + d*x^(1/3)] - 510*b^3*d^3*n^3*x*Log[e + d*x^(1/3)] + 12*b^2*d^ 3*n^2*x*Log[d + e/x^(1/3)]*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(1/3))^n])*( 3*Log[e + d*x^(1/3)] - Log[x]) + 36*a^2*b*d^3*n*x*Log[x] - 132*a*b^2*d^3*n ^2*x*Log[x] + 170*b^3*d^3*n^3*x*Log[x] - 18*b^2*d^3*n^2*x*Log[d + e/x^(1/3 )]^2*(6*a - 11*b*n + 6*b*Log[c*(d + e/x^(1/3))^n] + 6*b*n*Log[e + d*x^(1/3 )] - 2*b*n*Log[x]) + 18*b^2*Log[c*(d + e/x^(1/3))^n]^2*(e*(-6*a*e^2 + 2*b* e^2*n - 3*b*d*e*n*x^(1/3) + 6*b*d^2*n*x^(2/3)) - 6*b*d^3*n*x*Log[e + d*x^( 1/3)] + 2*b*d^3*n*x*Log[x]) - 6*b*Log[c*(d + e/x^(1/3))^n]*(18*a^2*e^3 - 6 *a*b*e*n*(2*e^2 - 3*d*e*x^(1/3) + 6*d^2*x^(2/3)) + b^2*e*n^2*(4*e^2 - 15*d *e*x^(1/3) + 66*d^2*x^(2/3)) + 6*b*d^3*n*(6*a - 11*b*n)*x*Log[e + d*x^(1/3 )] + 2*b*d^3*n*(-6*a + 11*b*n)*x*Log[x]))/(36*e^3*x)
Time = 1.24 (sec) , antiderivative size = 444, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^{2/3}}d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle -3 \int \left (\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^2}-\frac {2 d \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^2}+\frac {d^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^2}\right )d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {2 b^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 e^3}-\frac {3 b^2 d n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^3}+\frac {6 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}-\frac {3 b d^2 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}+\frac {d^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}-\frac {b n \left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{3 e^3}+\frac {3 b d n \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 e^3}+\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{3 e^3}-\frac {d \left (d+\frac {e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac {6 b^3 d^2 n^2 \left (d+\frac {e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )}{e^3}-\frac {6 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}-\frac {2 b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^3}{27 e^3}+\frac {3 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{4 e^3}\right )\) |
Input:
Int[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,x]
Output:
-3*((3*b^3*d*n^3*(d + e/x^(1/3))^2)/(4*e^3) - (2*b^3*n^3*(d + e/x^(1/3))^3 )/(27*e^3) + (6*a*b^2*d^2*n^2)/(e^2*x^(1/3)) - (6*b^3*d^2*n^3)/(e^2*x^(1/3 )) + (6*b^3*d^2*n^2*(d + e/x^(1/3))*Log[c*(d + e/x^(1/3))^n])/e^3 - (3*b^2 *d*n^2*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n]))/(2*e^3) + (2*b^ 2*n^2*(d + e/x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n]))/(9*e^3) - (3*b*d ^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 + (3*b*d*n*(d + e/x^(1/3))^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(2*e^3) - (b*n*(d + e/ x^(1/3))^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/(3*e^3) + (d^2*(d + e/x^(1/ 3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 - (d*(d + e/x^(1/3))^2*(a + b* Log[c*(d + e/x^(1/3))^n])^3)/e^3 + ((d + e/x^(1/3))^3*(a + b*Log[c*(d + e/ x^(1/3))^n])^3)/(3*e^3))
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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{3}}{x^{2}}d x\]
Input:
int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x^2,x)
Output:
int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x^2,x)
Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (384) = 768\).
Time = 0.16 (sec) , antiderivative size = 814, normalized size of antiderivative = 1.86 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x^2,x, algorithm="fricas")
Output:
1/36*(8*b^3*e^3*n^3 - 24*a*b^2*e^3*n^2 + 36*a^2*b*e^3*n - 36*a^3*e^3 + 36* (b^3*e^3*x - b^3*e^3)*log(c)^3 - 36*(b^3*d^3*n^3*x + b^3*e^3*n^3)*log((d*x + e*x^(2/3))/x)^3 + 36*(b^3*e^3*n - 3*a*b^2*e^3 - (b^3*e^3*n - 3*a*b^2*e^ 3)*x)*log(c)^2 + 18*(6*b^3*d^2*e*n^3*x^(2/3) - 3*b^3*d*e^2*n^3*x^(1/3) + 2 *b^3*e^3*n^3 - 6*a*b^2*e^3*n^2 + (11*b^3*d^3*n^3 - 6*a*b^2*d^3*n^2)*x - 6* (b^3*d^3*n^2*x + b^3*e^3*n^2)*log(c))*log((d*x + e*x^(2/3))/x)^2 - 4*(2*b^ 3*e^3*n^3 - 6*a*b^2*e^3*n^2 + 9*a^2*b*e^3*n - 9*a^3*e^3)*x - 12*(2*b^3*e^3 *n^2 - 6*a*b^2*e^3*n + 9*a^2*b*e^3 - (2*b^3*e^3*n^2 - 6*a*b^2*e^3*n + 9*a^ 2*b*e^3)*x)*log(c) - 6*(4*b^3*e^3*n^3 - 12*a*b^2*e^3*n^2 + 18*a^2*b*e^3*n + 18*(b^3*d^3*n*x + b^3*e^3*n)*log(c)^2 + (85*b^3*d^3*n^3 - 66*a*b^2*d^3*n ^2 + 18*a^2*b*d^3*n)*x - 6*(2*b^3*e^3*n^2 - 6*a*b^2*e^3*n + (11*b^3*d^3*n^ 2 - 6*a*b^2*d^3*n)*x)*log(c) + 6*(11*b^3*d^2*e*n^3 - 6*b^3*d^2*e*n^2*log(c ) - 6*a*b^2*d^2*e*n^2)*x^(2/3) - 3*(5*b^3*d*e^2*n^3 - 6*b^3*d*e^2*n^2*log( c) - 6*a*b^2*d*e^2*n^2)*x^(1/3))*log((d*x + e*x^(2/3))/x) + 6*(85*b^3*d^2* e*n^3 + 18*b^3*d^2*e*n*log(c)^2 - 66*a*b^2*d^2*e*n^2 + 18*a^2*b*d^2*e*n - 6*(11*b^3*d^2*e*n^2 - 6*a*b^2*d^2*e*n)*log(c))*x^(2/3) - 3*(19*b^3*d*e^2*n ^3 + 18*b^3*d*e^2*n*log(c)^2 - 30*a*b^2*d*e^2*n^2 + 18*a^2*b*d*e^2*n - 6*( 5*b^3*d*e^2*n^2 - 6*a*b^2*d*e^2*n)*log(c))*x^(1/3))/(e^3*x)
\[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}}{x^{2}}\, dx \] Input:
integrate((a+b*ln(c*(d+e/x**(1/3))**n))**3/x**2,x)
Output:
Integral((a + b*log(c*(d + e/x**(1/3))**n))**3/x**2, x)
Time = 0.09 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.46 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x^2,x, algorithm="maxima")
Output:
-1/2*a^2*b*e*n*(6*d^3*log(d*x^(1/3) + e)/e^4 - 2*d^3*log(x)/e^4 - (6*d^2*x ^(2/3) - 3*d*e*x^(1/3) + 2*e^2)/(e^3*x)) - b^3*log(c*(d + e/x^(1/3))^n)^3/ x - 1/6*(6*e*n*(6*d^3*log(d*x^(1/3) + e)/e^4 - 2*d^3*log(x)/e^4 - (6*d^2*x ^(2/3) - 3*d*e*x^(1/3) + 2*e^2)/(e^3*x))*log(c*(d + e/x^(1/3))^n) - (18*d^ 3*x*log(d*x^(1/3) + e)^2 + 2*d^3*x*log(x)^2 - 22*d^3*x*log(x) - 66*d^2*e*x ^(2/3) + 15*d*e^2*x^(1/3) - 4*e^3 - 6*(2*d^3*x*log(x) - 11*d^3*x)*log(d*x^ (1/3) + e))*n^2/(e^3*x))*a*b^2 - 1/108*(54*e*n*(6*d^3*log(d*x^(1/3) + e)/e ^4 - 2*d^3*log(x)/e^4 - (6*d^2*x^(2/3) - 3*d*e*x^(1/3) + 2*e^2)/(e^3*x))*l og(c*(d + e/x^(1/3))^n)^2 + e*n*((108*d^3*x*log(d*x^(1/3) + e)^3 - 4*d^3*x *log(x)^3 + 66*d^3*x*log(x)^2 - 510*d^3*x*log(x) - 1530*d^2*e*x^(2/3) + 17 1*d*e^2*x^(1/3) - 24*e^3 - 54*(2*d^3*x*log(x) - 11*d^3*x)*log(d*x^(1/3) + e)^2 + 18*(2*d^3*x*log(x)^2 - 22*d^3*x*log(x) + 85*d^3*x)*log(d*x^(1/3) + e))*n^2/(e^4*x) - 18*(18*d^3*x*log(d*x^(1/3) + e)^2 + 2*d^3*x*log(x)^2 - 2 2*d^3*x*log(x) - 66*d^2*e*x^(2/3) + 15*d*e^2*x^(1/3) - 4*e^3 - 6*(2*d^3*x* log(x) - 11*d^3*x)*log(d*x^(1/3) + e))*n*log(c*(d + e/x^(1/3))^n)/(e^4*x)) )*b^3 - 3*a*b^2*log(c*(d + e/x^(1/3))^n)^2/x - 3*a^2*b*log(c*(d + e/x^(1/3 ))^n)/x - a^3/x
Leaf count of result is larger than twice the leaf count of optimal. 846 vs. \(2 (384) = 768\).
Time = 0.22 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.93 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx=\text {Too large to display} \] Input:
integrate((a+b*log(c*(d+e/x^(1/3))^n))^3/x^2,x, algorithm="giac")
Output:
-1/36*(36*(3*(d*x^(1/3) + e)*b^3*d^2*n^3/(e^2*x^(1/3)) - 3*(d*x^(1/3) + e) ^2*b^3*d*n^3/(e^2*x^(2/3)) + (d*x^(1/3) + e)^3*b^3*n^3/(e^2*x))*log((d*x^( 1/3) + e)/x^(1/3))^3 - 18*(2*(b^3*n^3 - 3*b^3*n^2*log(c) - 3*a*b^2*n^2)*(d *x^(1/3) + e)^3/(e^2*x) - 9*(b^3*d*n^3 - 2*b^3*d*n^2*log(c) - 2*a*b^2*d*n^ 2)*(d*x^(1/3) + e)^2/(e^2*x^(2/3)) + 18*(b^3*d^2*n^3 - b^3*d^2*n^2*log(c) - a*b^2*d^2*n^2)*(d*x^(1/3) + e)/(e^2*x^(1/3)))*log((d*x^(1/3) + e)/x^(1/3 ))^2 + 6*(2*(2*b^3*n^3 - 6*b^3*n^2*log(c) + 9*b^3*n*log(c)^2 - 6*a*b^2*n^2 + 18*a*b^2*n*log(c) + 9*a^2*b*n)*(d*x^(1/3) + e)^3/(e^2*x) - 27*(b^3*d*n^ 3 - 2*b^3*d*n^2*log(c) + 2*b^3*d*n*log(c)^2 - 2*a*b^2*d*n^2 + 4*a*b^2*d*n* log(c) + 2*a^2*b*d*n)*(d*x^(1/3) + e)^2/(e^2*x^(2/3)) + 54*(2*b^3*d^2*n^3 - 2*b^3*d^2*n^2*log(c) + b^3*d^2*n*log(c)^2 - 2*a*b^2*d^2*n^2 + 2*a*b^2*d^ 2*n*log(c) + a^2*b*d^2*n)*(d*x^(1/3) + e)/(e^2*x^(1/3)))*log((d*x^(1/3) + e)/x^(1/3)) - 4*(2*b^3*n^3 - 6*b^3*n^2*log(c) + 9*b^3*n*log(c)^2 - 9*b^3*l og(c)^3 - 6*a*b^2*n^2 + 18*a*b^2*n*log(c) - 27*a*b^2*log(c)^2 + 9*a^2*b*n - 27*a^2*b*log(c) - 9*a^3)*(d*x^(1/3) + e)^3/(e^2*x) + 27*(3*b^3*d*n^3 - 6 *b^3*d*n^2*log(c) + 6*b^3*d*n*log(c)^2 - 4*b^3*d*log(c)^3 - 6*a*b^2*d*n^2 + 12*a*b^2*d*n*log(c) - 12*a*b^2*d*log(c)^2 + 6*a^2*b*d*n - 12*a^2*b*d*log (c) - 4*a^3*d)*(d*x^(1/3) + e)^2/(e^2*x^(2/3)) - 108*(6*b^3*d^2*n^3 - 6*b^ 3*d^2*n^2*log(c) + 3*b^3*d^2*n*log(c)^2 - b^3*d^2*log(c)^3 - 6*a*b^2*d^2*n ^2 + 6*a*b^2*d^2*n*log(c) - 3*a*b^2*d^2*log(c)^2 + 3*a^2*b*d^2*n - 3*a^...
Time = 25.58 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.30 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx=\frac {\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{2\,e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{4\,e}}{x^{2/3}}-{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^3\,\left (\frac {b^3}{x}+\frac {b^3\,d^3}{e^3}\right )-{\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )}^2\,\left (\frac {b^2\,\left (3\,a-b\,n\right )}{x}-\frac {\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{2\,e}-\frac {9\,a\,b^2\,d}{2\,e}}{x^{2/3}}+\frac {d\,\left (6\,a\,b^2\,d^2-11\,b^3\,d^2\,n\right )}{2\,e^3}+\frac {d\,\left (\frac {3\,b^2\,d\,\left (3\,a-b\,n\right )}{e}-\frac {9\,a\,b^2\,d}{e}\right )}{e\,x^{1/3}}\right )-\frac {a^3-a^2\,b\,n+\frac {2\,a\,b^2\,n^2}{3}-\frac {2\,b^3\,n^3}{9}}{x}-\frac {\frac {d\,\left (\frac {d\,\left (3\,a^3-3\,a^2\,b\,n+2\,a\,b^2\,n^2-\frac {2\,b^3\,n^3}{3}\right )}{e}-\frac {d\,\left (6\,a^3-6\,a\,b^2\,n^2+5\,b^3\,n^3\right )}{2\,e}\right )}{e}+\frac {b^2\,d^2\,n^2\,\left (6\,a-11\,b\,n\right )}{e^2}}{x^{1/3}}-\frac {\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\,\left (\frac {\frac {d\,\left (b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )\right )}{e}+6\,b^3\,d^2\,n^2}{e\,x^{1/3}}-\frac {b\,d\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )-3\,b\,d\,e\,\left (3\,a^2-b^2\,n^2\right )}{2\,e\,x^{2/3}}+\frac {b\,e\,\left (9\,a^2-6\,a\,b\,n+2\,b^2\,n^2\right )}{3\,x}\right )}{e}-\frac {\ln \left (d+\frac {e}{x^{1/3}}\right )\,\left (18\,a^2\,b\,d^3\,n-66\,a\,b^2\,d^3\,n^2+85\,b^3\,d^3\,n^3\right )}{6\,e^3} \] Input:
int((a + b*log(c*(d + e/x^(1/3))^n))^3/x^2,x)
Output:
((d*(3*a^3 - (2*b^3*n^3)/3 + 2*a*b^2*n^2 - 3*a^2*b*n))/(2*e) - (d*(6*a^3 + 5*b^3*n^3 - 6*a*b^2*n^2))/(4*e))/x^(2/3) - log(c*(d + e/x^(1/3))^n)^3*(b^ 3/x + (b^3*d^3)/e^3) - log(c*(d + e/x^(1/3))^n)^2*((b^2*(3*a - b*n))/x - ( (3*b^2*d*(3*a - b*n))/(2*e) - (9*a*b^2*d)/(2*e))/x^(2/3) + (d*(6*a*b^2*d^2 - 11*b^3*d^2*n))/(2*e^3) + (d*((3*b^2*d*(3*a - b*n))/e - (9*a*b^2*d)/e))/ (e*x^(1/3))) - (a^3 - (2*b^3*n^3)/9 + (2*a*b^2*n^2)/3 - a^2*b*n)/x - ((d*( (d*(3*a^3 - (2*b^3*n^3)/3 + 2*a*b^2*n^2 - 3*a^2*b*n))/e - (d*(6*a^3 + 5*b^ 3*n^3 - 6*a*b^2*n^2))/(2*e)))/e + (b^2*d^2*n^2*(6*a - 11*b*n))/e^2)/x^(1/3 ) - (log(c*(d + e/x^(1/3))^n)*(((d*(b*d*e*(9*a^2 + 2*b^2*n^2 - 6*a*b*n) - 3*b*d*e*(3*a^2 - b^2*n^2)))/e + 6*b^3*d^2*n^2)/(e*x^(1/3)) - (b*d*e*(9*a^2 + 2*b^2*n^2 - 6*a*b*n) - 3*b*d*e*(3*a^2 - b^2*n^2))/(2*e*x^(2/3)) + (b*e* (9*a^2 + 2*b^2*n^2 - 6*a*b*n))/(3*x)))/e - (log(d + e/x^(1/3))*(85*b^3*d^3 *n^3 - 66*a*b^2*d^3*n^2 + 18*a^2*b*d^3*n))/(6*e^3)
Time = 0.16 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.60 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^2} \, dx =\text {Too large to display} \] Input:
int((a+b*log(c*(d+e/x^(1/3))^n))^3/x^2,x)
Output:
(108*x**(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d**2*e*n + 216 *x**(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b**2*d**2*e*n - 396*x**( 2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**3*d**2*e*n**2 + 108*x**(2/3) *a**2*b*d**2*e*n - 396*x**(2/3)*a*b**2*d**2*e*n**2 + 510*x**(2/3)*b**3*d** 2*e*n**3 - 54*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d*e** 2*n - 108*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b**2*d*e**2*n + 90*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**3*d*e**2*n**2 - 54*x **(1/3)*a**2*b*d*e**2*n + 90*x**(1/3)*a*b**2*d*e**2*n**2 - 57*x**(1/3)*b** 3*d*e**2*n**3 - 36*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**3*b**3*d**3*x - 36*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**3*b**3*e**3 - 108*log(((x**(1/3) *d + e)**n*c)/x**(n/3))**2*a*b**2*d**3*x - 108*log(((x**(1/3)*d + e)**n*c) /x**(n/3))**2*a*b**2*e**3 + 198*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b **3*d**3*n*x + 36*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*e**3*n - 1 08*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a**2*b*d**3*x - 108*log(((x**(1/3 )*d + e)**n*c)/x**(n/3))*a**2*b*e**3 + 396*log(((x**(1/3)*d + e)**n*c)/x** (n/3))*a*b**2*d**3*n*x + 72*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b**2*e **3*n - 510*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**3*d**3*n**2*x - 24*lo g(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**3*e**3*n**2 - 36*a**3*e**3 + 36*a** 2*b*e**3*n - 24*a*b**2*e**3*n**2 + 8*b**3*e**3*n**3)/(36*e**3*x)