\(\int \frac {(a+b \log (c (d+\frac {e}{\sqrt [3]{x}})^n))^3}{x^3} \, dx\) [507]

Optimal result

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} \]

[Out]

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+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+18*a*b^2*d^5*n^2/e^5/x^(1/3)-45/16*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*ln(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

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 28, 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 \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\frac {b^3 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^6}{72 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}{2 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}{4 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}{12 e^6}-\frac {18 b^3 d n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^5}{125 e^6}+\frac {3 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 {9 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 {18 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 {45 b^3 d^2 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^4}{64 e^6}-\frac {15 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 {45 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 {45 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}{9 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}{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}{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}{3 e^6}+\frac {45 b^3 d^4 n^3 \left (d+\frac {e}{\sqrt [3]{x}}\right )^2}{8 e^6}-\frac {15 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 {45 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 {45 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 {3 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 {9 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 {18 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 {18 b^3 d^5 n^3}{e^5 \sqrt [3]{x}}+\frac {18 a b^2 d^5 n^2}{e^5 \sqrt [3]{x}} \]

[In]

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

[Out]

(45*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)/(9*e^6) + (45*b^3*d^2*n^3*(d +
e/x^(1/3))^4)/(64*e^6) - (18*b^3*d*n^3*(d + e/x^(1/3))^5)/(125*e^6) + (b^3*n^3*(d + e/x^(1/3))^6)/(72*e^6) +
(18*a*b^2*d^5*n^2)/(e^5*x^(1/3)) - (18*b^3*d^5*n^3)/(e^5*x^(1/3)) + (18*b^3*d^5*n^2*(d + e/x^(1/3))*Log[c*(d +
e/x^(1/3))^n])/e^6 - (45*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]))/(3*e^6) - (45*b^2*d^2*n^2*(d + e/x^(1/3))^4*(a + b*L
og[c*(d + e/x^(1/3))^n]))/(16*e^6) + (18*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]))/(12*e^6) - (9*b*d^5*n*(d + e/x^(1/3))*(a + b*
Log[c*(d + e/x^(1/3))^n])^2)/e^6 + (45*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)/e^6 + (45*b*d^2*n*(d + e/x^(1/3))^4*(a + b*
Log[c*(d + e/x^(1/3))^n])^2)/(8*e^6) - (9*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)/(4*e^6) + (3*d^5*(d + e/x^(1/3))*(a + b*Log[c*(d
+ e/x^(1/3))^n])^3)/e^6 - (15*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)/e^6 - (15*d^2*(d + e/x^(1/3))^4*(a + b*Log[c*(d + e/x^(1/3))
^n])^3)/(2*e^6) + (3*d*(d + e/x^(1/3))^5*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^6 - ((d + e/x^(1/3))^6*(a + b*L
og[c*(d + e/x^(1/3))^n])^3)/(2*e^6)

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}& = -\left (3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\left (3 \text {Subst}\left (\int \left (-\frac {d^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {5 d^4 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {10 d^3 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {10 d^2 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}-\frac {5 d (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}+\frac {(d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3}{e^5}\right ) \, dx,x,\frac {1}{\sqrt [3]{x}}\right )\right ) \\ & = -\frac {3 \text {Subst}\left (\int (d+e x)^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {(15 d) \text {Subst}\left (\int (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}-\frac {\left (30 d^2\right ) \text {Subst}\left (\int (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {\left (30 d^3\right ) \text {Subst}\left (\int (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}-\frac {\left (15 d^4\right ) \text {Subst}\left (\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5}+\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c (d+e x)^n\right )\right )^3 \, dx,x,\frac {1}{\sqrt [3]{x}}\right )}{e^5} \\ & = -\frac {3 \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {(15 d) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (30 d^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (30 d^3\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (15 d^4\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (3 d^5\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^3 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{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}+\frac {(3 b n) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {(9 b d n) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (45 b d^2 n\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {\left (30 b d^3 n\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}+\frac {\left (45 b d^4 n\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}-\frac {\left (9 b d^5 n\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{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}-\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int x^5 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {\left (18 b^2 d n^2\right ) \text {Subst}\left (\int x^4 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{5 e^6}-\frac {\left (45 b^2 d^2 n^2\right ) \text {Subst}\left (\int x^3 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{4 e^6}+\frac {\left (20 b^2 d^3 n^2\right ) \text {Subst}\left (\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6}-\frac {\left (45 b^2 d^4 n^2\right ) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{2 e^6}+\frac {\left (18 b^2 d^5 n^2\right ) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6} \\ & = \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 {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}+\frac {\left (18 b^3 d^5 n^2\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+\frac {e}{\sqrt [3]{x}}\right )}{e^6} \\ & = \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} \\ \end{align*}

Mathematica [A] (verified)

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} \]

[In]

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

[Out]

(-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) + 15
840*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 + 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[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 + 49*b*n - 20*b*Log[c*(d + e/x^(1/3))^n])*(3*Log[e + d*x^(1/3)] - Log
[x]) - 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[d + e/x^(1/3)]^2*(60*a - 147*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) + 1
5*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*e^2*x + 2610*d^4*e*x^(4/3) - 8820*d^5*x^(5/3)) - 60*a*b*e*n*(10
*e^5 - 12*d*e^4*x^(1/3) + 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)) + 180*b*d^6*n
*(-20*a + 49*b*n)*x^2*Log[e + d*x^(1/3)] + 60*b*d^6*n*(20*a - 49*b*n)*x^2*Log[x]))/(72000*e^6*x^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 1404, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 - 600*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 - 2
0*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*n^2)*log(c))*x^(2/3) - 6*(44*b^3*d*e^5*n^3 - 120*a*b^2*d*e^5*n^
2 - 15*(29*b^3*d^4*e^2*n^3 - 20*a*b^2*d^4*e^2*n^2)*x + 60*(5*b^3*d^4*e^2*n^2*x - 2*b^3*d*e^5*n^2)*log(c))*x^(1
/3))*log((d*x + e*x^(2/3))/x) + 15*(919*b^3*d^2*e^4*n^3 - 2220*a*b^2*d^2*e^4*n^2 + 1800*a^2*b*d^2*e^4*n - 1800
*(4*b^3*d^5*e*n*x - b^3*d^2*e^4*n)*log(c)^2 - 4*(13489*b^3*d^5*e*n^3 - 8820*a*b^2*d^5*e*n^2 + 1800*a^2*b*d^5*e
*n)*x - 60*(37*b^3*d^2*e^4*n^2 - 60*a*b^2*d^2*e^4*n - 12*(49*b^3*d^5*e*n^2 - 20*a*b^2*d^5*e*n)*x)*log(c))*x^(2
/3) - 6*(728*b^3*d*e^5*n^3 - 2640*a*b^2*d*e^5*n^2 + 3600*a^2*b*d*e^5*n - 1800*(5*b^3*d^4*e^2*n*x - 2*b^3*d*e^5
*n)*log(c)^2 - 5*(4669*b^3*d^4*e^2*n^3 - 5220*a*b^2*d^4*e^2*n^2 + 1800*a^2*b*d^4*e^2*n)*x - 60*(44*b^3*d*e^5*n
^2 - 120*a*b^2*d*e^5*n - 15*(29*b^3*d^4*e^2*n^2 - 20*a*b^2*d^4*e^2*n)*x)*log(c))*x^(1/3))/(e^6*x^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 864, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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
+ 1/216000*(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*(200*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 - 60*(20*d^6*x^2*log(x) - 147*d^6*x^2)*log(d*x^(1/3) + e))*n*log
(c*(d + e/x^(1/3))^n)/(e^7*x^2)))*b^3 - 1/2*b^3*log(c*(d + e/x^(1/3))^n)^3/x^2 - 3/2*a*b^2*log(c*(d + e/x^(1/3
))^n)^2/x^2 - 3/2*a^2*b*log(c*(d + e/x^(1/3))^n)/x^2 - 1/2*a^3/x^2

Giac [B] (verification not implemented)

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 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

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 - 6
0*(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) + 13500*(b^3*d^4*n^3 - 2*b^3*d^4*n^2*log(c) + 2*b^3*d^4*n*log(c)^2 -
2*a*b^2*d^4*n^2 + 4*a*b^2*d^4*n*log(c) + 2*a^2*b*d^4*n)*(d*x^(1/3) + e)^2/(e^5*x^(2/3)) - 10800*(2*b^3*d^5*n^3
- 2*b^3*d^5*n^2*log(c) + b^3*d^5*n*log(c)^2 - 2*a*b^2*d^5*n^2 + 2*a*b^2*d^5*n*log(c) + a^2*b*d^5*n)*(d*x^(1/3
) + e)/(e^5*x^(1/3)))*log((d*x^(1/3) + e)/x^(1/3)) + 1000*(b^3*n^3 - 6*b^3*n^2*log(c) + 18*b^3*n*log(c)^2 - 36
*b^3*log(c)^3 - 6*a*b^2*n^2 + 36*a*b^2*n*log(c) - 108*a*b^2*log(c)^2 + 18*a^2*b*n - 108*a^2*b*log(c) - 36*a^3)
*(d*x^(1/3) + e)^6/(e^5*x^2) - 1728*(6*b^3*d*n^3 - 30*b^3*d*n^2*log(c) + 75*b^3*d*n*log(c)^2 - 125*b^3*d*log(c
)^3 - 30*a*b^2*d*n^2 + 150*a*b^2*d*n*log(c) - 375*a*b^2*d*log(c)^2 + 75*a^2*b*d*n - 375*a^2*b*d*log(c) - 125*a
^3*d)*(d*x^(1/3) + e)^5/(e^5*x^(5/3)) + 16875*(3*b^3*d^2*n^3 - 12*b^3*d^2*n^2*log(c) + 24*b^3*d^2*n*log(c)^2 -
32*b^3*d^2*log(c)^3 - 12*a*b^2*d^2*n^2 + 48*a*b^2*d^2*n*log(c) - 96*a*b^2*d^2*log(c)^2 + 24*a^2*b*d^2*n - 96*
a^2*b*d^2*log(c) - 32*a^3*d^2)*(d*x^(1/3) + e)^4/(e^5*x^(4/3)) - 80000*(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 - 9*b^3*d^3*log(c)^3 - 6*a*b^2*d^3*n^2 + 18*a*b^2*d^3*n*log(c) - 27*a*b^2*d^3*log(c)^2 +
9*a^2*b*d^3*n - 27*a^2*b*d^3*log(c) - 9*a^3*d^3)*(d*x^(1/3) + e)^3/(e^5*x) + 135000*(3*b^3*d^4*n^3 - 6*b^3*d^
4*n^2*log(c) + 6*b^3*d^4*n*log(c)^2 - 4*b^3*d^4*log(c)^3 - 6*a*b^2*d^4*n^2 + 12*a*b^2*d^4*n*log(c) - 12*a*b^2*
d^4*log(c)^2 + 6*a^2*b*d^4*n - 12*a^2*b*d^4*log(c) - 4*a^3*d^4)*(d*x^(1/3) + e)^2/(e^5*x^(2/3)) - 216000*(6*b^
3*d^5*n^3 - 6*b^3*d^5*n^2*log(c) + 3*b^3*d^5*n*log(c)^2 - b^3*d^5*log(c)^3 - 6*a*b^2*d^5*n^2 + 6*a*b^2*d^5*n*l
og(c) - 3*a*b^2*d^5*log(c)^2 + 3*a^2*b*d^5*n - 3*a^2*b*d^5*log(c) - a^3*d^5)*(d*x^(1/3) + e)/(e^5*x^(1/3)))/e

Mupad [B] (verification not implemented)

Time = 9.28 (sec) , antiderivative size = 992, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(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*log(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))/(8
0*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*(d + e/x^(1/3))^n)^2)/(2*e^5*x^(1/3)) + (147*b^3*d^5*n^2*log(c*(
d + e/x^(1/3))^n))/(20*e^5*x^(1/3)) + (19*a*b^2*d^3*n^2)/(20*e^3*x) - (37*a*b^2*d^2*n^2)/(80*e^2*x^(4/3)) - (8
7*a*b^2*d^4*n^2)/(40*e^4*x^(2/3)) + (147*a*b^2*d^5*n^2)/(20*e^5*x^(1/3)) - (3*a^2*b*d*n)/(10*e*x^(5/3)) - (3*a
*b^2*d*n*log(c*(d + e/x^(1/3))^n))/(5*e*x^(5/3)) - (a*b^2*d^3*n*log(c*(d + e/x^(1/3))^n))/(e^3*x) + (3*a*b^2*d
^2*n*log(c*(d + e/x^(1/3))^n))/(4*e^2*x^(4/3)) + (3*a*b^2*d^4*n*log(c*(d + e/x^(1/3))^n))/(2*e^4*x^(2/3)) - (3
*a*b^2*d^5*n*log(c*(d + e/x^(1/3))^n))/(e^5*x^(1/3))