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

Optimal. Leaf size=907 \[ \text{result too large to display} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.0013, antiderivative size = 907, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304} \[ \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}} \]

Antiderivative was successfully verified.

[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 2454

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])

Rule 2401

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 2389

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 2296

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 2295

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

Rule 2390

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 2305

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 2304

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
]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{x^3} \, dx &=-\left (3 \operatorname{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 \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 1.63296, size = 962, normalized size = 1.06 \[ \frac{-72000 b^3 n^3 x^2 \log ^3\left (d+\frac{e}{\sqrt [3]{x}}\right ) d^6+809340 b^3 n^3 x^2 \log \left (\sqrt [3]{x} d+e\right ) d^6-529200 a b^2 n^2 x^2 \log \left (\sqrt [3]{x} d+e\right ) d^6+108000 a^2 b n x^2 \log \left (\sqrt [3]{x} d+e\right ) d^6+3600 b^2 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 (\sqrt [3]{x} d+e\right )-\log (x)\right ) d^6-269780 b^3 n^3 x^2 \log (x) d^6+176400 a b^2 n^2 x^2 \log (x) d^6-36000 a^2 b n x^2 \log (x) d^6+1800 b^2 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 (\sqrt [3]{x} d+e\right )-20 b n \log (x)\right ) d^6-809340 b^3 e n^3 x^{5/3} d^5+529200 a b^2 e n^2 x^{5/3} d^5-108000 a^2 b e n x^{5/3} d^5+140070 b^3 e^2 n^3 x^{4/3} d^4-156600 a b^2 e^2 n^2 x^{4/3} d^4+54000 a^2 b e^2 n x^{4/3} d^4-41180 b^3 e^3 n^3 x d^3+68400 a b^2 e^3 n^2 x d^3-36000 a^2 b e^3 n x d^3+13785 b^3 e^4 n^3 x^{2/3} d^2-33300 a b^2 e^4 n^2 x^{2/3} d^2+27000 a^2 b e^4 n x^{2/3} d^2-4368 b^3 e^5 n^3 \sqrt [3]{x} d+15840 a b^2 e^5 n^2 \sqrt [3]{x} d-21600 a^2 b e^5 n \sqrt [3]{x} d-36000 a^3 e^6+1000 b^3 e^6 n^3-36000 b^3 e^6 \log ^3\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-6000 a b^2 e^6 n^2+18000 a^2 b e^6 n+1800 b^2 \log ^2\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \left (60 b n x^2 \log \left (\sqrt [3]{x} d+e\right ) d^6-20 b n x^2 \log (x) d^6+e \left (-60 b n x^{5/3} d^5+30 b e n x^{4/3} d^4-20 b e^2 n x d^3+15 b e^3 n x^{2/3} d^2-12 b e^4 n \sqrt [3]{x} d-60 a e^5+10 b e^5 n\right )\right )-60 b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \left (180 b n (49 b n-20 a) x^2 \log \left (\sqrt [3]{x} d+e\right ) d^6+60 b n (20 a-49 b n) x^2 \log (x) d^6+1800 a^2 e^6+b^2 e n^2 \left (-8820 x^{5/3} d^5+2610 e x^{4/3} d^4-1140 e^2 x d^3+555 e^3 x^{2/3} d^2-264 e^4 \sqrt [3]{x} d+100 e^5\right )-60 a b e n \left (-60 x^{5/3} d^5+30 e x^{4/3} d^4-20 e^2 x d^3+15 e^3 x^{2/3} d^2-12 e^4 \sqrt [3]{x} d+10 e^5\right )\right )}{72000 e^6 x^2} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 0.353, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt [3]{x}}}} \right ) ^{n} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 1.17795, size = 1166, normalized size = 1.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.42146, size = 3106, normalized size = 3.42 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{1}{3}}}\right )}^{n}\right ) + a\right )}^{3}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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