3.506 \(\int \frac{(a+b \log (c (d+\frac{e}{\sqrt [3]{x}})^n))^3}{x^2} \, dx\)

Optimal. Leaf size=438 \[ -\frac{2 b^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}+\frac{9 b^2 d n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^3}-\frac{18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac{9 b d^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac{3 d^2 \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac{b n \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac{9 b d n \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 e^3}-\frac{\left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac{3 d \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}-\frac{18 b^3 d^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac{18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}+\frac{2 b^3 n^3 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac{9 b^3 d n^3 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}{4 e^3} \]

[Out]

(-9*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)/(9*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))*Log[c*(d + e/x^(1/3))^n])/e^3 + (9*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]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 - (9*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)/e^3 - (3*d^2*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 + (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)/e^3

________________________________________________________________________________________

Rubi [A]  time = 0.452248, antiderivative size = 438, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{2 b^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 e^3}+\frac{9 b^2 d n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 e^3}-\frac{18 a b^2 d^2 n^2}{e^2 \sqrt [3]{x}}+\frac{9 b d^2 n \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac{3 d^2 \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac{b n \left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{e^3}-\frac{9 b d n \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{2 e^3}-\frac{\left (d+\frac{e}{\sqrt [3]{x}}\right )^3 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}+\frac{3 d \left (d+\frac{e}{\sqrt [3]{x}}\right )^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )\right )^3}{e^3}-\frac{18 b^3 d^2 n^2 \left (d+\frac{e}{\sqrt [3]{x}}\right ) \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )}{e^3}+\frac{18 b^3 d^2 n^3}{e^2 \sqrt [3]{x}}+\frac{2 b^3 n^3 \left (d+\frac{e}{\sqrt [3]{x}}\right )^3}{9 e^3}-\frac{9 b^3 d n^3 \left (d+\frac{e}{\sqrt [3]{x}}\right )^2}{4 e^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,x]

[Out]

(-9*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)/(9*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))*Log[c*(d + e/x^(1/3))^n])/e^3 + (9*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]))/(3*e^3) + (9*b*d^2*n*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/e^3 - (9*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)/e^3 - (3*d^2*(d + e/x^(1/3))*(a + b*Log[c*(d + e/x^(1/3))^n])^3)/e^3 + (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)/e^3

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

Mathematica [A]  time = 0.794028, size = 666, normalized size = 1.52 \[ \frac{-6 b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \left (18 a^2 e^3-6 a b e n \left (6 d^2 x^{2/3}-3 d e \sqrt [3]{x}+2 e^2\right )+6 b d^3 n x (6 a-11 b n) \log \left (d \sqrt [3]{x}+e\right )+2 b d^3 n x \log (x) (11 b n-6 a)+b^2 e n^2 \left (66 d^2 x^{2/3}-15 d e \sqrt [3]{x}+4 e^2\right )\right )+108 a^2 b d^2 e n x^{2/3}-108 a^2 b d^3 n x \log \left (d \sqrt [3]{x}+e\right )+36 a^2 b d^3 n x \log (x)-54 a^2 b d e^2 n \sqrt [3]{x}+36 a^2 b e^3 n-36 a^3 e^3+18 b^2 \log ^2\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right ) \left (e \left (-6 a e^2+6 b d^2 n x^{2/3}-3 b d e n \sqrt [3]{x}+2 b e^2 n\right )-6 b d^3 n x \log \left (d \sqrt [3]{x}+e\right )+2 b d^3 n x \log (x)\right )-18 b^2 d^3 n^2 x \log ^2\left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (6 a+6 b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+6 b n \log \left (d \sqrt [3]{x}+e\right )-2 b n \log (x)-11 b n\right )+12 b^2 d^3 n^2 x \log \left (d+\frac{e}{\sqrt [3]{x}}\right ) \left (3 \log \left (d \sqrt [3]{x}+e\right )-\log (x)\right ) \left (6 a+6 b \log \left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )-11 b n\right )-396 a b^2 d^2 e n^2 x^{2/3}+396 a b^2 d^3 n^2 x \log \left (d \sqrt [3]{x}+e\right )-132 a b^2 d^3 n^2 x \log (x)+90 a b^2 d e^2 n^2 \sqrt [3]{x}-24 a b^2 e^3 n^2-36 b^3 e^3 \log ^3\left (c \left (d+\frac{e}{\sqrt [3]{x}}\right )^n\right )+510 b^3 d^2 e n^3 x^{2/3}+72 b^3 d^3 n^3 x \log ^3\left (d+\frac{e}{\sqrt [3]{x}}\right )-510 b^3 d^3 n^3 x \log \left (d \sqrt [3]{x}+e\right )+170 b^3 d^3 n^3 x \log (x)-57 b^3 d e^2 n^3 \sqrt [3]{x}+8 b^3 e^3 n^3}{36 e^3 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(1/3))^n])^3/x^2,x]

[Out]

(-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*Lo
g[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)

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Maple [F]  time = 0.348, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

int((a+b*ln(c*(d+e/x^(1/3))^n))^3/x^2,x)

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Maxima [A]  time = 1.16536, size = 864, normalized size = 1.97 \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^2,x, algorithm="maxima")

[Out]

-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))*log(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) + 171*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 - 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*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

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Fricas [B]  time = 2.22042, size = 1781, normalized size = 4.07 \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^2,x, algorithm="fricas")

[Out]

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)

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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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(1/3))^n) + a)^3/x^2, x)