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

Optimal. Leaf size=569 \[ -\frac{3 b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}+\frac{5 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{d}{d+\frac{e}{\sqrt{x}}}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{d^4}-\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (3,\frac{d}{d+\frac{e}{\sqrt{x}}}\right )}{d^4}-\frac{5 b^2 e^4 n^2 \log \left (1-\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}-\frac{3 b^2 e^4 n^2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{5 b^2 e^3 n^2 \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}+\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{3 b e^4 n \log \left (1-\frac{d}{d+\frac{e}{\sqrt{x}}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}+\frac{3 b e^3 n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3+\frac{b^3 e^3 n^3 \sqrt{x}}{2 d^3}-\frac{b^3 e^4 n^3 \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4} \]

[Out]

(b^3*e^3*n^3*Sqrt[x])/(2*d^3) - (b^3*e^4*n^3*Log[d + e/Sqrt[x]])/(2*d^4) - (5*b^2*e^3*n^2*(d + e/Sqrt[x])*Sqrt
[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (b^2*e^2*n^2*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^2) - (5*
b^2*e^4*n^2*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (3*b*e^3*n*(d + e/Sqrt[x])*
Sqrt[x]*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d^4) - (3*b*e^2*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*d^2)
+ (b*e*n*x^(3/2)*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(2*d) + (3*b*e^4*n*Log[1 - d/(d + e/Sqrt[x])]*(a + b*Log
[c*(d + e/Sqrt[x])^n])^2)/(2*d^4) + (x^2*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/2 - (3*b^2*e^4*n^2*(a + b*Log[c*(
d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))])/d^4 - (3*b^3*e^4*n^3*Log[x])/(2*d^4) + (5*b^3*e^4*n^3*PolyLog[2, d/(
d + e/Sqrt[x])])/(2*d^4) - (3*b^2*e^4*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, d/(d + e/Sqrt[x])])/d^4
- (3*b^3*e^4*n^3*PolyLog[2, 1 + e/(d*Sqrt[x])])/d^4 - (3*b^3*e^4*n^3*PolyLog[3, d/(d + e/Sqrt[x])])/d^4

________________________________________________________________________________________

Rubi [A]  time = 1.49154, antiderivative size = 546, normalized size of antiderivative = 0.96, number of steps used = 35, number of rules used = 17, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.773, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31, 44} \[ \frac{3 b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{d^4}-\frac{11 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right )}{d^4}-\frac{11 b^2 e^4 n^2 \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}-\frac{5 b^2 e^3 n^2 \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}+\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 d^4}+\frac{5 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^4}+\frac{3 b e^4 n \log \left (-\frac{e}{d \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}+\frac{3 b e^3 n \sqrt{x} \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3+\frac{b^3 e^3 n^3 \sqrt{x}}{2 d^3}-\frac{b^3 e^4 n^3 \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e^3*n^3*Sqrt[x])/(2*d^3) - (b^3*e^4*n^3*Log[d + e/Sqrt[x]])/(2*d^4) - (5*b^2*e^3*n^2*(d + e/Sqrt[x])*Sqrt
[x]*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^4) + (b^2*e^2*n^2*x*(a + b*Log[c*(d + e/Sqrt[x])^n]))/(2*d^2) + (5*
b*e^4*n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*d^4) + (3*b*e^3*n*(d + e/Sqrt[x])*Sqrt[x]*(a + b*Log[c*(d + e/S
qrt[x])^n])^2)/(2*d^4) - (3*b*e^2*n*x*(a + b*Log[c*(d + e/Sqrt[x])^n])^2)/(4*d^2) + (b*e*n*x^(3/2)*(a + b*Log[
c*(d + e/Sqrt[x])^n])^2)/(2*d) - (e^4*(a + b*Log[c*(d + e/Sqrt[x])^n])^3)/(2*d^4) + (x^2*(a + b*Log[c*(d + e/S
qrt[x])^n])^3)/2 - (11*b^2*e^4*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*Log[-(e/(d*Sqrt[x]))])/(2*d^4) + (3*b*e^4*
n*(a + b*Log[c*(d + e/Sqrt[x])^n])^2*Log[-(e/(d*Sqrt[x]))])/(2*d^4) - (3*b^3*e^4*n^3*Log[x])/(2*d^4) - (11*b^3
*e^4*n^3*PolyLog[2, 1 + e/(d*Sqrt[x])])/(2*d^4) + (3*b^2*e^4*n^2*(a + b*Log[c*(d + e/Sqrt[x])^n])*PolyLog[2, 1
+ e/(d*Sqrt[x])])/d^4 - (3*b^3*e^4*n^3*PolyLog[3, 1 + e/(d*Sqrt[x])])/d^4

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 2398

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
f + g*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n])^p)/(g*(q + 1)), x] - Dist[(b*e*n*p)/(g*(q + 1)), Int[((f + g*x)^(q
+ 1)*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2347

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[((
d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2344

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2318

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])
^p)/(d*(d + e*x)), x] - Dist[(b*n*p)/d, Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[(x*(d + e*x^r)^(q
+ 1)*(a + b*Log[c*x^n]))/d, x] - Dist[(b*n)/d, Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3 \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x^5} \, dx,x,\frac{1}{\sqrt{x}}\right )\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{1}{2} (3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{x^4 (d+e x)} \, dx,x,\frac{1}{\sqrt{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{1}{2} (3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{\sqrt{x}}\right )\\ &=\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{(3 b n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^4} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d}+\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d}\\ &=\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3+\frac{(3 b e n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}-\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d}\\ &=-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{\left (3 b e^2 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^3}+\frac{\left (3 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^3}-\frac{\left (b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^3} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^2}+\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}\\ &=\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{3 b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3+\frac{\left (3 b e^3 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{\left (3 b e^4 n\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}+\frac{\left (b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^3}+\frac{\left (3 b^2 e^2 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{\left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^3}-\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^3}-\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x \left (-\frac{d}{e}+\frac{x}{e}\right )} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^3}-\frac{\left (b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (-\frac{d}{e}+\frac{x}{e}\right )^2} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}\\ &=-\frac{5 b^2 e^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}+\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{3 b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{d^4}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{2 d^4}-\frac{\left (3 e^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}-\frac{\left (b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}-\frac{\left (3 b^2 e^3 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}+\frac{\left (b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}+\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c x^n\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{\left (3 b^2 e^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}-\frac{\left (b^3 e^2 n^3\right ) \operatorname{Subst}\left (\int \left (\frac{e^2}{d (d-x)^2}+\frac{e^2}{d^2 (d-x)}+\frac{e^2}{d^2 x}\right ) \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^2}+\frac{\left (b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}+\frac{\left (3 b^3 e^3 n^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{d}{e}+\frac{x}{e}} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}\\ &=\frac{b^3 e^3 n^3 \sqrt{x}}{2 d^3}-\frac{b^3 e^4 n^3 \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{5 b^2 e^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}+\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{5 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^4}+\frac{3 b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 d^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{11 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{2 d^4}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac{3 b^3 e^4 n^3 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^4}+\frac{3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^4}+\frac{\left (b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}+\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{\left (3 b^3 e^4 n^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+\frac{e}{\sqrt{x}}\right )}{d^4}\\ &=\frac{b^3 e^3 n^3 \sqrt{x}}{2 d^3}-\frac{b^3 e^4 n^3 \log \left (d+\frac{e}{\sqrt{x}}\right )}{2 d^4}-\frac{5 b^2 e^3 n^2 \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^4}+\frac{b^2 e^2 n^2 x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )}{2 d^2}+\frac{5 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^4}+\frac{3 b e^3 n \left (d+\frac{e}{\sqrt{x}}\right ) \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{4 d^2}+\frac{b e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2}{2 d}-\frac{e^4 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3}{2 d^4}+\frac{1}{2} x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^3-\frac{11 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \log \left (-\frac{e}{d \sqrt{x}}\right )}{2 d^4}+\frac{3 b e^4 n \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right )^2 \log \left (-\frac{e}{d \sqrt{x}}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4}-\frac{11 b^3 e^4 n^3 \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{2 d^4}+\frac{3 b^2 e^4 n^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )\right ) \text{Li}_2\left (1+\frac{e}{d \sqrt{x}}\right )}{d^4}-\frac{3 b^3 e^4 n^3 \text{Li}_3\left (1+\frac{e}{d \sqrt{x}}\right )}{d^4}\\ \end{align*}

Mathematica [A]  time = 0.977875, size = 777, normalized size = 1.37 \[ \frac{-2 b^2 n^2 \left (-6 e^4 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+3 \left (e^4-d^4 x^2\right ) \log ^2\left (d+\frac{e}{\sqrt{x}}\right )-e \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (-3 d^2 e x+2 d^3 x^{3/2}+6 d e^2 \sqrt{x}+6 e^3 \log \left (-\frac{e}{d \sqrt{x}}\right )+11 e^3\right )+e^2 \left (d^2 (-x)+11 e^2 \log \left (-\frac{e}{d \sqrt{x}}\right )+5 d e \sqrt{x}\right )\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )+b^3 n^3 \left (-2 e^4 \left (6 \text{PolyLog}\left (3,\frac{e}{d \sqrt{x}}+1\right )-6 \log \left (d+\frac{e}{\sqrt{x}}\right ) \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )+\left (\log \left (d+\frac{e}{\sqrt{x}}\right )-3 \log \left (-\frac{e}{d \sqrt{x}}\right )\right ) \log ^2\left (d+\frac{e}{\sqrt{x}}\right )\right )+11 e^4 \left (\log \left (d+\frac{e}{\sqrt{x}}\right ) \left (\log \left (d+\frac{e}{\sqrt{x}}\right )-2 \log \left (-\frac{e}{d \sqrt{x}}\right )\right )-2 \text{PolyLog}\left (2,\frac{e}{d \sqrt{x}}+1\right )\right )+d^2 e^2 x \left (2-3 \log \left (d+\frac{e}{\sqrt{x}}\right )\right ) \log \left (d+\frac{e}{\sqrt{x}}\right )+2 d^4 x^2 \log ^3\left (d+\frac{e}{\sqrt{x}}\right )+2 d^3 e x^{3/2} \log ^2\left (d+\frac{e}{\sqrt{x}}\right )+2 d e^3 \sqrt{x} \left (3 \log ^2\left (d+\frac{e}{\sqrt{x}}\right )-5 \log \left (d+\frac{e}{\sqrt{x}}\right )+1\right )+12 e^4 \left (\log \left (-\frac{e}{d \sqrt{x}}\right )-\log \left (d+\frac{e}{\sqrt{x}}\right )\right )\right )-3 b d^2 e^2 n x \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2+2 d^4 x^2 \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^3+6 b d^4 n x^2 \log \left (d+\frac{e}{\sqrt{x}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2+2 b d^3 e n x^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2+6 b d e^3 n \sqrt{x} \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2-6 b e^4 n \log \left (d \sqrt{x}+e\right ) \left (a+b \log \left (c \left (d+\frac{e}{\sqrt{x}}\right )^n\right )-b n \log \left (d+\frac{e}{\sqrt{x}}\right )\right )^2}{4 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*b*d*e^3*n*Sqrt[x]*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^2 - 3*b*d^2*e^2*n*x*(a - b*n*Lo
g[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^2 + 2*b*d^3*e*n*x^(3/2)*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(
d + e/Sqrt[x])^n])^2 + 6*b*d^4*n*x^2*Log[d + e/Sqrt[x]]*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^
n])^2 + 2*d^4*x^2*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^3 - 6*b*e^4*n*(a - b*n*Log[d + e/S
qrt[x]] + b*Log[c*(d + e/Sqrt[x])^n])^2*Log[e + d*Sqrt[x]] - 2*b^2*n^2*(a - b*n*Log[d + e/Sqrt[x]] + b*Log[c*(
d + e/Sqrt[x])^n])*(3*(e^4 - d^4*x^2)*Log[d + e/Sqrt[x]]^2 + e^2*(5*d*e*Sqrt[x] - d^2*x + 11*e^2*Log[-(e/(d*Sq
rt[x]))]) - e*Log[d + e/Sqrt[x]]*(11*e^3 + 6*d*e^2*Sqrt[x] - 3*d^2*e*x + 2*d^3*x^(3/2) + 6*e^3*Log[-(e/(d*Sqrt
[x]))]) - 6*e^4*PolyLog[2, 1 + e/(d*Sqrt[x])]) + b^3*n^3*(d^2*e^2*x*(2 - 3*Log[d + e/Sqrt[x]])*Log[d + e/Sqrt[
x]] + 2*d^3*e*x^(3/2)*Log[d + e/Sqrt[x]]^2 + 2*d^4*x^2*Log[d + e/Sqrt[x]]^3 + 2*d*e^3*Sqrt[x]*(1 - 5*Log[d + e
/Sqrt[x]] + 3*Log[d + e/Sqrt[x]]^2) + 12*e^4*(-Log[d + e/Sqrt[x]] + Log[-(e/(d*Sqrt[x]))]) + 11*e^4*(Log[d + e
/Sqrt[x]]*(Log[d + e/Sqrt[x]] - 2*Log[-(e/(d*Sqrt[x]))]) - 2*PolyLog[2, 1 + e/(d*Sqrt[x])]) - 2*e^4*(Log[d + e
/Sqrt[x]]^2*(Log[d + e/Sqrt[x]] - 3*Log[-(e/(d*Sqrt[x]))]) - 6*Log[d + e/Sqrt[x]]*PolyLog[2, 1 + e/(d*Sqrt[x])
] + 6*PolyLog[3, 1 + e/(d*Sqrt[x])])))/(4*d^4)

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Maple [F]  time = 0.427, size = 0, normalized size = 0. \begin{align*} \int x \left ( a+b\ln \left ( c \left ( d+{e{\frac{1}{\sqrt{x}}}} \right ) ^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b^{3} x^{2} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{3} - \int \frac{4 \,{\left (b^{3} d x^{2} + b^{3} e x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{3} - 4 \,{\left (b^{3} d \log \left (c\right )^{3} + 3 \, a b^{2} d \log \left (c\right )^{2} + 3 \, a^{2} b d \log \left (c\right ) + a^{3} d\right )} x^{2} + 3 \,{\left (b^{3} d n x^{2} - 4 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} - 4 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{3}{2}} + 4 \,{\left (b^{3} d x^{2} + b^{3} e x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right )^{2} - 12 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} - 4 \,{\left (b^{3} e \log \left (c\right )^{3} + 3 \, a b^{2} e \log \left (c\right )^{2} + 3 \, a^{2} b e \log \left (c\right ) + a^{3} e\right )} x^{\frac{3}{2}} - 12 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{2} +{\left (b^{3} d x^{2} + b^{3} e x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )^{2} +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x^{\frac{3}{2}} - 2 \,{\left ({\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} x^{2} +{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )\right )} \log \left ({\left (d \sqrt{x} + e\right )}^{n}\right ) + 12 \,{\left ({\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} x^{2} +{\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x^{\frac{3}{2}}\right )} \log \left (x^{\frac{1}{2} \, n}\right )}{4 \,{\left (d x + e \sqrt{x}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^3*x^2*log((d*sqrt(x) + e)^n)^3 - integrate(1/4*(4*(b^3*d*x^2 + b^3*e*x^(3/2))*log(x^(1/2*n))^3 - 4*(b^3*
d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^2 + 3*(b^3*d*n*x^2 - 4*(b^3*d*log(c) + a*b^2*d)*
x^2 - 4*(b^3*e*log(c) + a*b^2*e)*x^(3/2) + 4*(b^3*d*x^2 + b^3*e*x^(3/2))*log(x^(1/2*n)))*log((d*sqrt(x) + e)^n
)^2 - 12*((b^3*d*log(c) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(3/2))*log(x^(1/2*n))^2 - 4*(b^3*e*log(c)^
3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(3/2) - 12*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)
*x^2 + (b^3*d*x^2 + b^3*e*x^(3/2))*log(x^(1/2*n))^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(3/2) -
2*((b^3*d*log(c) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(3/2))*log(x^(1/2*n)))*log((d*sqrt(x) + e)^n) + 1
2*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(3/2))*
log(x^(1/2*n)))/(d*x + e*sqrt(x)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{3} x \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{3} + 3 \, a b^{2} x \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right )^{2} + 3 \, a^{2} b x \log \left (c \left (\frac{d x + e \sqrt{x}}{x}\right )^{n}\right ) + a^{3} x, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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