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

Optimal. Leaf size=573 \[ \frac{3 b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4}-\frac{5 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{d}{d+e \sqrt{x}}\right )}{2 d^4}+\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{d^4}+\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (3,\frac{d}{d+e \sqrt{x}}\right )}{d^4}+\frac{5 b^2 e^4 n^2 \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^4}+\frac{3 b^2 e^4 n^2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4}+\frac{5 b^2 e^3 n^2 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^4 \sqrt{x}}-\frac{b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}-\frac{3 b e^4 n \log \left (1-\frac{d}{d+e \sqrt{x}}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4 \sqrt{x}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 d^2 x}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{2 x^2}-\frac{b^3 e^3 n^3}{2 d^3 \sqrt{x}}+\frac{b^3 e^4 n^3 \log \left (d+e \sqrt{x}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.49695, antiderivative size = 550, normalized size of antiderivative = 0.96, number of steps used = 35, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31, 44} \[ -\frac{3 b^2 e^4 n^2 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{d^4}+\frac{11 b^3 e^4 n^3 \text{PolyLog}\left (2,\frac{e \sqrt{x}}{d}+1\right )}{2 d^4}+\frac{3 b^3 e^4 n^3 \text{PolyLog}\left (3,\frac{e \sqrt{x}}{d}+1\right )}{d^4}+\frac{11 b^2 e^4 n^2 \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^4}+\frac{5 b^2 e^3 n^2 \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^4 \sqrt{x}}-\frac{b^2 e^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )}{2 d^2 x}+\frac{e^4 \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{2 d^4}-\frac{5 b e^4 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 d^4}-\frac{3 b e^4 n \log \left (-\frac{e \sqrt{x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4}-\frac{3 b e^3 n \left (d+e \sqrt{x}\right ) \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d^4 \sqrt{x}}+\frac{3 b e^2 n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{4 d^2 x}-\frac{b e n \left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2}{2 d x^{3/2}}-\frac{\left (a+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3}{2 x^2}-\frac{b^3 e^3 n^3}{2 d^3 \sqrt{x}}+\frac{b^3 e^4 n^3 \log \left (d+e \sqrt{x}\right )}{2 d^4}-\frac{3 b^3 e^4 n^3 \log (x)}{2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 1.05605, size = 841, normalized size = 1.47 \[ -\frac{2 \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^3 d^4+6 b n \log \left (d+e \sqrt{x}\right ) \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 d^4+2 b e n \sqrt{x} \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 d^3-3 b e^2 n x \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 d^2+6 b e^3 n x^{3/2} \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 d-6 b e^4 n x^2 \log \left (d+e \sqrt{x}\right ) \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2+3 b e^4 n x^2 \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right )^2 \log (x)-2 b^2 n^2 \left (a-b n \log \left (d+e \sqrt{x}\right )+b \log \left (c \left (d+e \sqrt{x}\right )^n\right )\right ) \left (-6 x^2 \text{PolyLog}\left (2,\frac{\sqrt{x} e}{d}+1\right ) e^4+x \left (-d^2+5 e \sqrt{x} d+11 e^2 x \log \left (-\frac{e \sqrt{x}}{d}\right )\right ) e^2-3 \left (d^4-e^4 x^2\right ) \log ^2\left (d+e \sqrt{x}\right )-\log \left (d+e \sqrt{x}\right ) \left (11 x^2 e^4+6 x^2 \log \left (-\frac{e \sqrt{x}}{d}\right ) e^4+6 d x^{3/2} e^3-3 d^2 x e^2+2 d^3 \sqrt{x} e\right )\right )+b^3 n^3 \left (2 \log ^3\left (d+e \sqrt{x}\right ) d^4+2 e \sqrt{x} \log ^2\left (d+e \sqrt{x}\right ) d^3+e^2 x \left (2-3 \log \left (d+e \sqrt{x}\right )\right ) \log \left (d+e \sqrt{x}\right ) d^2+2 e^3 x^{3/2} \left (3 \log ^2\left (d+e \sqrt{x}\right )-5 \log \left (d+e \sqrt{x}\right )+1\right ) d+12 e^4 x^2 \left (\log \left (-\frac{e \sqrt{x}}{d}\right )-\log \left (d+e \sqrt{x}\right )\right )+11 e^4 x^2 \left (\log \left (d+e \sqrt{x}\right ) \left (\log \left (d+e \sqrt{x}\right )-2 \log \left (-\frac{e \sqrt{x}}{d}\right )\right )-2 \text{PolyLog}\left (2,\frac{\sqrt{x} e}{d}+1\right )\right )-2 e^4 x^2 \left (\left (\log \left (d+e \sqrt{x}\right )-3 \log \left (-\frac{e \sqrt{x}}{d}\right )\right ) \log ^2\left (d+e \sqrt{x}\right )-6 \text{PolyLog}\left (2,\frac{\sqrt{x} e}{d}+1\right ) \log \left (d+e \sqrt{x}\right )+6 \text{PolyLog}\left (3,\frac{\sqrt{x} e}{d}+1\right )\right )\right )}{4 d^4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*b*d^3*e*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*L
og[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^2 + 6*b*d*e^3*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*Log[d + e*Sqrt[x]]*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])
^2 - 6*b*e^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*(a -
b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])^3 + 3*b*e^4*n*x^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*
(d + e*Sqrt[x])^n])^2*Log[x] - 2*b^2*n^2*(a - b*n*Log[d + e*Sqrt[x]] + b*Log[c*(d + e*Sqrt[x])^n])*(-3*(d^4 -
e^4*x^2)*Log[d + e*Sqrt[x]]^2 + e^2*x*(-d^2 + 5*d*e*Sqrt[x] + 11*e^2*x*Log[-((e*Sqrt[x])/d)]) - Log[d + e*Sqrt
[x]]*(2*d^3*e*Sqrt[x] - 3*d^2*e^2*x + 6*d*e^3*x^(3/2) + 11*e^4*x^2 + 6*e^4*x^2*Log[-((e*Sqrt[x])/d)]) - 6*e^4*
x^2*PolyLog[2, 1 + (e*Sqrt[x])/d]) + 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*Sqrt[x]*Log[d + e*Sqrt[x]]^2 + 2*d^4*Log[d + e*Sqrt[x]]^3 + 2*d*e^3*x^(3/2)*(1 - 5*Log[d + e*Sqrt[x]] + 3*Lo
g[d + e*Sqrt[x]]^2) + 12*e^4*x^2*(-Log[d + e*Sqrt[x]] + Log[-((e*Sqrt[x])/d)]) + 11*e^4*x^2*(Log[d + e*Sqrt[x]
]*(Log[d + e*Sqrt[x]] - 2*Log[-((e*Sqrt[x])/d)]) - 2*PolyLog[2, 1 + (e*Sqrt[x])/d]) - 2*e^4*x^2*(Log[d + e*Sqr
t[x]]^2*(Log[d + e*Sqrt[x]] - 3*Log[-((e*Sqrt[x])/d)]) - 6*Log[d + e*Sqrt[x]]*PolyLog[2, 1 + (e*Sqrt[x])/d] +
6*PolyLog[3, 1 + (e*Sqrt[x])/d])))/(4*d^4*x^2)

________________________________________________________________________________________

Maple [F]  time = 0.098, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ( a+b\ln \left ( c \left ( d+e\sqrt{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^(1/2))^n))^3/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b^{3} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right )^{3}}{2 \, x^{2}} + \int \frac{3 \,{\left (b^{3} e n x + 4 \,{\left (b^{3} e \log \left (c\right ) + a b^{2} e\right )} x + 4 \,{\left (b^{3} d \log \left (c\right ) + a b^{2} d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{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 + 12 \,{\left ({\left (b^{3} e \log \left (c\right )^{2} + 2 \, a b^{2} e \log \left (c\right ) + a^{2} b e\right )} x +{\left (b^{3} d \log \left (c\right )^{2} + 2 \, a b^{2} d \log \left (c\right ) + a^{2} b d\right )} \sqrt{x}\right )} \log \left ({\left (e \sqrt{x} + d\right )}^{n}\right ) + 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 )} \sqrt{x}}{4 \,{\left (e x^{4} + d x^{\frac{7}{2}}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b^3*log((e*sqrt(x) + d)^n)^3/x^2 + integrate(1/4*(3*(b^3*e*n*x + 4*(b^3*e*log(c) + a*b^2*e)*x + 4*(b^3*d*
log(c) + a*b^2*d)*sqrt(x))*log((e*sqrt(x) + d)^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 + 12*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*
b*d)*sqrt(x))*log((e*sqrt(x) + d)^n) + 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)*sqrt
(x))/(e*x^4 + d*x^(7/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log((e*sqrt(x) + d)^n*c)^3 + 3*a*b^2*log((e*sqrt(x) + d)^n*c)^2 + 3*a^2*b*log((e*sqrt(x) + d)^n*
c) + a^3)/x^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**(1/2))**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 ({\left (e \sqrt{x} + d\right )}^{n} c\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^(1/2))^n))^3/x^3,x, algorithm="giac")

[Out]

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