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

Optimal. Leaf size=673 \[ \frac{12 b^2 f^2 n^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{24 b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{6 b f^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{12 b^3 f^2 n^3 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{24 b^3 f^2 n^3 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{PolyLog}\left (4,-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}+\frac{f^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}+\frac{3 b f^2 n \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{6 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}+\frac{6 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{12 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{3 b^3 f^2 n^3 \log (x)}{e^2}-\frac{90 b^3 f n^3}{e \sqrt{x}} \]

[Out]

(-90*b^3*f*n^3)/(e*Sqrt[x]) + (6*b^3*f^2*n^3*Log[e + f*Sqrt[x]])/e^2 - (6*b^3*n^3*Log[d*(e + f*Sqrt[x])])/x -
(12*b^3*f^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (3*b^3*f^2*n^3*Log[x])/e^2 + (3*b^3*f^2*n^3*Lo
g[x]^2)/(2*e^2) - (42*b^2*f*n^2*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (6*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*(a + b*Log
[c*x^n]))/e^2 - (6*b^2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (3*b^2*f^2*n^2*Log[x]*(a + b*Log[c*x
^n]))/e^2 - (9*b*f*n*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x
+ (3*b*f^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(2*e^2) - (f*(a +
b*Log[c*x^n])^3)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(
a + b*Log[c*x^n])^3)/e^2 - (f^2*(a + b*Log[c*x^n])^4)/(8*b*e^2*n) - (12*b^3*f^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])
/e])/e^2 + (12*b^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 + (6*b*f^2*n*(a + b*Log[c*x^n]
)^2*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (24*b^3*f^2*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 - (24*b^2*f^2*n^2*(a
+ b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 + (48*b^3*f^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^2

________________________________________________________________________________________

Rubi [A]  time = 1.18155, antiderivative size = 673, normalized size of antiderivative = 1., number of steps used = 34, number of rules used = 19, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.679, Rules used = {2454, 2395, 44, 2377, 2305, 2304, 2375, 2337, 2374, 2383, 6589, 2376, 2394, 2315, 2301, 2366, 12, 2302, 30} \[ \frac{12 b^2 f^2 n^2 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{24 b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{6 b f^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{12 b^3 f^2 n^3 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{24 b^3 f^2 n^3 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{PolyLog}\left (4,-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}+\frac{f^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}+\frac{3 b f^2 n \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{6 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}+\frac{6 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{12 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{3 b^3 f^2 n^3 \log (x)}{e^2}-\frac{90 b^3 f n^3}{e \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-90*b^3*f*n^3)/(e*Sqrt[x]) + (6*b^3*f^2*n^3*Log[e + f*Sqrt[x]])/e^2 - (6*b^3*n^3*Log[d*(e + f*Sqrt[x])])/x -
(12*b^3*f^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (3*b^3*f^2*n^3*Log[x])/e^2 + (3*b^3*f^2*n^3*Lo
g[x]^2)/(2*e^2) - (42*b^2*f*n^2*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (6*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*(a + b*Log
[c*x^n]))/e^2 - (6*b^2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (3*b^2*f^2*n^2*Log[x]*(a + b*Log[c*x
^n]))/e^2 - (9*b*f*n*(a + b*Log[c*x^n])^2)/(e*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x
+ (3*b*f^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(2*e^2) - (f*(a +
b*Log[c*x^n])^3)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(
a + b*Log[c*x^n])^3)/e^2 - (f^2*(a + b*Log[c*x^n])^4)/(8*b*e^2*n) - (12*b^3*f^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])
/e])/e^2 + (12*b^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 + (6*b*f^2*n*(a + b*Log[c*x^n]
)^2*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (24*b^3*f^2*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 - (24*b^2*f^2*n^2*(a
+ b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/e^2 + (48*b^3*f^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^2

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 2395

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

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])

Rule 2377

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[
Dist[(a + b*Log[c*x^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && IGtQ[p, 0] &&
RationalQ[m] && RationalQ[q] && NeQ[q, -1] && (EqQ[p, 1] || (FractionQ[m] && IntegerQ[(q + 1)/m]) || (IGtQ[q,
0] && IntegerQ[(q + 1)/m] && EqQ[d*e, 1]))

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
]

Rule 2375

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

Rule 2337

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

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 2383

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

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 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2394

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

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 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^2} \, dx &=-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-(3 b n) \int \left (-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e x^{3/2}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2}-\frac{f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 x}\right ) \, dx\\ &=-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac{f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}+(3 b n) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+\frac{(3 b f n) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx}{e}+\frac{\left (3 b f^2 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}-\frac{\left (3 b f^2 n\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{e^2}\\ &=-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}+\frac{3 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{3 b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^3 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{2 e^2}-\frac{\left (3 b f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx}{2 e^2}-\left (6 b^2 n^2\right ) \int \left (-\frac{f \left (a+b \log \left (c x^n\right )\right )}{e x^{3/2}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 x}\right ) \, dx+\frac{\left (12 b^2 f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e}\\ &=-\frac{48 b^3 f n^3}{e \sqrt{x}}-\frac{24 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}+\frac{3 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{3 b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \int \frac{\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx}{2 e^2}-\frac{\left (3 b f^2 n\right ) \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{e^2}+\left (6 b^2 n^2\right ) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx+\frac{\left (6 b^2 f n^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e}+\frac{\left (3 b^2 f^2 n^2\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}-\frac{\left (6 b^2 f^2 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}\\ &=-\frac{72 b^3 f n^3}{e \sqrt{x}}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{f^2 \operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\frac{\left (3 b f^3 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (e+f \sqrt{x}\right ) \sqrt{x}} \, dx}{2 e^2}-\frac{\left (3 b^2 f^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx}{e^2}-\frac{\left (12 b^2 f^2 n^2\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^2}-\left (6 b^3 n^3\right ) \int \left (-\frac{f}{e x^{3/2}}+\frac{f^2 \log \left (e+f \sqrt{x}\right )}{e^2 x}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right )}{x^2}-\frac{f^2 \log (x)}{2 e^2 x}\right ) \, dx\\ &=-\frac{84 b^3 f n^3}{e \sqrt{x}}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{3 b f^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{\left (3 b f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}-\frac{\left (6 b^2 f^2 n^2\right ) \int \frac{\log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}+\left (6 b^3 n^3\right ) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right )}{x^2} \, dx+\frac{\left (3 b^3 f^2 n^3\right ) \int \frac{\log (x)}{x} \, dx}{e^2}-\frac{\left (6 b^3 f^2 n^3\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{e^2}+\frac{\left (24 b^3 f^2 n^3\right ) \int \frac{\text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^2}\\ &=-\frac{84 b^3 f n^3}{e \sqrt{x}}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{3 b f^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{\left (3 f^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 e^2}+\left (12 b^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (d (e+f x))}{x^3} \, dx,x,\sqrt{x}\right )-\frac{\left (12 b^3 f^2 n^3\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^2}-\frac{\left (12 b^3 f^2 n^3\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{84 b^3 f n^3}{e \sqrt{x}}-\frac{6 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{12 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{3 b f^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}+\frac{12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^3 f^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\left (6 b^3 f n^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (e+f x)} \, dx,x,\sqrt{x}\right )+\frac{\left (12 b^3 f^3 n^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{84 b^3 f n^3}{e \sqrt{x}}-\frac{6 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{12 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{3 b f^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac{12 b^3 f^2 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^3 f^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\left (6 b^3 f n^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{e x^2}-\frac{f}{e^2 x}+\frac{f^2}{e^2 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{90 b^3 f n^3}{e \sqrt{x}}+\frac{6 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{6 b^3 n^3 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{12 b^3 f^2 n^3 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{3 b^3 f^2 n^3 \log (x)}{e^2}+\frac{3 b^3 f^2 n^3 \log ^2(x)}{2 e^2}-\frac{42 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{6 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{6 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{3 b^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{9 b f n \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{3 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{3 b f^2 n \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^3}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}+\frac{f^2 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{e^2}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b e^2 n}-\frac{12 b^3 f^2 n^3 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{12 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{6 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^3 f^2 n^3 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{24 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{48 b^3 f^2 n^3 \text{Li}_4\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}\\ \end{align*}

Mathematica [A]  time = 1.09267, size = 976, normalized size = 1.45 \[ -\frac{b^3 \left (6 f^2 x \text{PolyLog}\left (2,-\frac{e}{f \sqrt{x}}\right ) \log ^2(x)+f \sqrt{x} \left (e \log ^3(x)-f \sqrt{x} \log \left (\frac{e}{f \sqrt{x}}+1\right ) \log ^3(x)+6 e \log ^2(x)+24 e \log (x)+24 f \sqrt{x} \text{PolyLog}\left (3,-\frac{e}{f \sqrt{x}}\right ) \log (x)+48 e+48 f \sqrt{x} \text{PolyLog}\left (4,-\frac{e}{f \sqrt{x}}\right )\right )\right ) n^3+b^2 f \sqrt{x} \left (a+b n-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\frac{1}{2} f \sqrt{x} \log ^3(x)+3 e \log ^2(x)-3 f \sqrt{x} \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log ^2(x)+12 e \log (x)-12 f \sqrt{x} \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \log (x)+24 e+24 f \sqrt{x} \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )\right ) n^2+3 b f \sqrt{x} \left (a^2+2 b n a+2 b \left (\log \left (c x^n\right )-n \log (x)\right ) a+2 b^2 n^2+b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2+2 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right )\right ) \left (\frac{1}{4} f \sqrt{x} \log ^2(x)+\left (e-f \sqrt{x} \log \left (\frac{\sqrt{x} f}{e}+1\right )\right ) \log (x)+2 e-2 f \sqrt{x} \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )\right ) n+e^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a^3+3 b n a^2+6 b^2 n^2 a+6 b^3 n^3+b^3 \log ^3\left (c x^n\right )+3 b^2 (a+b n) \log ^2\left (c x^n\right )+3 b \left (a^2+2 b n a+2 b^2 n^2\right ) \log \left (c x^n\right )\right )-f^2 x \log \left (e+f \sqrt{x}\right ) \left (a^3+3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right ) a+6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3+3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+\frac{1}{2} f^2 x \log (x) \left (a^3+3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right ) a+6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3+3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )+e f \sqrt{x} \left (a^3+3 b n a^2+3 b \left (\log \left (c x^n\right )-n \log (x)\right ) a^2+6 b^2 n^2 a+3 b^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 a+6 b^2 n \left (\log \left (c x^n\right )-n \log (x)\right ) a+6 b^3 n^3+b^3 \left (\log \left (c x^n\right )-n \log (x)\right )^3+3 b^3 n \left (\log \left (c x^n\right )-n \log (x)\right )^2+6 b^3 n^2 \left (\log \left (c x^n\right )-n \log (x)\right )\right )}{e^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((e^2*Log[d*(e + f*Sqrt[x])]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*b*(a^2 + 2*a*b*n + 2*b^2*n^2)*Log
[c*x^n] + 3*b^2*(a + b*n)*Log[c*x^n]^2 + b^3*Log[c*x^n]^3) + e*f*Sqrt[x]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^
3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + L
og[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) +
Log[c*x^n])^3) - f^2*x*Log[e + f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) +
Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x
]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) + (f^2*x*Log[x]*
(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log
[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x])
+ Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3))/2 + 3*b*f*n*Sqrt[x]*(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*a*b*(-
(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(2*e + (e -
f*Sqrt[x]*Log[1 + (f*Sqrt[x])/e])*Log[x] + (f*Sqrt[x]*Log[x]^2)/4 - 2*f*Sqrt[x]*PolyLog[2, -((f*Sqrt[x])/e)])
+ b^2*f*n^2*Sqrt[x]*(a + b*n - b*n*Log[x] + b*Log[c*x^n])*(24*e + 12*e*Log[x] + 3*e*Log[x]^2 - 3*f*Sqrt[x]*Log
[1 + (f*Sqrt[x])/e]*Log[x]^2 + (f*Sqrt[x]*Log[x]^3)/2 - 12*f*Sqrt[x]*Log[x]*PolyLog[2, -((f*Sqrt[x])/e)] + 24*
f*Sqrt[x]*PolyLog[3, -((f*Sqrt[x])/e)]) + b^3*n^3*(6*f^2*x*Log[x]^2*PolyLog[2, -(e/(f*Sqrt[x]))] + f*Sqrt[x]*(
48*e + 24*e*Log[x] + 6*e*Log[x]^2 + e*Log[x]^3 - f*Sqrt[x]*Log[1 + e/(f*Sqrt[x])]*Log[x]^3 + 24*f*Sqrt[x]*Log[
x]*PolyLog[3, -(e/(f*Sqrt[x]))] + 48*f*Sqrt[x]*PolyLog[4, -(e/(f*Sqrt[x]))])))/(e^2*x))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}}{{x}^{2}}\ln \left ( d \left ( e+f\sqrt{x} \right ) \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a^3)*log(d*f*sqrt(x) + d*e)/x^2, 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*x**n))**3*ln(d*(e+f*x**(1/2)))/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 x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + e\right )} d\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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