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 {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 {12 b^2 f^2 n^2 \text {Li}_2\left (-\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 {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\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 {6 b f^2 n \text {Li}_2\left (-\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 )^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 {12 b^3 f^2 n^3 \text {Li}_2\left (\frac {\sqrt {x} f}{e}+1\right )}{e^2}-\frac {24 b^3 f^2 n^3 \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 {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]

-3*b^3*f^2*n^3*ln(x)/e^2+3/2*b^3*f^2*n^3*ln(x)^2/e^2-3*b^2*f^2*n^2*ln(x)*(a+b*ln(c*x^n))/e^2-1/2*f^2*(a+b*ln(c
*x^n))^3/e^2-1/8*f^2*(a+b*ln(c*x^n))^4/b/e^2/n+6*b^3*f^2*n^3*ln(e+f*x^(1/2))/e^2+6*b^2*f^2*n^2*(a+b*ln(c*x^n))
*ln(e+f*x^(1/2))/e^2-12*b^3*f^2*n^3*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e^2-6*b^3*n^3*ln(d*(e+f*x^(1/2)))/x-6*b^2
*n^2*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2)))/x-3*b*n*(a+b*ln(c*x^n))^2*ln(d*(e+f*x^(1/2)))/x-(a+b*ln(c*x^n))^3*ln(
d*(e+f*x^(1/2)))/x+3*b*f^2*n*(a+b*ln(c*x^n))^2*ln(1+f*x^(1/2)/e)/e^2+f^2*(a+b*ln(c*x^n))^3*ln(1+f*x^(1/2)/e)/e
^2+12*b^2*f^2*n^2*(a+b*ln(c*x^n))*polylog(2,-f*x^(1/2)/e)/e^2+6*b*f^2*n*(a+b*ln(c*x^n))^2*polylog(2,-f*x^(1/2)
/e)/e^2-12*b^3*f^2*n^3*polylog(2,1+f*x^(1/2)/e)/e^2-24*b^3*f^2*n^3*polylog(3,-f*x^(1/2)/e)/e^2-24*b^2*f^2*n^2*
(a+b*ln(c*x^n))*polylog(3,-f*x^(1/2)/e)/e^2+48*b^3*f^2*n^3*polylog(4,-f*x^(1/2)/e)/e^2-90*b^3*f*n^3/e/x^(1/2)-
42*b^2*f*n^2*(a+b*ln(c*x^n))/e/x^(1/2)-9*b*f*n*(a+b*ln(c*x^n))^2/e/x^(1/2)-f*(a+b*ln(c*x^n))^3/e/x^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.18, antiderivative size = 673, normalized size of antiderivative = 1.00, 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 12

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

Rule 30

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

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*}

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Mathematica [A]  time = 1.19, size = 976, normalized size = 1.45 \[ -\frac {b^3 \left (6 f^2 x \text {Li}_2\left (-\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 {Li}_3\left (-\frac {e}{f \sqrt {x}}\right ) \log (x)+48 e+48 f \sqrt {x} \text {Li}_4\left (-\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 {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) \log (x)+24 e+24 f \sqrt {x} \text {Li}_3\left (-\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 {Li}_2\left (-\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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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt {x} + d e\right )}{x^{2}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}}\,{d x} \]

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)

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maple [F]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \sqrt {x}+e \right ) d \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x^{2}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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