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

Optimal. Leaf size=441 \[ \frac{4 b f^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{4 b^2 f^2 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{8 b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac{f^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{2 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}+\frac{2 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{4 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b^2 f^2 n^2 \log (x)}{e^2}-\frac{14 b^2 f n^2}{e \sqrt{x}} \]

[Out]

(-14*b^2*f*n^2)/(e*Sqrt[x]) + (2*b^2*f^2*n^2*Log[e + f*Sqrt[x]])/e^2 - (2*b^2*n^2*Log[d*(e + f*Sqrt[x])])/x -
(4*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b^2*f^2*n^2*Log[x])/e^2 + (b^2*f^2*n^2*Log[x]^
2)/(2*e^2) - (6*b*f*n*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (2*b*f^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2
- (2*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (b*f^2*n*Log[x]*(a + b*Log[c*x^n]))/e^2 - (f*(a + b*Lo
g[c*x^n])^2)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(a +
b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(6*b*e^2*n) - (4*b^2*f^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/
e^2 + (4*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (8*b^2*f^2*n^2*PolyLog[3, -((f*Sqrt[x]
)/e)])/e^2

________________________________________________________________________________________

Rubi [A]  time = 0.632556, antiderivative size = 441, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 17, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.607, Rules used = {2454, 2395, 44, 2377, 2304, 2375, 2337, 2374, 6589, 2376, 2394, 2315, 2301, 2366, 12, 2302, 30} \[ \frac{4 b f^2 n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{4 b^2 f^2 n^2 \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{e^2}-\frac{8 b^2 f^2 n^2 \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{f^2 \left (a+b \log \left (c x^n\right )\right )^3}{6 b e^2 n}+\frac{f^2 \log \left (\frac{f \sqrt{x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}-\frac{2 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}+\frac{2 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{4 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b^2 f^2 n^2 \log (x)}{e^2}-\frac{14 b^2 f n^2}{e \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-14*b^2*f*n^2)/(e*Sqrt[x]) + (2*b^2*f^2*n^2*Log[e + f*Sqrt[x]])/e^2 - (2*b^2*n^2*Log[d*(e + f*Sqrt[x])])/x -
(4*b^2*f^2*n^2*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/e^2 - (b^2*f^2*n^2*Log[x])/e^2 + (b^2*f^2*n^2*Log[x]^
2)/(2*e^2) - (6*b*f*n*(a + b*Log[c*x^n]))/(e*Sqrt[x]) + (2*b*f^2*n*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/e^2
- (2*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x - (b*f^2*n*Log[x]*(a + b*Log[c*x^n]))/e^2 - (f*(a + b*Lo
g[c*x^n])^2)/(e*Sqrt[x]) - (Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x + (f^2*Log[1 + (f*Sqrt[x])/e]*(a +
b*Log[c*x^n])^2)/e^2 - (f^2*(a + b*Log[c*x^n])^3)/(6*b*e^2*n) - (4*b^2*f^2*n^2*PolyLog[2, 1 + (f*Sqrt[x])/e])/
e^2 + (4*b*f^2*n*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^2 - (8*b^2*f^2*n^2*PolyLog[3, -((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 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 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 )^2}{x^2} \, dx &=-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{\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 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}-(2 b n) \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{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}+\frac{f^2 \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}-\frac{\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 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 e^2}+(2 b n) \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{(2 b f n) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx}{e}+\frac{\left (b f^2 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{e^2}-\frac{\left (2 b f^2 n\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{8 b^2 f n^2}{e \sqrt{x}}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac{f^3 \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 (b f^2 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx}{e^2}-\left (2 b^2 n^2\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{12 b^2 f n^2}{e \sqrt{x}}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\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 \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 \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx}{2 e^2}-\frac{\left (2 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 )}{x} \, dx}{e^2}+\left (2 b^2 n^2\right ) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )\right )}{x^2} \, dx+\frac{\left (b^2 f^2 n^2\right ) \int \frac{\log (x)}{x} \, dx}{e^2}-\frac{\left (2 b^2 f^2 n^2\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{e^2}\\ &=-\frac{12 b^2 f n^2}{e \sqrt{x}}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\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 \log \left (1+\frac{f \sqrt{x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac{4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{f^2 \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b e^2 n}+\left (4 b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (d (e+f x))}{x^3} \, dx,x,\sqrt{x}\right )-\frac{\left (4 b^2 f^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{x} \, dx}{e^2}-\frac{\left (4 b^2 f^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{12 b^2 f n^2}{e \sqrt{x}}-\frac{2 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{4 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\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 \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}{6 b e^2 n}+\frac{4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{8 b^2 f^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\left (2 b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 (e+f x)} \, dx,x,\sqrt{x}\right )+\frac{\left (4 b^2 f^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{e^2}\\ &=-\frac{12 b^2 f n^2}{e \sqrt{x}}-\frac{2 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{4 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\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 \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}{6 b e^2 n}-\frac{4 b^2 f^2 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{8 b^2 f^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}+\left (2 b^2 f n^2\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{14 b^2 f n^2}{e \sqrt{x}}+\frac{2 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right )}{e^2}-\frac{2 b^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )}{x}-\frac{4 b^2 f^2 n^2 \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{b^2 f^2 n^2 \log (x)}{e^2}+\frac{b^2 f^2 n^2 \log ^2(x)}{2 e^2}-\frac{6 b f n \left (a+b \log \left (c x^n\right )\right )}{e \sqrt{x}}+\frac{2 b f^2 n \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{2 b n \log \left (d \left (e+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{b f^2 n \log (x) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac{f \left (a+b \log \left (c x^n\right )\right )^2}{e \sqrt{x}}-\frac{\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 \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}{6 b e^2 n}-\frac{4 b^2 f^2 n^2 \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{e^2}+\frac{4 b f^2 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}-\frac{8 b^2 f^2 n^2 \text{Li}_3\left (-\frac{f \sqrt{x}}{e}\right )}{e^2}\\ \end{align*}

Mathematica [A]  time = 0.500432, size = 821, normalized size = 1.86 \[ -\frac{\frac{1}{2} b^2 f^2 n^2 x \log ^3(x)-\frac{3}{2} b^2 f^2 n^2 x \log ^2(x)-\frac{3}{2} a b f^2 n x \log ^2(x)-3 b^2 f^2 n^2 x \log \left (e+f \sqrt{x}\right ) \log ^2(x)+3 b^2 f^2 n^2 x \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log ^2(x)-\frac{3}{2} b^2 f^2 n x \log \left (c x^n\right ) \log ^2(x)+\frac{3}{2} b^2 f^2 x \log ^2\left (c x^n\right ) \log (x)+\frac{3}{2} a^2 f^2 x \log (x)+3 b^2 f^2 n^2 x \log (x)+3 a b f^2 n x \log (x)+6 b^2 f^2 n^2 x \log \left (e+f \sqrt{x}\right ) \log (x)+6 a b f^2 n x \log \left (e+f \sqrt{x}\right ) \log (x)-6 b^2 f^2 n^2 x \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log (x)-6 a b f^2 n x \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log (x)+3 a b f^2 x \log \left (c x^n\right ) \log (x)+3 b^2 f^2 n x \log \left (c x^n\right ) \log (x)+6 b^2 f^2 n x \log \left (e+f \sqrt{x}\right ) \log \left (c x^n\right ) \log (x)-6 b^2 f^2 n x \log \left (\frac{\sqrt{x} f}{e}+1\right ) \log \left (c x^n\right ) \log (x)-3 b^2 f^2 x \log \left (e+f \sqrt{x}\right ) \log ^2\left (c x^n\right )+3 b^2 e^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \log ^2\left (c x^n\right )+3 b^2 e f \sqrt{x} \log ^2\left (c x^n\right )-3 a^2 f^2 x \log \left (e+f \sqrt{x}\right )-6 b^2 f^2 n^2 x \log \left (e+f \sqrt{x}\right )-6 a b f^2 n x \log \left (e+f \sqrt{x}\right )+3 a^2 e^2 \log \left (d \left (e+f \sqrt{x}\right )\right )+6 b^2 e^2 n^2 \log \left (d \left (e+f \sqrt{x}\right )\right )+6 a b e^2 n \log \left (d \left (e+f \sqrt{x}\right )\right )-6 a b f^2 x \log \left (e+f \sqrt{x}\right ) \log \left (c x^n\right )-6 b^2 f^2 n x \log \left (e+f \sqrt{x}\right ) \log \left (c x^n\right )+6 a b e^2 \log \left (d \left (e+f \sqrt{x}\right )\right ) \log \left (c x^n\right )+6 b^2 e^2 n \log \left (d \left (e+f \sqrt{x}\right )\right ) \log \left (c x^n\right )+6 a b e f \sqrt{x} \log \left (c x^n\right )+18 b^2 e f n \sqrt{x} \log \left (c x^n\right )-12 b f^2 n x \left (a+b n+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )+24 b^2 f^2 n^2 x \text{PolyLog}\left (3,-\frac{f \sqrt{x}}{e}\right )+42 b^2 e f n^2 \sqrt{x}+3 a^2 e f \sqrt{x}+18 a b e f n \sqrt{x}}{3 e^2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3*a^2*e*f*Sqrt[x] + 18*a*b*e*f*n*Sqrt[x] + 42*b^2*e*f*n^2*Sqrt[x] - 3*a^2*f^2*x*Log[e + f*Sqrt[x]] - 6*a*b*f
^2*n*x*Log[e + f*Sqrt[x]] - 6*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]] + 3*a^2*e^2*Log[d*(e + f*Sqrt[x])] + 6*a*b*e^2*
n*Log[d*(e + f*Sqrt[x])] + 6*b^2*e^2*n^2*Log[d*(e + f*Sqrt[x])] + (3*a^2*f^2*x*Log[x])/2 + 3*a*b*f^2*n*x*Log[x
] + 3*b^2*f^2*n^2*x*Log[x] + 6*a*b*f^2*n*x*Log[e + f*Sqrt[x]]*Log[x] + 6*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[
x] - 6*a*b*f^2*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - 6*b^2*f^2*n^2*x*Log[1 + (f*Sqrt[x])/e]*Log[x] - (3*a*b*f^2*
n*x*Log[x]^2)/2 - (3*b^2*f^2*n^2*x*Log[x]^2)/2 - 3*b^2*f^2*n^2*x*Log[e + f*Sqrt[x]]*Log[x]^2 + 3*b^2*f^2*n^2*x
*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 + (b^2*f^2*n^2*x*Log[x]^3)/2 + 6*a*b*e*f*Sqrt[x]*Log[c*x^n] + 18*b^2*e*f*n*Sq
rt[x]*Log[c*x^n] - 6*a*b*f^2*x*Log[e + f*Sqrt[x]]*Log[c*x^n] - 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[c*x^n] + 6
*a*b*e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 6*b^2*e^2*n*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 3*a*b*f^2*x*Log[x
]*Log[c*x^n] + 3*b^2*f^2*n*x*Log[x]*Log[c*x^n] + 6*b^2*f^2*n*x*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] - 6*b^2*f^
2*n*x*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] - (3*b^2*f^2*n*x*Log[x]^2*Log[c*x^n])/2 + 3*b^2*e*f*Sqrt[x]*Log
[c*x^n]^2 - 3*b^2*f^2*x*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 + 3*b^2*e^2*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + (3*b
^2*f^2*x*Log[x]*Log[c*x^n]^2)/2 - 12*b*f^2*n*x*(a + b*n + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)] + 24*b^2*
f^2*n^2*x*PolyLog[3, -((f*Sqrt[x])/e)])/(3*e^2*x)

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{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))^2*ln(d*(e+f*x^(1/2)))/x^2,x)

[Out]

int((a+b*ln(c*x^n))^2*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 )}^{2} \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))^2*log(d*(e+f*x^(1/2)))/x^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)^2*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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\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))^2*log(d*(e+f*x^(1/2)))/x^2,x, algorithm="fricas")

[Out]

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

[Out]

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