3.58 \(\int \frac{\log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))^2}{x^3} \, dx\)

Optimal. Leaf size=555 \[ 2 b d^4 f^4 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+b^2 d^4 f^4 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+\frac{1}{2} d^4 f^4 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b d^4 f^4 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{21 b^2 d^3 f^3 n^2}{4 \sqrt{x}}+\frac{7 b^2 d^2 f^2 n^2}{8 x}+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)+\frac{1}{4} b^2 d^4 f^4 n^2 \log \left (d f \sqrt{x}+1\right )-\frac{1}{8} b^2 d^4 f^4 n^2 \log (x)-\frac{37 b^2 d f n^2}{108 x^{3/2}}-\frac{b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{4 x^2} \]

[Out]

(-37*b^2*d*f*n^2)/(108*x^(3/2)) + (7*b^2*d^2*f^2*n^2)/(8*x) - (21*b^2*d^3*f^3*n^2)/(4*Sqrt[x]) + (b^2*d^4*f^4*
n^2*Log[1 + d*f*Sqrt[x]])/4 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (b^2*d^4*f^4*n^2*Log[x])/8 + (b^2*d^4*f
^4*n^2*Log[x]^2)/8 - (7*b*d*f*n*(a + b*Log[c*x^n]))/(18*x^(3/2)) + (3*b*d^2*f^2*n*(a + b*Log[c*x^n]))/(4*x) -
(5*b*d^3*f^3*n*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*
n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*x^2) - (b*d^4*f^4*n*Log[x]*(a + b*Log[c*x^n]))/4 - (d*f*(a + b*L
og[c*x^n])^2)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^2)/(4*x) - (d^3*f^3*(a + b*Log[c*x^n])^2)/(2*Sqrt[x])
+ (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*x^2)
- (d^4*f^4*(a + b*Log[c*x^n])^3)/(12*b*n) + b^2*d^4*f^4*n^2*PolyLog[2, -(d*f*Sqrt[x])] + 2*b*d^4*f^4*n*(a + b*
Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 4*b^2*d^4*f^4*n^2*PolyLog[3, -(d*f*Sqrt[x])]

________________________________________________________________________________________

Rubi [A]  time = 0.550214, antiderivative size = 555, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2454, 2395, 44, 2377, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30} \[ 2 b d^4 f^4 n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )+b^2 d^4 f^4 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+\frac{1}{2} d^4 f^4 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{1}{2} b d^4 f^4 n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{21 b^2 d^3 f^3 n^2}{4 \sqrt{x}}+\frac{7 b^2 d^2 f^2 n^2}{8 x}+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)+\frac{1}{4} b^2 d^4 f^4 n^2 \log \left (d f \sqrt{x}+1\right )-\frac{1}{8} b^2 d^4 f^4 n^2 \log (x)-\frac{37 b^2 d f n^2}{108 x^{3/2}}-\frac{b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]

[Out]

(-37*b^2*d*f*n^2)/(108*x^(3/2)) + (7*b^2*d^2*f^2*n^2)/(8*x) - (21*b^2*d^3*f^3*n^2)/(4*Sqrt[x]) + (b^2*d^4*f^4*
n^2*Log[1 + d*f*Sqrt[x]])/4 - (b^2*n^2*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (b^2*d^4*f^4*n^2*Log[x])/8 + (b^2*d^4*f
^4*n^2*Log[x]^2)/8 - (7*b*d*f*n*(a + b*Log[c*x^n]))/(18*x^(3/2)) + (3*b*d^2*f^2*n*(a + b*Log[c*x^n]))/(4*x) -
(5*b*d^3*f^3*n*(a + b*Log[c*x^n]))/(2*Sqrt[x]) + (b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/2 - (b*
n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(2*x^2) - (b*d^4*f^4*n*Log[x]*(a + b*Log[c*x^n]))/4 - (d*f*(a + b*L
og[c*x^n])^2)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^2)/(4*x) - (d^3*f^3*(a + b*Log[c*x^n])^2)/(2*Sqrt[x])
+ (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(2*x^2)
- (d^4*f^4*(a + b*Log[c*x^n])^3)/(12*b*n) + b^2*d^4*f^4*n^2*PolyLog[2, -(d*f*Sqrt[x])] + 2*b*d^4*f^4*n*(a + b*
Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 4*b^2*d^4*f^4*n^2*PolyLog[3, -(d*f*Sqrt[x])]

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 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 2391

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

Rule 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 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 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 (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2-(2 b n) \int \left (-\frac{d f \left (a+b \log \left (c x^n\right )\right )}{6 x^{5/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{2 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^3}+\frac{d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac{d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )}{4 x}\right ) \, dx\\ &=-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2+(b n) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^3} \, dx+\frac{1}{3} (b d f n) \int \frac{a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac{1}{2} \left (b d^2 f^2 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx+\left (b d^3 f^3 n\right ) \int \frac{a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac{1}{2} \left (b d^4 f^4 n\right ) \int \frac{\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (b d^4 f^4 n\right ) \int \frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx\\ &=-\frac{4 b^2 d f n^2}{27 x^{3/2}}+\frac{b^2 d^2 f^2 n^2}{2 x}-\frac{4 b^2 d^3 f^3 n^2}{\sqrt{x}}-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-\frac{1}{2} \left (b d^4 f^4 n\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (b^2 n^2\right ) \int \left (-\frac{d f}{6 x^{5/2}}+\frac{d^2 f^2}{4 x^2}-\frac{d^3 f^3}{2 x^{3/2}}-\frac{\log \left (1+d f \sqrt{x}\right )}{2 x^3}+\frac{d^4 f^4 \log \left (1+d f \sqrt{x}\right )}{2 x}-\frac{d^4 f^4 \log (x)}{4 x}\right ) \, dx-\left (2 b^2 d^4 f^4 n^2\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{7 b^2 d f n^2}{27 x^{3/2}}+\frac{3 b^2 d^2 f^2 n^2}{4 x}-\frac{5 b^2 d^3 f^3 n^2}{\sqrt{x}}-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{1}{4} \left (d^4 f^4\right ) \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac{1}{2} \left (b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x^3} \, dx+\frac{1}{4} \left (b^2 d^4 f^4 n^2\right ) \int \frac{\log (x)}{x} \, dx-\frac{1}{2} \left (b^2 d^4 f^4 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx\\ &=-\frac{7 b^2 d f n^2}{27 x^{3/2}}+\frac{3 b^2 d^2 f^2 n^2}{4 x}-\frac{5 b^2 d^3 f^3 n^2}{\sqrt{x}}+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}+b^2 d^4 f^4 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )-\frac{\left (d^4 f^4\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}+\left (b^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+d f x)}{x^5} \, dx,x,\sqrt{x}\right )\\ &=-\frac{7 b^2 d f n^2}{27 x^{3/2}}+\frac{3 b^2 d^2 f^2 n^2}{4 x}-\frac{5 b^2 d^3 f^3 n^2}{\sqrt{x}}-\frac{b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{4 x^2}+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )+\frac{1}{4} \left (b^2 d f n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 (1+d f x)} \, dx,x,\sqrt{x}\right )\\ &=-\frac{7 b^2 d f n^2}{27 x^{3/2}}+\frac{3 b^2 d^2 f^2 n^2}{4 x}-\frac{5 b^2 d^3 f^3 n^2}{\sqrt{x}}-\frac{b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{4 x^2}+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )+\frac{1}{4} \left (b^2 d f n^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{x^4}-\frac{d f}{x^3}+\frac{d^2 f^2}{x^2}-\frac{d^3 f^3}{x}+\frac{d^4 f^4}{1+d f x}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{37 b^2 d f n^2}{108 x^{3/2}}+\frac{7 b^2 d^2 f^2 n^2}{8 x}-\frac{21 b^2 d^3 f^3 n^2}{4 \sqrt{x}}+\frac{1}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt{x}\right )-\frac{b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{4 x^2}-\frac{1}{8} b^2 d^4 f^4 n^2 \log (x)+\frac{1}{8} b^2 d^4 f^4 n^2 \log ^2(x)-\frac{7 b d f n \left (a+b \log \left (c x^n\right )\right )}{18 x^{3/2}}+\frac{3 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )}{4 x}-\frac{5 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )}{2 \sqrt{x}}+\frac{1}{2} b d^4 f^4 n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 x^2}-\frac{1}{4} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac{d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{3/2}}+\frac{d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x}-\frac{d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 \sqrt{x}}+\frac{1}{2} d^4 f^4 \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{\log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 x^2}-\frac{d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3}{12 b n}+b^2 d^4 f^4 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )+2 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )-4 b^2 d^4 f^4 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )\\ \end{align*}

Mathematica [A]  time = 0.522909, size = 881, normalized size = 1.59 \[ -\frac{18 b^2 d^4 n^2 x^2 \log ^3(x) f^4-27 b^2 d^4 n^2 x^2 \log ^2(x) f^4-54 a b d^4 n x^2 \log ^2(x) f^4-108 b^2 d^4 x^2 \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right ) f^4+54 b^2 d^4 x^2 \log (x) \log ^2\left (c x^n\right ) f^4-108 a^2 d^4 x^2 \log \left (d \sqrt{x} f+1\right ) f^4-54 b^2 d^4 n^2 x^2 \log \left (d \sqrt{x} f+1\right ) f^4-108 a b d^4 n x^2 \log \left (d \sqrt{x} f+1\right ) f^4+54 a^2 d^4 x^2 \log (x) f^4+27 b^2 d^4 n^2 x^2 \log (x) f^4+54 a b d^4 n x^2 \log (x) f^4-54 b^2 d^4 n x^2 \log ^2(x) \log \left (c x^n\right ) f^4-216 a b d^4 x^2 \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) f^4-108 b^2 d^4 n x^2 \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) f^4+108 a b d^4 x^2 \log (x) \log \left (c x^n\right ) f^4+54 b^2 d^4 n x^2 \log (x) \log \left (c x^n\right ) f^4-216 b d^4 n x^2 \left (2 a+b n+2 b \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right ) f^4+864 b^2 d^4 n^2 x^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) f^4+108 b^2 d^3 x^{3/2} \log ^2\left (c x^n\right ) f^3+108 a^2 d^3 x^{3/2} f^3+1134 b^2 d^3 n^2 x^{3/2} f^3+540 a b d^3 n x^{3/2} f^3+216 a b d^3 x^{3/2} \log \left (c x^n\right ) f^3+540 b^2 d^3 n x^{3/2} \log \left (c x^n\right ) f^3-54 b^2 d^2 x \log ^2\left (c x^n\right ) f^2-54 a^2 d^2 x f^2-189 b^2 d^2 n^2 x f^2-162 a b d^2 n x f^2-108 a b d^2 x \log \left (c x^n\right ) f^2-162 b^2 d^2 n x \log \left (c x^n\right ) f^2+36 b^2 d \sqrt{x} \log ^2\left (c x^n\right ) f+72 a b d \sqrt{x} \log \left (c x^n\right ) f+84 b^2 d n \sqrt{x} \log \left (c x^n\right ) f+74 b^2 d n^2 \sqrt{x} f+36 a^2 d \sqrt{x} f+84 a b d n \sqrt{x} f+108 b^2 \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right )+108 a^2 \log \left (d \sqrt{x} f+1\right )+54 b^2 n^2 \log \left (d \sqrt{x} f+1\right )+108 a b n \log \left (d \sqrt{x} f+1\right )+216 a b \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )+108 b^2 n \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )}{216 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/x^3,x]

[Out]

-(36*a^2*d*f*Sqrt[x] + 84*a*b*d*f*n*Sqrt[x] + 74*b^2*d*f*n^2*Sqrt[x] - 54*a^2*d^2*f^2*x - 162*a*b*d^2*f^2*n*x
- 189*b^2*d^2*f^2*n^2*x + 108*a^2*d^3*f^3*x^(3/2) + 540*a*b*d^3*f^3*n*x^(3/2) + 1134*b^2*d^3*f^3*n^2*x^(3/2) +
108*a^2*Log[1 + d*f*Sqrt[x]] + 108*a*b*n*Log[1 + d*f*Sqrt[x]] + 54*b^2*n^2*Log[1 + d*f*Sqrt[x]] - 108*a^2*d^4
*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 108*a*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] - 54*b^2*d^4*f^4*n^2*x^2*Log[1 + d*
f*Sqrt[x]] + 54*a^2*d^4*f^4*x^2*Log[x] + 54*a*b*d^4*f^4*n*x^2*Log[x] + 27*b^2*d^4*f^4*n^2*x^2*Log[x] - 54*a*b*
d^4*f^4*n*x^2*Log[x]^2 - 27*b^2*d^4*f^4*n^2*x^2*Log[x]^2 + 18*b^2*d^4*f^4*n^2*x^2*Log[x]^3 + 72*a*b*d*f*Sqrt[x
]*Log[c*x^n] + 84*b^2*d*f*n*Sqrt[x]*Log[c*x^n] - 108*a*b*d^2*f^2*x*Log[c*x^n] - 162*b^2*d^2*f^2*n*x*Log[c*x^n]
+ 216*a*b*d^3*f^3*x^(3/2)*Log[c*x^n] + 540*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] + 216*a*b*Log[1 + d*f*Sqrt[x]]*Lo
g[c*x^n] + 108*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 216*a*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 1
08*b^2*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 108*a*b*d^4*f^4*x^2*Log[x]*Log[c*x^n] + 54*b^2*d^4*f^4*
n*x^2*Log[x]*Log[c*x^n] - 54*b^2*d^4*f^4*n*x^2*Log[x]^2*Log[c*x^n] + 36*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 54*b^2*
d^2*f^2*x*Log[c*x^n]^2 + 108*b^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 + 108*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 10
8*b^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 54*b^2*d^4*f^4*x^2*Log[x]*Log[c*x^n]^2 - 216*b*d^4*f^4*n
*x^2*(2*a + b*n + 2*b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 864*b^2*d^4*f^4*n^2*x^2*PolyLog[3, -(d*f*Sqrt[x
])])/(216*x^2)

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Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{{x}^{3}}\ln \left ( d \left ({d}^{-1}+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*(1/d+f*x^(1/2)))/x^3,x)

[Out]

int((a+b*ln(c*x^n))^2*ln(d*(1/d+f*x^(1/2)))/x^3,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} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{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*(1/d+f*x^(1/2)))/x^3,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)^2*log((f*sqrt(x) + 1/d)*d)/x^3, 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} + 1\right )}{x^{3}}, 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*(1/d+f*x^(1/2)))/x^3,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) + 1)/x^3, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )}{x^{3}}\,{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*(1/d+f*x^(1/2)))/x^3,x, algorithm="giac")

[Out]

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