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

Optimal. Leaf size=849 \[ -\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b n}-\frac {1}{8} d^4 \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {1}{2} d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {3}{16} b^3 d^4 n^3 \log ^2(x) f^4+\frac {3}{4} b d^4 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4+\frac {3}{8} b^3 d^4 n^3 \log \left (d \sqrt {x} f+1\right ) f^4-\frac {3}{16} b^3 d^4 n^3 \log (x) f^4+\frac {3}{4} b^2 d^4 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4-\frac {3}{8} b^2 d^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4+\frac {3}{2} b^3 d^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right ) f^4+3 b d^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right ) f^4+3 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right ) f^4-6 b^3 d^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right ) f^4-12 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right ) f^4+24 b^3 d^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right ) f^4-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 \sqrt {x}}-\frac {15 b d^3 n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 \sqrt {x}}-\frac {63 b^2 d^3 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 \sqrt {x}}-\frac {255 b^3 d^3 n^3 f^3}{8 \sqrt {x}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 x}+\frac {9 b d^2 n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 x}+\frac {21 b^2 d^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 x}+\frac {45 b^3 d^2 n^3 f^2}{16 x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3 f}{6 x^{3/2}}-\frac {7 b d n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 x^{3/2}}-\frac {37 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 x^{3/2}}-\frac {175 b^3 d n^3 f}{216 x^{3/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3 b^3 n^3 \log \left (d \sqrt {x} f+1\right )}{8 x^2}-\frac {3 b^2 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \]

[Out]

-1/8*d^4*f^4*(a+b*ln(c*x^n))^3-1/2*(a+b*ln(c*x^n))^3*ln(1+d*f*x^(1/2))/x^2-1/6*d*f*(a+b*ln(c*x^n))^3/x^(3/2)+1
/4*d^2*f^2*(a+b*ln(c*x^n))^3/x-3/8*b^3*n^3*ln(1+d*f*x^(1/2))/x^2+1/2*d^4*f^4*(a+b*ln(c*x^n))^3*ln(1+d*f*x^(1/2
))-1/2*d^3*f^3*(a+b*ln(c*x^n))^3/x^(1/2)-175/216*b^3*d*f*n^3/x^(3/2)+45/16*b^3*d^2*f^2*n^3/x-3/8*b^2*d^4*f^4*n
^2*ln(x)*(a+b*ln(c*x^n))-7/12*b*d*f*n*(a+b*ln(c*x^n))^2/x^(3/2)+9/8*b*d^2*f^2*n*(a+b*ln(c*x^n))^2/x+3/4*b^2*d^
4*f^4*n^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))+3/4*b*d^4*f^4*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))+3*b^2*d^4*f^4*
n^2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))+3*b*d^4*f^4*n*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^(1/2))-12*b^2*d^4
*f^4*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))-63/4*b^2*d^3*f^3*n^2*(a+b*ln(c*x^n))/x^(1/2)-15/4*b*d^3*f^3*n
*(a+b*ln(c*x^n))^2/x^(1/2)-3/16*b^3*d^4*f^4*n^3*ln(x)+3/16*b^3*d^4*f^4*n^3*ln(x)^2-1/16*d^4*f^4*(a+b*ln(c*x^n)
)^4/b/n+3/8*b^3*d^4*f^4*n^3*ln(1+d*f*x^(1/2))-3/4*b^2*n^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/x^2-3/4*b*n*(a+b*l
n(c*x^n))^2*ln(1+d*f*x^(1/2))/x^2+3/2*b^3*d^4*f^4*n^3*polylog(2,-d*f*x^(1/2))-6*b^3*d^4*f^4*n^3*polylog(3,-d*f
*x^(1/2))+24*b^3*d^4*f^4*n^3*polylog(4,-d*f*x^(1/2))-255/8*b^3*d^3*f^3*n^3/x^(1/2)-37/36*b^2*d*f*n^2*(a+b*ln(c
*x^n))/x^(3/2)+21/8*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))/x

________________________________________________________________________________________

Rubi [A]  time = 1.14, antiderivative size = 849, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2454, 2395, 44, 2377, 2305, 2304, 2376, 2391, 2301, 2374, 6589, 2366, 12, 2302, 30, 2383} \[ -\frac {d^4 \left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b n}-\frac {1}{8} d^4 \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {1}{2} d^4 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4+\frac {3}{16} b^3 d^4 n^3 \log ^2(x) f^4+\frac {3}{4} b d^4 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4+\frac {3}{8} b^3 d^4 n^3 \log \left (d \sqrt {x} f+1\right ) f^4-\frac {3}{16} b^3 d^4 n^3 \log (x) f^4+\frac {3}{4} b^2 d^4 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) f^4-\frac {3}{8} b^2 d^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4+\frac {3}{2} b^3 d^4 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4+3 b d^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4+3 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-d f \sqrt {x}\right ) f^4-6 b^3 d^4 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4-12 b^2 d^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-d f \sqrt {x}\right ) f^4+24 b^3 d^4 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right ) f^4-\frac {d^3 \left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 \sqrt {x}}-\frac {15 b d^3 n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 \sqrt {x}}-\frac {63 b^2 d^3 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 \sqrt {x}}-\frac {255 b^3 d^3 n^3 f^3}{8 \sqrt {x}}+\frac {d^2 \left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 x}+\frac {9 b d^2 n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 x}+\frac {21 b^2 d^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 x}+\frac {45 b^3 d^2 n^3 f^2}{16 x}-\frac {d \left (a+b \log \left (c x^n\right )\right )^3 f}{6 x^{3/2}}-\frac {7 b d n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 x^{3/2}}-\frac {37 b^2 d n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 x^{3/2}}-\frac {175 b^3 d n^3 f}{216 x^{3/2}}-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3 b^3 n^3 \log \left (d \sqrt {x} f+1\right )}{8 x^2}-\frac {3 b^2 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-175*b^3*d*f*n^3)/(216*x^(3/2)) + (45*b^3*d^2*f^2*n^3)/(16*x) - (255*b^3*d^3*f^3*n^3)/(8*Sqrt[x]) + (3*b^3*d^
4*f^4*n^3*Log[1 + d*f*Sqrt[x]])/8 - (3*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(8*x^2) - (3*b^3*d^4*f^4*n^3*Log[x])/16 +
(3*b^3*d^4*f^4*n^3*Log[x]^2)/16 - (37*b^2*d*f*n^2*(a + b*Log[c*x^n]))/(36*x^(3/2)) + (21*b^2*d^2*f^2*n^2*(a +
b*Log[c*x^n]))/(8*x) - (63*b^2*d^3*f^3*n^2*(a + b*Log[c*x^n]))/(4*Sqrt[x]) + (3*b^2*d^4*f^4*n^2*Log[1 + d*f*S
qrt[x]]*(a + b*Log[c*x^n]))/4 - (3*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*x^2) - (3*b^2*d^4*f^4*n
^2*Log[x]*(a + b*Log[c*x^n]))/8 - (7*b*d*f*n*(a + b*Log[c*x^n])^2)/(12*x^(3/2)) + (9*b*d^2*f^2*n*(a + b*Log[c*
x^n])^2)/(8*x) - (15*b*d^3*f^3*n*(a + b*Log[c*x^n])^2)/(4*Sqrt[x]) + (3*b*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*(a +
b*Log[c*x^n])^2)/4 - (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(4*x^2) - (d^4*f^4*(a + b*Log[c*x^n])^3
)/8 - (d*f*(a + b*Log[c*x^n])^3)/(6*x^(3/2)) + (d^2*f^2*(a + b*Log[c*x^n])^3)/(4*x) - (d^3*f^3*(a + b*Log[c*x^
n])^3)/(2*Sqrt[x]) + (d^4*f^4*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[
c*x^n])^3)/(2*x^2) - (d^4*f^4*(a + b*Log[c*x^n])^4)/(16*b*n) + (3*b^3*d^4*f^4*n^3*PolyLog[2, -(d*f*Sqrt[x])])/
2 + 3*b^2*d^4*f^4*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 3*b*d^4*f^4*n*(a + b*Log[c*x^n])^2*PolyL
og[2, -(d*f*Sqrt[x])] - 6*b^3*d^4*f^4*n^3*PolyLog[3, -(d*f*Sqrt[x])] - 12*b^2*d^4*f^4*n^2*(a + b*Log[c*x^n])*P
olyLog[3, -(d*f*Sqrt[x])] + 24*b^3*d^4*f^4*n^3*PolyLog[4, -(d*f*Sqrt[x])]

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 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 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 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 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 (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x^3} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )^2}{6 x^{5/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^2}{2 x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{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}{2 x}-\frac {d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{4 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {1}{4} d^4 f^4 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{2} (3 b n) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^3} \, dx+\frac {1}{2} (b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{5/2}} \, dx-\frac {1}{4} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx+\frac {1}{2} \left (3 b d^3 f^3 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac {1}{4} \left (3 b d^4 f^4 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\frac {1}{2} \left (3 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 )^2}{x} \, dx\\ &=-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3}{8} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{4} \left (3 b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (3 b^2 n^2\right ) \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 {1}{3} \left (2 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{2} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\left (6 b^2 d^3 f^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (6 b^2 d^4 f^4 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {8 b^3 d f n^3}{27 x^{3/2}}+\frac {3 b^3 d^2 f^2 n^3}{2 x}-\frac {24 b^3 d^3 f^3 n^3}{\sqrt {x}}-\frac {4 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {3 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{2 x}-\frac {12 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3}{8} b d^4 f^4 n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \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 )^3}{x} \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \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}{2} \left (b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{5/2}} \, dx-\frac {1}{4} \left (3 b^2 d^2 f^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx+\frac {1}{2} \left (3 b^2 d^3 f^3 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\frac {1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\frac {1}{2} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx+\left (12 b^3 d^4 f^4 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {14 b^3 d f n^3}{27 x^{3/2}}+\frac {9 b^3 d^2 f^2 n^3}{4 x}-\frac {30 b^3 d^3 f^3 n^3}{\sqrt {x}}-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {\left (d^4 f^4\right ) \operatorname {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {1}{4} \left (3 b^2 d^4 f^4 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\frac {1}{2} \left (3 b^3 n^3\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 (3 b^3 d^4 f^4 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{8} \left (3 b d^4 f^4 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\frac {1}{4} \left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^3} \, dx+\frac {1}{8} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac {\log (x)}{x} \, dx-\frac {1}{4} \left (3 b^3 d^4 f^4 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{8} \left (3 d^4 f^4\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\frac {1}{2} \left (3 b^3 n^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+d f x)}{x^5} \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\frac {1}{8} \left (3 b^3 d f n^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^4 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {37 b^3 d f n^3}{54 x^{3/2}}+\frac {21 b^3 d^2 f^2 n^3}{8 x}-\frac {63 b^3 d^3 f^3 n^3}{2 \sqrt {x}}-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\frac {1}{8} \left (3 b^3 d f n^3\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 {175 b^3 d f n^3}{216 x^{3/2}}+\frac {45 b^3 d^2 f^2 n^3}{16 x}-\frac {255 b^3 d^3 f^3 n^3}{8 \sqrt {x}}+\frac {3}{8} b^3 d^4 f^4 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 x^2}-\frac {3}{16} b^3 d^4 f^4 n^3 \log (x)+\frac {3}{16} b^3 d^4 f^4 n^3 \log ^2(x)-\frac {37 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 x^{3/2}}+\frac {21 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 x}-\frac {63 b^2 d^3 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 \sqrt {x}}+\frac {3}{4} b^2 d^4 f^4 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {3 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3}{8} b^2 d^4 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {7 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{12 x^{3/2}}+\frac {9 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 x}-\frac {15 b d^3 f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 \sqrt {x}}+\frac {3}{4} b d^4 f^4 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {1}{8} d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{6 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )^3}{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 )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b n}+\frac {3}{2} b^3 d^4 f^4 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+3 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+3 b d^4 f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-6 b^3 d^4 f^4 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-12 b^2 d^4 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+24 b^3 d^4 f^4 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )\\ \end {align*}

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Mathematica [B]  time = 1.07, size = 2009, normalized size = 2.37 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(a^3*d*f)/x^(3/2) - (7*a^2*b*d*f*n)/(12*x^(3/2)) - (37*a*b^2*d*f*n^2)/(36*x^(3/2)) - (175*b^3*d*f*n^3)/(2
16*x^(3/2)) + (a^3*d^2*f^2)/(4*x) + (9*a^2*b*d^2*f^2*n)/(8*x) + (21*a*b^2*d^2*f^2*n^2)/(8*x) + (45*b^3*d^2*f^2
*n^3)/(16*x) - (a^3*d^3*f^3)/(2*Sqrt[x]) - (15*a^2*b*d^3*f^3*n)/(4*Sqrt[x]) - (63*a*b^2*d^3*f^3*n^2)/(4*Sqrt[x
]) - (255*b^3*d^3*f^3*n^3)/(8*Sqrt[x]) + (a^3*d^4*f^4*Log[1 + d*f*Sqrt[x]])/2 + (3*a^2*b*d^4*f^4*n*Log[1 + d*f
*Sqrt[x]])/4 + (3*a*b^2*d^4*f^4*n^2*Log[1 + d*f*Sqrt[x]])/4 + (3*b^3*d^4*f^4*n^3*Log[1 + d*f*Sqrt[x]])/8 - (a^
3*Log[1 + d*f*Sqrt[x]])/(2*x^2) - (3*a^2*b*n*Log[1 + d*f*Sqrt[x]])/(4*x^2) - (3*a*b^2*n^2*Log[1 + d*f*Sqrt[x]]
)/(4*x^2) - (3*b^3*n^3*Log[1 + d*f*Sqrt[x]])/(8*x^2) - (a^3*d^4*f^4*Log[x])/4 - (3*a^2*b*d^4*f^4*n*Log[x])/8 -
(3*a*b^2*d^4*f^4*n^2*Log[x])/8 - (3*b^3*d^4*f^4*n^3*Log[x])/16 + (3*a^2*b*d^4*f^4*n*Log[x]^2)/8 + (3*a*b^2*d^
4*f^4*n^2*Log[x]^2)/8 + (3*b^3*d^4*f^4*n^3*Log[x]^2)/16 - (a*b^2*d^4*f^4*n^2*Log[x]^3)/4 - (b^3*d^4*f^4*n^3*Lo
g[x]^3)/8 + (b^3*d^4*f^4*n^3*Log[1 + 1/(d*f*Sqrt[x])]*Log[x]^3)/2 - (b^3*d^4*f^4*n^3*Log[1 + d*f*Sqrt[x]]*Log[
x]^3)/2 + (b^3*d^4*f^4*n^3*Log[x]^4)/8 - (a^2*b*d*f*Log[c*x^n])/(2*x^(3/2)) - (7*a*b^2*d*f*n*Log[c*x^n])/(6*x^
(3/2)) - (37*b^3*d*f*n^2*Log[c*x^n])/(36*x^(3/2)) + (3*a^2*b*d^2*f^2*Log[c*x^n])/(4*x) + (9*a*b^2*d^2*f^2*n*Lo
g[c*x^n])/(4*x) + (21*b^3*d^2*f^2*n^2*Log[c*x^n])/(8*x) - (3*a^2*b*d^3*f^3*Log[c*x^n])/(2*Sqrt[x]) - (15*a*b^2
*d^3*f^3*n*Log[c*x^n])/(2*Sqrt[x]) - (63*b^3*d^3*f^3*n^2*Log[c*x^n])/(4*Sqrt[x]) + (3*a^2*b*d^4*f^4*Log[1 + d*
f*Sqrt[x]]*Log[c*x^n])/2 + (3*a*b^2*d^4*f^4*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n])/2 + (3*b^3*d^4*f^4*n^2*Log[1 +
d*f*Sqrt[x]]*Log[c*x^n])/4 - (3*a^2*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n])/(2*x^2) - (3*a*b^2*n*Log[1 + d*f*Sqrt[x
]]*Log[c*x^n])/(2*x^2) - (3*b^3*n^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n])/(4*x^2) - (3*a^2*b*d^4*f^4*Log[x]*Log[c*x
^n])/4 - (3*a*b^2*d^4*f^4*n*Log[x]*Log[c*x^n])/4 - (3*b^3*d^4*f^4*n^2*Log[x]*Log[c*x^n])/8 + (3*a*b^2*d^4*f^4*
n*Log[x]^2*Log[c*x^n])/4 + (3*b^3*d^4*f^4*n^2*Log[x]^2*Log[c*x^n])/8 - (b^3*d^4*f^4*n^2*Log[x]^3*Log[c*x^n])/4
- (a*b^2*d*f*Log[c*x^n]^2)/(2*x^(3/2)) - (7*b^3*d*f*n*Log[c*x^n]^2)/(12*x^(3/2)) + (3*a*b^2*d^2*f^2*Log[c*x^n
]^2)/(4*x) + (9*b^3*d^2*f^2*n*Log[c*x^n]^2)/(8*x) - (3*a*b^2*d^3*f^3*Log[c*x^n]^2)/(2*Sqrt[x]) - (15*b^3*d^3*f
^3*n*Log[c*x^n]^2)/(4*Sqrt[x]) + (3*a*b^2*d^4*f^4*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2)/2 + (3*b^3*d^4*f^4*n*Log[
1 + d*f*Sqrt[x]]*Log[c*x^n]^2)/4 - (3*a*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2)/(2*x^2) - (3*b^3*n*Log[1 + d*f*
Sqrt[x]]*Log[c*x^n]^2)/(4*x^2) - (3*a*b^2*d^4*f^4*Log[x]*Log[c*x^n]^2)/4 - (3*b^3*d^4*f^4*n*Log[x]*Log[c*x^n]^
2)/8 + (3*b^3*d^4*f^4*n*Log[x]^2*Log[c*x^n]^2)/8 - (b^3*d*f*Log[c*x^n]^3)/(6*x^(3/2)) + (b^3*d^2*f^2*Log[c*x^n
]^3)/(4*x) - (b^3*d^3*f^3*Log[c*x^n]^3)/(2*Sqrt[x]) + (b^3*d^4*f^4*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3)/2 - (b^3
*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3)/(2*x^2) - (b^3*d^4*f^4*Log[x]*Log[c*x^n]^3)/4 - 3*b^3*d^4*f^4*n^3*Log[x]^2
*PolyLog[2, -(1/(d*f*Sqrt[x]))] + (3*b*d^4*f^4*n*(2*a^2 + 2*a*b*n + b^2*n^2 - 2*b^2*n^2*Log[x]^2 + 2*b*(2*a +
b*n)*Log[c*x^n] + 2*b^2*Log[c*x^n]^2)*PolyLog[2, -(d*f*Sqrt[x])])/2 - 12*b^3*d^4*f^4*n^3*Log[x]*PolyLog[3, -(1
/(d*f*Sqrt[x]))] - 12*a*b^2*d^4*f^4*n^2*PolyLog[3, -(d*f*Sqrt[x])] - 6*b^3*d^4*f^4*n^3*PolyLog[3, -(d*f*Sqrt[x
])] + 12*b^3*d^4*f^4*n^3*Log[x]*PolyLog[3, -(d*f*Sqrt[x])] - 12*b^3*d^4*f^4*n^2*Log[c*x^n]*PolyLog[3, -(d*f*Sq
rt[x])] - 24*b^3*d^4*f^4*n^3*PolyLog[4, -(1/(d*f*Sqrt[x]))]

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fricas [F]  time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt {x} + 1\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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