3.1.59 \(\int x \log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^3 \, dx\) [59]

Optimal. Leaf size=858 \[ -\frac {255 b^3 n^3 \sqrt {x}}{8 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {45 b^3 n^3 x}{16 d^2 f^2}-\frac {175 b^3 n^3 x^{3/2}}{216 d f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 d^4 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.63, antiderivative size = 858, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 13, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {2504, 2442, 45, 2424, 2333, 2332, 2342, 2341, 2421, 2430, 6724, 2423, 2438} \begin {gather*} \frac {3}{8} n^3 x^2 b^3-\frac {175 n^3 x^{3/2} b^3}{216 d f}+\frac {45 n^3 x b^3}{16 d^2 f^2}+\frac {3 n^3 \log \left (d \sqrt {x} f+1\right ) b^3}{8 d^4 f^4}-\frac {3}{8} n^3 x^2 \log \left (d \sqrt {x} f+1\right ) b^3-\frac {9 n^2 x \log \left (c x^n\right ) b^3}{4 d^2 f^2}-\frac {3 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b^3}{2 d^4 f^4}-\frac {6 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) b^3}{d^4 f^4}-\frac {24 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right ) b^3}{d^4 f^4}-\frac {255 n^3 \sqrt {x} b^3}{8 d^3 f^3}-\frac {9 a n^2 x b^2}{4 d^2 f^2}-\frac {9}{16} n^2 x^2 \left (a+b \log \left (c x^n\right )\right ) b^2+\frac {37 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right ) b^2}{36 d f}-\frac {3 n^2 x \left (a+b \log \left (c x^n\right )\right ) b^2}{8 d^2 f^2}-\frac {3 n^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^4 f^4}+\frac {3}{4} n^2 x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right ) b^2+\frac {63 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) b^2}{4 d^3 f^3}+\frac {3 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {12 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-d f \sqrt {x}\right ) b^2}{d^4 f^4}+\frac {3}{8} n x^2 \left (a+b \log \left (c x^n\right )\right )^2 b-\frac {7 n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2 b}{12 d f}+\frac {9 n x \left (a+b \log \left (c x^n\right )\right )^2 b}{8 d^2 f^2}-\frac {3}{4} n x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b+\frac {3 n \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^4 f^4}-\frac {15 n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 b}{4 d^3 f^3}-\frac {3 n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) b}{d^4 f^4}-\frac {1}{8} x^2 \left (a+b \log \left (c x^n\right )\right )^3+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {1}{2} x^2 \log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (d \sqrt {x} f+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 d^4 f^4}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2341

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 2342

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2423

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 2424

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 2430

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[PolyLo
g[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 2438

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 2442

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 2504

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 6724

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 x \log \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3 \, dx &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-(3 b n) \int \left (-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{4 d^2 f^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{6 d f}-\frac {1}{8} x \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 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2\right ) \, dx\\ &=\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {1}{8} (3 b n) \int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx-\frac {1}{2} (3 b n) \int x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2 \, dx+\frac {(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} \, dx}{2 d^4 f^4}-\frac {(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}} \, dx}{2 d^3 f^3}+\frac {(3 b n) \int \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{4 d^2 f^2}-\frac {(b n) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2 \, dx}{2 d f}\\ &=-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\left (3 b^2 n^2\right ) \int \left (-\frac {a+b \log \left (c x^n\right )}{4 d^2 f^2}+\frac {a+b \log \left (c x^n\right )}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{6 d f}-\frac {1}{8} x \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )\right ) \, dx+\frac {\left (6 b^2 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}{d^4 f^4}+\frac {\left (6 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{d^3 f^3}-\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d^2 f^2}+\frac {\left (2 b^2 n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{3 d f}\\ &=-\frac {24 b^3 n^3 \sqrt {x}}{d^3 f^3}-\frac {3 a b^2 n^2 x}{2 d^2 f^2}-\frac {8 b^3 n^3 x^{3/2}}{27 d f}+\frac {3}{32} b^3 n^3 x^2+\frac {12 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{d^3 f^3}+\frac {4 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 d f}-\frac {3}{16} b^2 n^2 x^2 \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{8} \left (3 b^2 n^2\right ) \int x \left (a+b \log \left (c x^n\right )\right ) \, dx+\frac {1}{2} \left (3 b^2 n^2\right ) \int x \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx-\frac {\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} \, dx}{2 d^4 f^4}+\frac {\left (3 b^2 n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{\sqrt {x}} \, dx}{2 d^3 f^3}-\frac {\left (3 b^2 n^2\right ) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{4 d^2 f^2}-\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{2 d^2 f^2}+\frac {\left (b^2 n^2\right ) \int \sqrt {x} \left (a+b \log \left (c x^n\right )\right ) \, dx}{2 d f}-\frac {\left (12 b^3 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac {30 b^3 n^3 \sqrt {x}}{d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {3 b^3 n^3 x}{2 d^2 f^2}-\frac {14 b^3 n^3 x^{3/2}}{27 d f}+\frac {3}{16} b^3 n^3 x^2-\frac {3 b^3 n^2 x \log \left (c x^n\right )}{2 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {\left (3 b^3 n^2\right ) \int \log \left (c x^n\right ) \, dx}{4 d^2 f^2}-\frac {1}{2} \left (3 b^3 n^3\right ) \int \left (-\frac {1}{4 d^2 f^2}+\frac {1}{2 d^3 f^3 \sqrt {x}}+\frac {\sqrt {x}}{6 d f}-\frac {x}{8}-\frac {\log \left (1+d f \sqrt {x}\right )}{2 d^4 f^4 x}+\frac {1}{2} x \log \left (1+d f \sqrt {x}\right )\right ) \, dx-\frac {\left (3 b^3 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx}{d^4 f^4}\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{4} \left (3 b^3 n^3\right ) \int x \log \left (1+d f \sqrt {x}\right ) \, dx+\frac {\left (3 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx}{4 d^4 f^4}\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {1}{2} \left (3 b^3 n^3\right ) \text {Subst}\left (\int x^3 \log (1+d f x) \, dx,x,\sqrt {x}\right )\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {x^4}{1+d f x} \, dx,x,\sqrt {x}\right )\\ &=-\frac {63 b^3 n^3 \sqrt {x}}{2 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {21 b^3 n^3 x}{8 d^2 f^2}-\frac {37 b^3 n^3 x^{3/2}}{54 d f}+\frac {9}{32} b^3 n^3 x^2-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {1}{8} \left (3 b^3 d f n^3\right ) \text {Subst}\left (\int \left (-\frac {1}{d^4 f^4}+\frac {x}{d^3 f^3}-\frac {x^2}{d^2 f^2}+\frac {x^3}{d f}+\frac {1}{d^4 f^4 (1+d f x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {255 b^3 n^3 \sqrt {x}}{8 d^3 f^3}-\frac {9 a b^2 n^2 x}{4 d^2 f^2}+\frac {45 b^3 n^3 x}{16 d^2 f^2}-\frac {175 b^3 n^3 x^{3/2}}{216 d f}+\frac {3}{8} b^3 n^3 x^2+\frac {3 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{8 d^4 f^4}-\frac {3}{8} b^3 n^3 x^2 \log \left (1+d f \sqrt {x}\right )-\frac {9 b^3 n^2 x \log \left (c x^n\right )}{4 d^2 f^2}+\frac {63 b^2 n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{4 d^3 f^3}-\frac {3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 f^2}+\frac {37 b^2 n^2 x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{36 d f}-\frac {9}{16} b^2 n^2 x^2 \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 d^4 f^4}+\frac {3}{4} b^2 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {15 b n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{4 d^3 f^3}+\frac {9 b n x \left (a+b \log \left (c x^n\right )\right )^2}{8 d^2 f^2}-\frac {7 b n x^{3/2} \left (a+b \log \left (c x^n\right )\right )^2}{12 d f}+\frac {3}{8} b n x^2 \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 d^4 f^4}-\frac {3}{4} b n x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {\sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{2 d^3 f^3}-\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{4 d^2 f^2}+\frac {x^{3/2} \left (a+b \log \left (c x^n\right )\right )^3}{6 d f}-\frac {1}{8} x^2 \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 d^4 f^4}+\frac {1}{2} x^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {3 b^3 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )}{2 d^4 f^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {6 b^3 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}+\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )}{d^4 f^4}-\frac {24 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{d^4 f^4}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 1432, normalized size = 1.67 \begin {gather*} \frac {216 a^3 d f \sqrt {x}-1620 a^2 b d f n \sqrt {x}+6804 a b^2 d f n^2 \sqrt {x}-13770 b^3 d f n^3 \sqrt {x}-108 a^3 d^2 f^2 x+486 a^2 b d^2 f^2 n x-1134 a b^2 d^2 f^2 n^2 x+1215 b^3 d^2 f^2 n^3 x+72 a^3 d^3 f^3 x^{3/2}-252 a^2 b d^3 f^3 n x^{3/2}+444 a b^2 d^3 f^3 n^2 x^{3/2}-350 b^3 d^3 f^3 n^3 x^{3/2}-54 a^3 d^4 f^4 x^2+162 a^2 b d^4 f^4 n x^2-243 a b^2 d^4 f^4 n^2 x^2+162 b^3 d^4 f^4 n^3 x^2-216 a^3 \log \left (1+d f \sqrt {x}\right )+324 a^2 b n \log \left (1+d f \sqrt {x}\right )-324 a b^2 n^2 \log \left (1+d f \sqrt {x}\right )+162 b^3 n^3 \log \left (1+d f \sqrt {x}\right )+216 a^3 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right )-324 a^2 b d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right )+324 a b^2 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right )-162 b^3 d^4 f^4 n^3 x^2 \log \left (1+d f \sqrt {x}\right )+648 a^2 b d f \sqrt {x} \log \left (c x^n\right )-3240 a b^2 d f n \sqrt {x} \log \left (c x^n\right )+6804 b^3 d f n^2 \sqrt {x} \log \left (c x^n\right )-324 a^2 b d^2 f^2 x \log \left (c x^n\right )+972 a b^2 d^2 f^2 n x \log \left (c x^n\right )-1134 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+216 a^2 b d^3 f^3 x^{3/2} \log \left (c x^n\right )-504 a b^2 d^3 f^3 n x^{3/2} \log \left (c x^n\right )+444 b^3 d^3 f^3 n^2 x^{3/2} \log \left (c x^n\right )-162 a^2 b d^4 f^4 x^2 \log \left (c x^n\right )+324 a b^2 d^4 f^4 n x^2 \log \left (c x^n\right )-243 b^3 d^4 f^4 n^2 x^2 \log \left (c x^n\right )-648 a^2 b \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a b^2 n \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-324 b^3 n^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a^2 b d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )-648 a b^2 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+324 b^3 d^4 f^4 n^2 x^2 \log \left (1+d f \sqrt {x}\right ) \log \left (c x^n\right )+648 a b^2 d f \sqrt {x} \log ^2\left (c x^n\right )-1620 b^3 d f n \sqrt {x} \log ^2\left (c x^n\right )-324 a b^2 d^2 f^2 x \log ^2\left (c x^n\right )+486 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )+216 a b^2 d^3 f^3 x^{3/2} \log ^2\left (c x^n\right )-252 b^3 d^3 f^3 n x^{3/2} \log ^2\left (c x^n\right )-162 a b^2 d^4 f^4 x^2 \log ^2\left (c x^n\right )+162 b^3 d^4 f^4 n x^2 \log ^2\left (c x^n\right )-648 a b^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+324 b^3 n \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+648 a b^2 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )-324 b^3 d^4 f^4 n x^2 \log \left (1+d f \sqrt {x}\right ) \log ^2\left (c x^n\right )+216 b^3 d f \sqrt {x} \log ^3\left (c x^n\right )-108 b^3 d^2 f^2 x \log ^3\left (c x^n\right )+72 b^3 d^3 f^3 x^{3/2} \log ^3\left (c x^n\right )-54 b^3 d^4 f^4 x^2 \log ^3\left (c x^n\right )-216 b^3 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )+216 b^3 d^4 f^4 x^2 \log \left (1+d f \sqrt {x}\right ) \log ^3\left (c x^n\right )-648 b n \left (2 a^2-2 a b n+b^2 n^2-2 b (-2 a+b n) \log \left (c x^n\right )+2 b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+2592 b^2 n^2 \left (2 a-b n+2 b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-10368 b^3 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )}{432 d^4 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(216*a^3*d*f*Sqrt[x] - 1620*a^2*b*d*f*n*Sqrt[x] + 6804*a*b^2*d*f*n^2*Sqrt[x] - 13770*b^3*d*f*n^3*Sqrt[x] - 108
*a^3*d^2*f^2*x + 486*a^2*b*d^2*f^2*n*x - 1134*a*b^2*d^2*f^2*n^2*x + 1215*b^3*d^2*f^2*n^3*x + 72*a^3*d^3*f^3*x^
(3/2) - 252*a^2*b*d^3*f^3*n*x^(3/2) + 444*a*b^2*d^3*f^3*n^2*x^(3/2) - 350*b^3*d^3*f^3*n^3*x^(3/2) - 54*a^3*d^4
*f^4*x^2 + 162*a^2*b*d^4*f^4*n*x^2 - 243*a*b^2*d^4*f^4*n^2*x^2 + 162*b^3*d^4*f^4*n^3*x^2 - 216*a^3*Log[1 + d*f
*Sqrt[x]] + 324*a^2*b*n*Log[1 + d*f*Sqrt[x]] - 324*a*b^2*n^2*Log[1 + d*f*Sqrt[x]] + 162*b^3*n^3*Log[1 + d*f*Sq
rt[x]] + 216*a^3*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]] - 324*a^2*b*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]] + 324*a*b^2*d
^4*f^4*n^2*x^2*Log[1 + d*f*Sqrt[x]] - 162*b^3*d^4*f^4*n^3*x^2*Log[1 + d*f*Sqrt[x]] + 648*a^2*b*d*f*Sqrt[x]*Log
[c*x^n] - 3240*a*b^2*d*f*n*Sqrt[x]*Log[c*x^n] + 6804*b^3*d*f*n^2*Sqrt[x]*Log[c*x^n] - 324*a^2*b*d^2*f^2*x*Log[
c*x^n] + 972*a*b^2*d^2*f^2*n*x*Log[c*x^n] - 1134*b^3*d^2*f^2*n^2*x*Log[c*x^n] + 216*a^2*b*d^3*f^3*x^(3/2)*Log[
c*x^n] - 504*a*b^2*d^3*f^3*n*x^(3/2)*Log[c*x^n] + 444*b^3*d^3*f^3*n^2*x^(3/2)*Log[c*x^n] - 162*a^2*b*d^4*f^4*x
^2*Log[c*x^n] + 324*a*b^2*d^4*f^4*n*x^2*Log[c*x^n] - 243*b^3*d^4*f^4*n^2*x^2*Log[c*x^n] - 648*a^2*b*Log[1 + d*
f*Sqrt[x]]*Log[c*x^n] + 648*a*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 324*b^3*n^2*Log[1 + d*f*Sqrt[x]]*Log[c*x
^n] + 648*a^2*b*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 648*a*b^2*d^4*f^4*n*x^2*Log[1 + d*f*Sqrt[x]]*Log
[c*x^n] + 324*b^3*d^4*f^4*n^2*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 648*a*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 - 1620*
b^3*d*f*n*Sqrt[x]*Log[c*x^n]^2 - 324*a*b^2*d^2*f^2*x*Log[c*x^n]^2 + 486*b^3*d^2*f^2*n*x*Log[c*x^n]^2 + 216*a*b
^2*d^3*f^3*x^(3/2)*Log[c*x^n]^2 - 252*b^3*d^3*f^3*n*x^(3/2)*Log[c*x^n]^2 - 162*a*b^2*d^4*f^4*x^2*Log[c*x^n]^2
+ 162*b^3*d^4*f^4*n*x^2*Log[c*x^n]^2 - 648*a*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 324*b^3*n*Log[1 + d*f*Sqr
t[x]]*Log[c*x^n]^2 + 648*a*b^2*d^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 324*b^3*d^4*f^4*n*x^2*Log[1 + d
*f*Sqrt[x]]*Log[c*x^n]^2 + 216*b^3*d*f*Sqrt[x]*Log[c*x^n]^3 - 108*b^3*d^2*f^2*x*Log[c*x^n]^3 + 72*b^3*d^3*f^3*
x^(3/2)*Log[c*x^n]^3 - 54*b^3*d^4*f^4*x^2*Log[c*x^n]^3 - 216*b^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3 + 216*b^3*d
^4*f^4*x^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3 - 648*b*n*(2*a^2 - 2*a*b*n + b^2*n^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])] + 2592*b^2*n^2*(2*a - b*n + 2*b*Log[c*x^n])*PolyLog[3, -(d*
f*Sqrt[x])] - 10368*b^3*n^3*PolyLog[4, -(d*f*Sqrt[x])])/(432*d^4*f^4)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int x \left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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