3.2.32 \(\int \frac {\log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3}{x^3} \, dx\) [132]

Optimal. Leaf size=914 \[ -\frac {175 b^3 f n^3}{216 e x^{3/2}}+\frac {45 b^3 f^2 n^3}{16 e^2 x}-\frac {255 b^3 f^3 n^3}{8 e^3 \sqrt {x}}+\frac {3 b^3 f^4 n^3 \log \left (e+f \sqrt {x}\right )}{8 e^4}-\frac {3 b^3 n^3 \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}-\frac {3 b^3 f^4 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{2 e^4}-\frac {3 b^3 f^4 n^3 \log (x)}{16 e^4}+\frac {3 b^3 f^4 n^3 \log ^2(x)}{16 e^4}-\frac {37 b^2 f n^2 \left (a+b \log \left (c x^n\right )\right )}{36 e x^{3/2}}+\frac {21 b^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2 x}-\frac {63 b^2 f^3 n^2 \left (a+b \log \left (c x^n\right )\right )}{4 e^3 \sqrt {x}}+\frac {3 b^2 f^4 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}-\frac {3 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2}-\frac {3 b^2 f^4 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{8 e^4}-\frac {7 b f n \left (a+b \log \left (c x^n\right )\right )^2}{12 e x^{3/2}}+\frac {9 b f^2 n \left (a+b \log \left (c x^n\right )\right )^2}{8 e^2 x}-\frac {15 b f^3 n \left (a+b \log \left (c x^n\right )\right )^2}{4 e^3 \sqrt {x}}-\frac {3 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}+\frac {3 b f^4 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^3}{8 e^4}-\frac {f \left (a+b \log \left (c x^n\right )\right )^3}{6 e x^{3/2}}+\frac {f^2 \left (a+b \log \left (c x^n\right )\right )^3}{4 e^2 x}-\frac {f^3 \left (a+b \log \left (c x^n\right )\right )^3}{2 e^3 \sqrt {x}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}+\frac {f^4 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 e^4}-\frac {f^4 \left (a+b \log \left (c x^n\right )\right )^4}{16 b e^4 n}-\frac {3 b^3 f^4 n^3 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{2 e^4}+\frac {3 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {3 b f^4 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {6 b^3 f^4 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}-\frac {12 b^2 f^4 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{e^4}+\frac {24 b^3 f^4 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.01, antiderivative size = 914, normalized size of antiderivative = 1.00, number of steps used = 40, number of rules used = 19, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.679, Rules used = {2504, 2442, 46, 2424, 2342, 2341, 2422, 2375, 2421, 2430, 6724, 2423, 2441, 2352, 2338, 2413, 12, 2339, 30} \begin {gather*} -\frac {\left (a+b \log \left (c x^n\right )\right )^4 f^4}{16 b e^4 n}+\frac {\log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3 f^4}{2 e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^4}{8 e^4}+\frac {3 b^3 n^3 \log ^2(x) f^4}{16 e^4}+\frac {3 b n \log \left (\frac {\sqrt {x} f}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2 f^4}{4 e^4}+\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) f^4}{8 e^4}-\frac {3 b^3 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right ) f^4}{2 e^4}-\frac {3 b^3 n^3 \log (x) f^4}{16 e^4}+\frac {3 b^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right ) f^4}{4 e^4}-\frac {3 b^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right ) f^4}{8 e^4}-\frac {3 b^3 n^3 \text {PolyLog}\left (2,\frac {\sqrt {x} f}{e}+1\right ) f^4}{2 e^4}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {6 b^3 n^3 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {12 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}+\frac {24 b^3 n^3 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) f^4}{e^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^3}{2 e^3 \sqrt {x}}-\frac {15 b n \left (a+b \log \left (c x^n\right )\right )^2 f^3}{4 e^3 \sqrt {x}}-\frac {63 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^3}{4 e^3 \sqrt {x}}-\frac {255 b^3 n^3 f^3}{8 e^3 \sqrt {x}}+\frac {\left (a+b \log \left (c x^n\right )\right )^3 f^2}{4 e^2 x}+\frac {9 b n \left (a+b \log \left (c x^n\right )\right )^2 f^2}{8 e^2 x}+\frac {21 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f^2}{8 e^2 x}+\frac {45 b^3 n^3 f^2}{16 e^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 f}{6 e x^{3/2}}-\frac {7 b n \left (a+b \log \left (c x^n\right )\right )^2 f}{12 e x^{3/2}}-\frac {37 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) f}{36 e x^{3/2}}-\frac {175 b^3 n^3 f}{216 e x^{3/2}}-\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{2 x^2}-\frac {3 b n \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 x^2}-\frac {3 b^3 n^3 \log \left (d \left (e+f \sqrt {x}\right )\right )}{8 x^2}-\frac {3 b^2 n^2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-175*b^3*f*n^3)/(216*e*x^(3/2)) + (45*b^3*f^2*n^3)/(16*e^2*x) - (255*b^3*f^3*n^3)/(8*e^3*Sqrt[x]) + (3*b^3*f^
4*n^3*Log[e + f*Sqrt[x]])/(8*e^4) - (3*b^3*n^3*Log[d*(e + f*Sqrt[x])])/(8*x^2) - (3*b^3*f^4*n^3*Log[e + f*Sqrt
[x]]*Log[-((f*Sqrt[x])/e)])/(2*e^4) - (3*b^3*f^4*n^3*Log[x])/(16*e^4) + (3*b^3*f^4*n^3*Log[x]^2)/(16*e^4) - (3
7*b^2*f*n^2*(a + b*Log[c*x^n]))/(36*e*x^(3/2)) + (21*b^2*f^2*n^2*(a + b*Log[c*x^n]))/(8*e^2*x) - (63*b^2*f^3*n
^2*(a + b*Log[c*x^n]))/(4*e^3*Sqrt[x]) + (3*b^2*f^4*n^2*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(4*e^4) - (3*b^
2*n^2*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]))/(4*x^2) - (3*b^2*f^4*n^2*Log[x]*(a + b*Log[c*x^n]))/(8*e^4) -
(7*b*f*n*(a + b*Log[c*x^n])^2)/(12*e*x^(3/2)) + (9*b*f^2*n*(a + b*Log[c*x^n])^2)/(8*e^2*x) - (15*b*f^3*n*(a +
b*Log[c*x^n])^2)/(4*e^3*Sqrt[x]) - (3*b*n*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2)/(4*x^2) + (3*b*f^4*n*L
og[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/(4*e^4) - (f^4*(a + b*Log[c*x^n])^3)/(8*e^4) - (f*(a + b*Log[c*x^n
])^3)/(6*e*x^(3/2)) + (f^2*(a + b*Log[c*x^n])^3)/(4*e^2*x) - (f^3*(a + b*Log[c*x^n])^3)/(2*e^3*Sqrt[x]) - (Log
[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3)/(2*x^2) + (f^4*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^3)/(2*e^4)
- (f^4*(a + b*Log[c*x^n])^4)/(16*b*e^4*n) - (3*b^3*f^4*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/(2*e^4) + (3*b^2*f^4
*n^2*(a + b*Log[c*x^n])*PolyLog[2, -((f*Sqrt[x])/e)])/e^4 + (3*b*f^4*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sq
rt[x])/e)])/e^4 - (6*b^3*f^4*n^3*PolyLog[3, -((f*Sqrt[x])/e)])/e^4 - (12*b^2*f^4*n^2*(a + b*Log[c*x^n])*PolyLo
g[3, -((f*Sqrt[x])/e)])/e^4 + (24*b^3*f^4*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/e^4

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2338

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 2339

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 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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2375

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[f^m*Log[1 + e*(x^r/d)]*((a + b*Log[c*x^n])^p/(e*r)), x] - Dist[b*f^m*n*(p/(e*r)), Int[Log[1 + e*(x^r/d)]*((
a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

Rule 2413

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 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 2422

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[Log[d*(e + f*x^m)^r]*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1))), x] - Dist[f*m*(r/(b*n*(p + 1))), Int[x
^(m - 1)*((a + b*Log[c*x^n])^(p + 1)/(e + f*x^m)), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]

Rule 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 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

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

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Mathematica [A]
time = 1.10, size = 1549, normalized size = 1.69 \begin {gather*} -\frac {54 e^4 \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+6 b \left (2 a^2+2 a b n+b^2 n^2\right ) \log \left (c x^n\right )+6 b^2 (2 a+b n) \log ^2\left (c x^n\right )+4 b^3 \log ^3\left (c x^n\right )\right )+18 e^3 f \sqrt {x} \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-27 e^2 f^2 x \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+54 e f^3 x^{3/2} \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-54 f^4 x^2 \log \left (e+f \sqrt {x}\right ) \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+27 f^4 x^2 \log (x) \left (4 a^3+6 a^2 b n+6 a b^2 n^2+3 b^3 n^3+12 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )+12 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+6 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+4 b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )+18 b f n \sqrt {x} \left (2 a^2+2 a b n+b^2 n^2+4 a b \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 n \left (-n \log (x)+\log \left (c x^n\right )\right )+2 b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (e \left (4 e^2-9 e f \sqrt {x}+36 f^2 x\right )+3 \left (2 e^3-3 e^2 f \sqrt {x}+6 e f^2 x-6 f^3 x^{3/2} \log \left (1+\frac {f \sqrt {x}}{e}\right )\right ) \log (x)+\frac {9}{2} f^3 x^{3/2} \log ^2(x)-36 f^3 x^{3/2} \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )\right )-6 b^2 f n^2 \sqrt {x} \left (-2 a-b n+2 b n \log (x)-2 b \log \left (c x^n\right )\right ) \left (16 e^3-54 e^2 f \sqrt {x}+432 e f^2 x+24 e^3 \log (x)-54 e^2 f \sqrt {x} \log (x)+216 e f^2 x \log (x)+18 e^3 \log ^2(x)-27 e^2 f \sqrt {x} \log ^2(x)+54 e f^2 x \log ^2(x)-54 f^3 x^{3/2} \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+9 f^3 x^{3/2} \log ^3(x)-216 f^3 x^{3/2} \log (x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+432 f^3 x^{3/2} \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\right )+4 b^3 n^3 \left (2 e f \sqrt {x} \left (16 e^2-81 e f \sqrt {x}+1296 f^2 x\right )+9 \left (e f \sqrt {x} \left (2 e^2-3 e f \sqrt {x}+6 f^2 x\right )-6 f^4 x^2 \log \left (1+\frac {e}{f \sqrt {x}}\right )\right ) \log ^3(x)+9 f \sqrt {x} \log ^2(x) \left (e \left (4 e^2-9 e f \sqrt {x}+36 f^2 x\right )+36 f^3 x^{3/2} \text {Li}_2\left (-\frac {e}{f \sqrt {x}}\right )\right )+6 f \sqrt {x} \log (x) \left (e \left (8 e^2-27 e f \sqrt {x}+216 f^2 x\right )+216 f^3 x^{3/2} \text {Li}_3\left (-\frac {e}{f \sqrt {x}}\right )\right )+2592 f^4 x^2 \text {Li}_4\left (-\frac {e}{f \sqrt {x}}\right )\right )}{432 e^4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/432*(54*e^4*Log[d*(e + f*Sqrt[x])]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 6*b*(2*a^2 + 2*a*b*n + b^
2*n^2)*Log[c*x^n] + 6*b^2*(2*a + b*n)*Log[c*x^n]^2 + 4*b^3*Log[c*x^n]^3) + 18*e^3*f*Sqrt[x]*(4*a^3 + 6*a^2*b*n
+ 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b
^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])
^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 27*e^2*f^2*x*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*
b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) +
12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n]
)^3) + 54*e*f^3*x^(3/2)*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 1
2*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^
n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) - 54*f^4*x^2*Log[e + f*Sqrt
[x]]*(4*a^3 + 6*a^2*b*n + 6*a*b^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[
x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(
n*Log[x]) + Log[c*x^n])^2 + 4*b^3*(-(n*Log[x]) + Log[c*x^n])^3) + 27*f^4*x^2*Log[x]*(4*a^3 + 6*a^2*b*n + 6*a*b
^2*n^2 + 3*b^3*n^3 + 12*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(
-(n*Log[x]) + Log[c*x^n]) + 12*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 6*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + 4*b
^3*(-(n*Log[x]) + Log[c*x^n])^3) + 18*b*f*n*Sqrt[x]*(2*a^2 + 2*a*b*n + b^2*n^2 + 4*a*b*(-(n*Log[x]) + Log[c*x^
n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*(-(n*Log[x]) + Log[c*x^n])^2)*(e*(4*e^2 - 9*e*f*Sqrt[x] + 36*
f^2*x) + 3*(2*e^3 - 3*e^2*f*Sqrt[x] + 6*e*f^2*x - 6*f^3*x^(3/2)*Log[1 + (f*Sqrt[x])/e])*Log[x] + (9*f^3*x^(3/2
)*Log[x]^2)/2 - 36*f^3*x^(3/2)*PolyLog[2, -((f*Sqrt[x])/e)]) - 6*b^2*f*n^2*Sqrt[x]*(-2*a - b*n + 2*b*n*Log[x]
- 2*b*Log[c*x^n])*(16*e^3 - 54*e^2*f*Sqrt[x] + 432*e*f^2*x + 24*e^3*Log[x] - 54*e^2*f*Sqrt[x]*Log[x] + 216*e*f
^2*x*Log[x] + 18*e^3*Log[x]^2 - 27*e^2*f*Sqrt[x]*Log[x]^2 + 54*e*f^2*x*Log[x]^2 - 54*f^3*x^(3/2)*Log[1 + (f*Sq
rt[x])/e]*Log[x]^2 + 9*f^3*x^(3/2)*Log[x]^3 - 216*f^3*x^(3/2)*Log[x]*PolyLog[2, -((f*Sqrt[x])/e)] + 432*f^3*x^
(3/2)*PolyLog[3, -((f*Sqrt[x])/e)]) + 4*b^3*n^3*(2*e*f*Sqrt[x]*(16*e^2 - 81*e*f*Sqrt[x] + 1296*f^2*x) + 9*(e*f
*Sqrt[x]*(2*e^2 - 3*e*f*Sqrt[x] + 6*f^2*x) - 6*f^4*x^2*Log[1 + e/(f*Sqrt[x])])*Log[x]^3 + 9*f*Sqrt[x]*Log[x]^2
*(e*(4*e^2 - 9*e*f*Sqrt[x] + 36*f^2*x) + 36*f^3*x^(3/2)*PolyLog[2, -(e/(f*Sqrt[x]))]) + 6*f*Sqrt[x]*Log[x]*(e*
(8*e^2 - 27*e*f*Sqrt[x] + 216*f^2*x) + 216*f^3*x^(3/2)*PolyLog[3, -(e/(f*Sqrt[x]))]) + 2592*f^4*x^2*PolyLog[4,
-(e/(f*Sqrt[x]))]))/(e^4*x^2)

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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