3.1.62 \(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))^3}{x^2} \, dx\) [62]

Optimal. Leaf size=610 \[ -\frac {90 b^3 d f n^3}{\sqrt {x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {3 b n \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right ) \]

[Out]

-3*b^3*d^2*f^2*n^3*ln(x)+3/2*b^3*d^2*f^2*n^3*ln(x)^2-3*b^2*d^2*f^2*n^2*ln(x)*(a+b*ln(c*x^n))-1/2*d^2*f^2*(a+b*
ln(c*x^n))^3-1/8*d^2*f^2*(a+b*ln(c*x^n))^4/b/n+6*b^3*d^2*f^2*n^3*ln(1+d*f*x^(1/2))-6*b^3*n^3*ln(1+d*f*x^(1/2))
/x+6*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))-6*b^2*n^2*(a+b*ln(c*x^n))*ln(1+d*f*x^(1/2))/x+3*b*d^2*f
^2*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))-3*b*n*(a+b*ln(c*x^n))^2*ln(1+d*f*x^(1/2))/x+d^2*f^2*(a+b*ln(c*x^n))^3
*ln(1+d*f*x^(1/2))-(a+b*ln(c*x^n))^3*ln(1+d*f*x^(1/2))/x+12*b^3*d^2*f^2*n^3*polylog(2,-d*f*x^(1/2))+12*b^2*d^2
*f^2*n^2*(a+b*ln(c*x^n))*polylog(2,-d*f*x^(1/2))+6*b*d^2*f^2*n*(a+b*ln(c*x^n))^2*polylog(2,-d*f*x^(1/2))-24*b^
3*d^2*f^2*n^3*polylog(3,-d*f*x^(1/2))-24*b^2*d^2*f^2*n^2*(a+b*ln(c*x^n))*polylog(3,-d*f*x^(1/2))+48*b^3*d^2*f^
2*n^3*polylog(4,-d*f*x^(1/2))-90*b^3*d*f*n^3/x^(1/2)-42*b^2*d*f*n^2*(a+b*ln(c*x^n))/x^(1/2)-9*b*d*f*n*(a+b*ln(
c*x^n))^2/x^(1/2)-d*f*(a+b*ln(c*x^n))^3/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2504, 2442, 46, 2424, 2342, 2341, 2423, 2438, 2338, 2421, 6724, 2413, 12, 2339, 30, 2430} \begin {gather*} 12 b^2 d^2 f^2 n^2 \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-24 b^2 d^2 f^2 n^2 \text {PolyLog}\left (3,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )+6 b d^2 f^2 n \text {PolyLog}\left (2,-d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2+12 b^3 d^2 f^2 n^3 \text {PolyLog}\left (2,-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {PolyLog}\left (3,-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {PolyLog}\left (4,-d f \sqrt {x}\right )+6 b^2 d^2 f^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}-\frac {1}{2} d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^3+d^2 f^2 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b d^2 f^2 n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}-\frac {3 b n \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)+6 b^3 d^2 f^2 n^3 \log \left (d f \sqrt {x}+1\right )-3 b^3 d^2 f^2 n^3 \log (x)-\frac {90 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (d f \sqrt {x}+1\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-90*b^3*d*f*n^3)/Sqrt[x] + 6*b^3*d^2*f^2*n^3*Log[1 + d*f*Sqrt[x]] - (6*b^3*n^3*Log[1 + d*f*Sqrt[x]])/x - 3*b^
3*d^2*f^2*n^3*Log[x] + (3*b^3*d^2*f^2*n^3*Log[x]^2)/2 - (42*b^2*d*f*n^2*(a + b*Log[c*x^n]))/Sqrt[x] + 6*b^2*d^
2*f^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]) - (6*b^2*n^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n]))/x - 3*
b^2*d^2*f^2*n^2*Log[x]*(a + b*Log[c*x^n]) - (9*b*d*f*n*(a + b*Log[c*x^n])^2)/Sqrt[x] + 3*b*d^2*f^2*n*Log[1 + d
*f*Sqrt[x]]*(a + b*Log[c*x^n])^2 - (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/x - (d^2*f^2*(a + b*Log[c
*x^n])^3)/2 - (d*f*(a + b*Log[c*x^n])^3)/Sqrt[x] + d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3 - (Log[1
+ d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/x - (d^2*f^2*(a + b*Log[c*x^n])^4)/(8*b*n) + 12*b^3*d^2*f^2*n^3*PolyLog[2
, -(d*f*Sqrt[x])] + 12*b^2*d^2*f^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] + 6*b*d^2*f^2*n*(a + b*Lo
g[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])] - 24*b^3*d^2*f^2*n^3*PolyLog[3, -(d*f*Sqrt[x])] - 24*b^2*d^2*f^2*n^2*(a
+ b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])] + 48*b^3*d^2*f^2*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 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 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 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 \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^2} \, dx &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {1}{2} d^2 f^2 \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}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^2}{2 x}\right ) \, dx\\ &=-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {1}{2} d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )^3+(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^2} \, dx+(3 b d f n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^{3/2}} \, dx+\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx-\left (3 b d^2 f^2 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 {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {3}{2} b d^2 f^2 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}{\sqrt {x}}+d^2 f^2 \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}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b n x} \, dx-\left (6 b^2 n^2\right ) \int \left (-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {d^2 f^2 \log (x) \left (a+b \log \left (c x^n\right )\right )}{2 x}\right ) \, dx+\left (12 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx-\left (12 b^2 d^2 f^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\\ &=-\frac {48 b^3 d f n^3}{\sqrt {x}}-\frac {24 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {3}{2} b d^2 f^2 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}{\sqrt {x}}+d^2 f^2 \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}{x}+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (d^2 f^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx+\left (6 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^2} \, dx+\left (6 b^2 d f n^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x^{3/2}} \, dx+\left (3 b^2 d^2 f^2 n^2\right ) \int \frac {\log (x) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx-\left (6 b^2 d^2 f^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+\left (24 b^3 d^2 f^2 n^3\right ) \int \frac {\text {Li}_3\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {72 b^3 d f n^3}{\sqrt {x}}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \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}{x}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {\left (d^2 f^2\right ) \text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (c x^n\right )\right )}{2 b n}-\left (3 b^2 d^2 f^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b n x} \, dx-\left (6 b^3 n^3\right ) \int \left (-\frac {d f}{x^{3/2}}-\frac {\log \left (1+d f \sqrt {x}\right )}{x^2}+\frac {d^2 f^2 \log \left (1+d f \sqrt {x}\right )}{x}-\frac {d^2 f^2 \log (x)}{2 x}\right ) \, dx-\left (12 b^3 d^2 f^2 n^3\right ) \int \frac {\text {Li}_2\left (-d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 b d^2 f^2 n\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx+\left (6 b^3 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x^2} \, dx+\left (3 b^3 d^2 f^2 n^3\right ) \int \frac {\log (x)}{x} \, dx-\left (6 b^3 d^2 f^2 n^3\right ) \int \frac {\log \left (1+d f \sqrt {x}\right )}{x} \, dx\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {d f \left (a+b \log \left (c x^n\right )\right )^3}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )-\frac {1}{2} \left (3 d^2 f^2\right ) \text {Subst}\left (\int x^2 \, dx,x,a+b \log \left (c x^n\right )\right )+\left (12 b^3 n^3\right ) \text {Subst}\left (\int \frac {\log (1+d f x)}{x^3} \, dx,x,\sqrt {x}\right )\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {1}{2} d^2 f^2 \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}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\left (6 b^3 d f n^3\right ) \text {Subst}\left (\int \frac {1}{x^2 (1+d f x)} \, dx,x,\sqrt {x}\right )\\ &=-\frac {84 b^3 d f n^3}{\sqrt {x}}-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {1}{2} d^2 f^2 \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}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )+\left (6 b^3 d f n^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}-\frac {d f}{x}+\frac {d^2 f^2}{1+d f x}\right ) \, dx,x,\sqrt {x}\right )\\ &=-\frac {90 b^3 d f n^3}{\sqrt {x}}+6 b^3 d^2 f^2 n^3 \log \left (1+d f \sqrt {x}\right )-\frac {6 b^3 n^3 \log \left (1+d f \sqrt {x}\right )}{x}-3 b^3 d^2 f^2 n^3 \log (x)+\frac {3}{2} b^3 d^2 f^2 n^3 \log ^2(x)-\frac {42 b^2 d f n^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt {x}}+6 b^2 d^2 f^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 n^2 \log \left (1+d f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x}-3 b^2 d^2 f^2 n^2 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b d f n \left (a+b \log \left (c x^n\right )\right )^2}{\sqrt {x}}+3 b d^2 f^2 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}{x}-\frac {1}{2} d^2 f^2 \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}{\sqrt {x}}+d^2 f^2 \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}{x}-\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )^4}{8 b n}+12 b^3 d^2 f^2 n^3 \text {Li}_2\left (-d f \sqrt {x}\right )+12 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f \sqrt {x}\right )+6 b d^2 f^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-d f \sqrt {x}\right )-24 b^3 d^2 f^2 n^3 \text {Li}_3\left (-d f \sqrt {x}\right )-24 b^2 d^2 f^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-d f \sqrt {x}\right )+48 b^3 d^2 f^2 n^3 \text {Li}_4\left (-d f \sqrt {x}\right )\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1455\) vs. \(2(610)=1220\).
time = 0.56, size = 1455, normalized size = 2.39 \begin {gather*} d^2 f^2 \log \left (1+d f \sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 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 )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-d^2 f^2 \log \left (\sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 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 )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )-\frac {\log \left (1+d f \sqrt {x}\right ) \left (a^3+3 a^2 b n+6 a b^2 n^2+6 b^3 n^3+3 a^2 b n \log (x)+6 a b^2 n^2 \log (x)+6 b^3 n^3 \log (x)+3 a b^2 n^2 \log ^2(x)+3 b^3 n^3 \log ^2(x)+b^3 n^3 \log ^3(x)+3 a^2 b \left (-n \log (x)+\log \left (c x^n\right )\right )+6 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 )+6 a b^2 n \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )+6 b^3 n^2 \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )+3 b^3 n^2 \log ^2(x) \left (-n \log (x)+\log \left (c x^n\right )\right )+3 a b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \left (-n \log (x)+\log \left (c x^n\right )\right )^2+3 b^3 n \log (x) \left (-n \log (x)+\log \left (c x^n\right )\right )^2+b^3 \left (-n \log (x)+\log \left (c x^n\right )\right )^3\right )}{x}+\frac {-a^3 d f-3 a^2 b d f n-6 a b^2 d f n^2-6 b^3 d f n^3-3 a^2 b d f \left (-n \log (x)+\log \left (c x^n\right )\right )-6 a b^2 d f n \left (-n \log (x)+\log \left (c x^n\right )\right )-6 b^3 d f n^2 \left (-n \log (x)+\log \left (c x^n\right )\right )-3 a b^2 d f \left (-n \log (x)+\log \left (c x^n\right )\right )^2-3 b^3 d f n \left (-n \log (x)+\log \left (c x^n\right )\right )^2-b^3 d f \left (-n \log (x)+\log \left (c x^n\right )\right )^3}{\sqrt {x}}+3 b d f n \left (a^2+2 a b n+2 b^2 n^2+2 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 )+b^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2\right ) \left (\left (-\frac {1}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right )-d f \log \left (\sqrt {x}\right )\right ) \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right )+2 \left (-\frac {1}{\sqrt {x}}-\frac {\log \left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log \left (\sqrt {x}\right )-\frac {1}{2} d f \log ^2\left (\sqrt {x}\right )+d f \text {Li}_2\left (-d f \sqrt {x}\right )\right )\right )+3 b^2 d f n^2 \left (a+b n+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right ) \left (\left (-\frac {1}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right )-d f \log \left (\sqrt {x}\right )\right ) \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right )^2+4 \left (-2 \log \left (\sqrt {x}\right )+\log (x)\right ) \left (-\frac {1}{\sqrt {x}}-\frac {\log \left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log \left (\sqrt {x}\right )-\frac {1}{2} d f \log ^2\left (\sqrt {x}\right )+d f \text {Li}_2\left (-d f \sqrt {x}\right )\right )+4 \left (-\frac {2}{\sqrt {x}}-\frac {2 \log \left (\sqrt {x}\right )}{\sqrt {x}}-\frac {\log ^2\left (\sqrt {x}\right )}{\sqrt {x}}+d f \log \left (1+d f \sqrt {x}\right ) \log ^2\left (\sqrt {x}\right )-\frac {1}{3} d f \log ^3\left (\sqrt {x}\right )+2 d f \log \left (\sqrt {x}\right ) \text {Li}_2\left (-d f \sqrt {x}\right )-2 d f \text {Li}_3\left (-d f \sqrt {x}\right )\right )\right )+\frac {b^3 d f n^3 \left (1+\frac {1}{d f \sqrt {x}}\right ) \left (2 \left (-d f \sqrt {x}+d^2 f^2 x \log \left (1+\frac {1}{d f \sqrt {x}}\right )\right ) \log ^3(x)-12 d f \sqrt {x} \log ^2(x) \left (1+d f \sqrt {x} \text {Li}_2\left (-\frac {1}{d f \sqrt {x}}\right )\right )-48 d f \sqrt {x} \log (x) \left (1+d f \sqrt {x} \text {Li}_3\left (-\frac {1}{d f \sqrt {x}}\right )\right )-96 d f \sqrt {x} \left (1+d f \sqrt {x} \text {Li}_4\left (-\frac {1}{d f \sqrt {x}}\right )\right )\right )}{2 \left (1+d f \sqrt {x}\right ) \sqrt {x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

d^2*f^2*Log[1 + d*f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) +
6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^
n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) - d^2*f^2*Log[Sqrt[x]]*(a^3 +
3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n
]) + 6*b^3*n^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]) + Log[
c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3) - (Log[1 + d*f*Sqrt[x]]*(a^3 + 3*a^2*b*n + 6*a*b^2*n^2 + 6*b^3*n
^3 + 3*a^2*b*n*Log[x] + 6*a*b^2*n^2*Log[x] + 6*b^3*n^3*Log[x] + 3*a*b^2*n^2*Log[x]^2 + 3*b^3*n^3*Log[x]^2 + b^
3*n^3*Log[x]^3 + 3*a^2*b*(-(n*Log[x]) + Log[c*x^n]) + 6*a*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*(-(n*Lo
g[x]) + Log[c*x^n]) + 6*a*b^2*n*Log[x]*(-(n*Log[x]) + Log[c*x^n]) + 6*b^3*n^2*Log[x]*(-(n*Log[x]) + Log[c*x^n]
) + 3*b^3*n^2*Log[x]^2*(-(n*Log[x]) + Log[c*x^n]) + 3*a*b^2*(-(n*Log[x]) + Log[c*x^n])^2 + 3*b^3*n*(-(n*Log[x]
) + Log[c*x^n])^2 + 3*b^3*n*Log[x]*(-(n*Log[x]) + Log[c*x^n])^2 + b^3*(-(n*Log[x]) + Log[c*x^n])^3))/x + (-(a^
3*d*f) - 3*a^2*b*d*f*n - 6*a*b^2*d*f*n^2 - 6*b^3*d*f*n^3 - 3*a^2*b*d*f*(-(n*Log[x]) + Log[c*x^n]) - 6*a*b^2*d*
f*n*(-(n*Log[x]) + Log[c*x^n]) - 6*b^3*d*f*n^2*(-(n*Log[x]) + Log[c*x^n]) - 3*a*b^2*d*f*(-(n*Log[x]) + Log[c*x
^n])^2 - 3*b^3*d*f*n*(-(n*Log[x]) + Log[c*x^n])^2 - b^3*d*f*(-(n*Log[x]) + Log[c*x^n])^3)/Sqrt[x] + 3*b*d*f*n*
(a^2 + 2*a*b*n + 2*b^2*n^2 + 2*a*b*(-(n*Log[x]) + Log[c*x^n]) + 2*b^2*n*(-(n*Log[x]) + Log[c*x^n]) + b^2*(-(n*
Log[x]) + Log[c*x^n])^2)*((-(1/Sqrt[x]) + d*f*Log[1 + d*f*Sqrt[x]] - d*f*Log[Sqrt[x]])*(-2*Log[Sqrt[x]] + Log[
x]) + 2*(-(1/Sqrt[x]) - Log[Sqrt[x]]/Sqrt[x] + d*f*Log[1 + d*f*Sqrt[x]]*Log[Sqrt[x]] - (d*f*Log[Sqrt[x]]^2)/2
+ d*f*PolyLog[2, -(d*f*Sqrt[x])])) + 3*b^2*d*f*n^2*(a + b*n + b*(-(n*Log[x]) + Log[c*x^n]))*((-(1/Sqrt[x]) + d
*f*Log[1 + d*f*Sqrt[x]] - d*f*Log[Sqrt[x]])*(-2*Log[Sqrt[x]] + Log[x])^2 + 4*(-2*Log[Sqrt[x]] + Log[x])*(-(1/S
qrt[x]) - Log[Sqrt[x]]/Sqrt[x] + d*f*Log[1 + d*f*Sqrt[x]]*Log[Sqrt[x]] - (d*f*Log[Sqrt[x]]^2)/2 + d*f*PolyLog[
2, -(d*f*Sqrt[x])]) + 4*(-2/Sqrt[x] - (2*Log[Sqrt[x]])/Sqrt[x] - Log[Sqrt[x]]^2/Sqrt[x] + d*f*Log[1 + d*f*Sqrt
[x]]*Log[Sqrt[x]]^2 - (d*f*Log[Sqrt[x]]^3)/3 + 2*d*f*Log[Sqrt[x]]*PolyLog[2, -(d*f*Sqrt[x])] - 2*d*f*PolyLog[3
, -(d*f*Sqrt[x])])) + (b^3*d*f*n^3*(1 + 1/(d*f*Sqrt[x]))*(2*(-(d*f*Sqrt[x]) + d^2*f^2*x*Log[1 + 1/(d*f*Sqrt[x]
)])*Log[x]^3 - 12*d*f*Sqrt[x]*Log[x]^2*(1 + d*f*Sqrt[x]*PolyLog[2, -(1/(d*f*Sqrt[x]))]) - 48*d*f*Sqrt[x]*Log[x
]*(1 + d*f*Sqrt[x]*PolyLog[3, -(1/(d*f*Sqrt[x]))]) - 96*d*f*Sqrt[x]*(1 + d*f*Sqrt[x]*PolyLog[4, -(1/(d*f*Sqrt[
x]))])))/(2*(1 + d*f*Sqrt[x])*Sqrt[x])

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (\frac {1}{d}+f \sqrt {x}\right )\right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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**2,x)

[Out]

Timed out

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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*(1/d+f*x^(1/2)))/x^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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^2} \,d x \end {gather*}

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^2,x)

[Out]

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

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