3.2.29 \(\int \log (d (e+f \sqrt {x})) (a+b \log (c x^n))^3 \, dx\) [129]

Optimal. Leaf size=639 \[ -\frac {90 b^3 e n^3 \sqrt {x}}{f}-6 a b^2 n^2 x+12 b^3 n^3 x+\frac {6 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right )}{f^2}-6 b^3 n^3 x \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {12 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-6 b^3 n^2 x \log \left (c x^n\right )+\frac {42 b^2 e n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {9 b e n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-3 b n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac {3 b e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{f}-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3-\frac {e^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}+\frac {12 b^3 e^2 n^3 \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {12 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {6 b e^2 n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {24 b^3 e^2 n^3 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^2}+\frac {24 b^2 e^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {48 b^3 e^2 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{f^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.58, antiderivative size = 639, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {2498, 272, 45, 2417, 2333, 2332, 2342, 2341, 2422, 2375, 2421, 2430, 6724, 2504, 2441, 2352} \begin {gather*} \frac {12 b^2 e^2 n^2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {24 b^2 e^2 n^2 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}-\frac {6 b e^2 n \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}+\frac {12 b^3 e^2 n^3 \text {PolyLog}\left (2,\frac {f \sqrt {x}}{e}+1\right )}{f^2}-\frac {24 b^3 e^2 n^3 \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {48 b^3 e^2 n^3 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )}{f^2}+6 b^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {6 b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{f^2}+\frac {42 b^2 e n^2 \sqrt {x} \left (a+b \log \left (c x^n\right )\right )}{f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x-3 b n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+x \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac {3 b e^2 n \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{f^2}-\frac {e^2 \log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{f^2}-\frac {9 b e n \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^2}{f}+\frac {e \sqrt {x} \left (a+b \log \left (c x^n\right )\right )^3}{f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac {1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )-6 b^3 n^3 x \log \left (d \left (e+f \sqrt {x}\right )\right )+\frac {6 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right )}{f^2}+\frac {12 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{f^2}-\frac {90 b^3 e n^3 \sqrt {x}}{f}+12 b^3 n^3 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-90*b^3*e*n^3*Sqrt[x])/f - 6*a*b^2*n^2*x + 12*b^3*n^3*x + (6*b^3*e^2*n^3*Log[e + f*Sqrt[x]])/f^2 - 6*b^3*n^3*
x*Log[d*(e + f*Sqrt[x])] + (12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/f^2 - 6*b^3*n^2*x*Log[c*x
^n] + (42*b^2*e*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/f - 3*b^2*n^2*x*(a + b*Log[c*x^n]) - (6*b^2*e^2*n^2*Log[e + f*
Sqrt[x]]*(a + b*Log[c*x^n]))/f^2 + 6*b^2*n^2*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n]) - (9*b*e*n*Sqrt[x]*(a
+ b*Log[c*x^n])^2)/f + 3*b*n*x*(a + b*Log[c*x^n])^2 - 3*b*n*x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (
3*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n])^2)/f^2 + (e*Sqrt[x]*(a + b*Log[c*x^n])^3)/f - (x*(a + b*Lo
g[c*x^n])^3)/2 + x*Log[d*(e + f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (e^2*Log[1 + (f*Sqrt[x])/e]*(a + b*Log[c*x^n]
)^3)/f^2 + (12*b^3*e^2*n^3*PolyLog[2, 1 + (f*Sqrt[x])/e])/f^2 + (12*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[2,
-((f*Sqrt[x])/e)])/f^2 - (6*b*e^2*n*(a + b*Log[c*x^n])^2*PolyLog[2, -((f*Sqrt[x])/e)])/f^2 - (24*b^3*e^2*n^3*P
olyLog[3, -((f*Sqrt[x])/e)])/f^2 + (24*b^2*e^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)])/f^2 - (48*
b^3*e^2*n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^2

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 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 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 2417

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], 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, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))

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 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 2498

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1522\) vs. \(2(639)=1278\).
time = 0.40, size = 1522, normalized size = 2.38 \begin {gather*} -\frac {-2 a^3 e f \sqrt {x}+18 a^2 b e f n \sqrt {x}-84 a b^2 e f n^2 \sqrt {x}+180 b^3 e f n^3 \sqrt {x}+a^3 f^2 x-6 a^2 b f^2 n x+18 a b^2 f^2 n^2 x-24 b^3 f^2 n^3 x+2 a^3 e^2 \log \left (e+f \sqrt {x}\right )-6 a^2 b e^2 n \log \left (e+f \sqrt {x}\right )+12 a b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right )-12 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right )-2 a^3 f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )+6 a^2 b f^2 n x \log \left (d \left (e+f \sqrt {x}\right )\right )-12 a b^2 f^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right )+12 b^3 f^2 n^3 x \log \left (d \left (e+f \sqrt {x}\right )\right )-6 a^2 b e^2 n \log \left (e+f \sqrt {x}\right ) \log (x)+12 a b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log (x)-12 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log (x)+6 a^2 b e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-12 a b^2 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+12 b^3 e^2 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+6 a b^2 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x)-6 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log ^2(x)-6 a b^2 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+6 b^3 e^2 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)-2 b^3 e^2 n^3 \log \left (e+f \sqrt {x}\right ) \log ^3(x)+2 b^3 e^2 n^3 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^3(x)-6 a^2 b e f \sqrt {x} \log \left (c x^n\right )+36 a b^2 e f n \sqrt {x} \log \left (c x^n\right )-84 b^3 e f n^2 \sqrt {x} \log \left (c x^n\right )+3 a^2 b f^2 x \log \left (c x^n\right )-12 a b^2 f^2 n x \log \left (c x^n\right )+18 b^3 f^2 n^2 x \log \left (c x^n\right )+6 a^2 b e^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-12 a b^2 e^2 n \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )+12 b^3 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log \left (c x^n\right )-6 a^2 b f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )+12 a b^2 f^2 n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )-12 b^3 f^2 n^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log \left (c x^n\right )-12 a b^2 e^2 n \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )+12 b^3 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log (x) \log \left (c x^n\right )+12 a b^2 e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )-12 b^3 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log \left (c x^n\right )+6 b^3 e^2 n^2 \log \left (e+f \sqrt {x}\right ) \log ^2(x) \log \left (c x^n\right )-6 b^3 e^2 n^2 \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x) \log \left (c x^n\right )-6 a b^2 e f \sqrt {x} \log ^2\left (c x^n\right )+18 b^3 e f n \sqrt {x} \log ^2\left (c x^n\right )+3 a b^2 f^2 x \log ^2\left (c x^n\right )-6 b^3 f^2 n x \log ^2\left (c x^n\right )+6 a b^2 e^2 \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 b^3 e^2 n \log \left (e+f \sqrt {x}\right ) \log ^2\left (c x^n\right )-6 a b^2 f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )+6 b^3 f^2 n x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^2\left (c x^n\right )-6 b^3 e^2 n \log \left (e+f \sqrt {x}\right ) \log (x) \log ^2\left (c x^n\right )+6 b^3 e^2 n \log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x) \log ^2\left (c x^n\right )-2 b^3 e f \sqrt {x} \log ^3\left (c x^n\right )+b^3 f^2 x \log ^3\left (c x^n\right )+2 b^3 e^2 \log \left (e+f \sqrt {x}\right ) \log ^3\left (c x^n\right )-2 b^3 f^2 x \log \left (d \left (e+f \sqrt {x}\right )\right ) \log ^3\left (c x^n\right )+12 b e^2 n \left (a^2-2 a b n+2 b^2 n^2+2 b (a-b n) \log \left (c x^n\right )+b^2 \log ^2\left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 e^2 n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+96 b^3 e^2 n^3 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )}{2 f^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(-2*a^3*e*f*Sqrt[x] + 18*a^2*b*e*f*n*Sqrt[x] - 84*a*b^2*e*f*n^2*Sqrt[x] + 180*b^3*e*f*n^3*Sqrt[x] + a^3*f
^2*x - 6*a^2*b*f^2*n*x + 18*a*b^2*f^2*n^2*x - 24*b^3*f^2*n^3*x + 2*a^3*e^2*Log[e + f*Sqrt[x]] - 6*a^2*b*e^2*n*
Log[e + f*Sqrt[x]] + 12*a*b^2*e^2*n^2*Log[e + f*Sqrt[x]] - 12*b^3*e^2*n^3*Log[e + f*Sqrt[x]] - 2*a^3*f^2*x*Log
[d*(e + f*Sqrt[x])] + 6*a^2*b*f^2*n*x*Log[d*(e + f*Sqrt[x])] - 12*a*b^2*f^2*n^2*x*Log[d*(e + f*Sqrt[x])] + 12*
b^3*f^2*n^3*x*Log[d*(e + f*Sqrt[x])] - 6*a^2*b*e^2*n*Log[e + f*Sqrt[x]]*Log[x] + 12*a*b^2*e^2*n^2*Log[e + f*Sq
rt[x]]*Log[x] - 12*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x] + 6*a^2*b*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 12*a*
b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x] + 12*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x] + 6*a*b^2*e^2*n^2*Log
[e + f*Sqrt[x]]*Log[x]^2 - 6*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x]^2 - 6*a*b^2*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*
Log[x]^2 + 6*b^3*e^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^2 - 2*b^3*e^2*n^3*Log[e + f*Sqrt[x]]*Log[x]^3 + 2*b^3*e
^2*n^3*Log[1 + (f*Sqrt[x])/e]*Log[x]^3 - 6*a^2*b*e*f*Sqrt[x]*Log[c*x^n] + 36*a*b^2*e*f*n*Sqrt[x]*Log[c*x^n] -
84*b^3*e*f*n^2*Sqrt[x]*Log[c*x^n] + 3*a^2*b*f^2*x*Log[c*x^n] - 12*a*b^2*f^2*n*x*Log[c*x^n] + 18*b^3*f^2*n^2*x*
Log[c*x^n] + 6*a^2*b*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 12*a*b^2*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n] + 12*b^3
*e^2*n^2*Log[e + f*Sqrt[x]]*Log[c*x^n] - 6*a^2*b*f^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] + 12*a*b^2*f^2*n*x*Lo
g[d*(e + f*Sqrt[x])]*Log[c*x^n] - 12*b^3*f^2*n^2*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n] - 12*a*b^2*e^2*n*Log[e +
f*Sqrt[x]]*Log[x]*Log[c*x^n] + 12*b^3*e^2*n^2*Log[e + f*Sqrt[x]]*Log[x]*Log[c*x^n] + 12*a*b^2*e^2*n*Log[1 + (f
*Sqrt[x])/e]*Log[x]*Log[c*x^n] - 12*b^3*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n] + 6*b^3*e^2*n^2*Log[e
+ f*Sqrt[x]]*Log[x]^2*Log[c*x^n] - 6*b^3*e^2*n^2*Log[1 + (f*Sqrt[x])/e]*Log[x]^2*Log[c*x^n] - 6*a*b^2*e*f*Sqr
t[x]*Log[c*x^n]^2 + 18*b^3*e*f*n*Sqrt[x]*Log[c*x^n]^2 + 3*a*b^2*f^2*x*Log[c*x^n]^2 - 6*b^3*f^2*n*x*Log[c*x^n]^
2 + 6*a*b^2*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 - 6*b^3*e^2*n*Log[e + f*Sqrt[x]]*Log[c*x^n]^2 - 6*a*b^2*f^2*x*
Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 + 6*b^3*f^2*n*x*Log[d*(e + f*Sqrt[x])]*Log[c*x^n]^2 - 6*b^3*e^2*n*Log[e +
f*Sqrt[x]]*Log[x]*Log[c*x^n]^2 + 6*b^3*e^2*n*Log[1 + (f*Sqrt[x])/e]*Log[x]*Log[c*x^n]^2 - 2*b^3*e*f*Sqrt[x]*Lo
g[c*x^n]^3 + b^3*f^2*x*Log[c*x^n]^3 + 2*b^3*e^2*Log[e + f*Sqrt[x]]*Log[c*x^n]^3 - 2*b^3*f^2*x*Log[d*(e + f*Sqr
t[x])]*Log[c*x^n]^3 + 12*b*e^2*n*(a^2 - 2*a*b*n + 2*b^2*n^2 + 2*b*(a - b*n)*Log[c*x^n] + b^2*Log[c*x^n]^2)*Pol
yLog[2, -((f*Sqrt[x])/e)] - 48*b^2*e^2*n^2*(a - b*n + b*Log[c*x^n])*PolyLog[3, -((f*Sqrt[x])/e)] + 96*b^3*e^2*
n^3*PolyLog[4, -((f*Sqrt[x])/e)])/f^2

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

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)

[Out]

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

[Out]

1/27*(27*b^3*x*e*log(d)*log(x^n)^3 - 81*((n*log(d) - log(c)*log(d))*b^3 - a*b^2*log(d))*x*e*log(x^n)^2 - 81*(2
*(n*log(d) - log(c)*log(d))*a*b^2 - (2*n^2*log(d) - 2*n*log(c)*log(d) + log(c)^2*log(d))*b^3 - a^2*b*log(d))*x
*e*log(x^n) - 27*(3*(n*log(d) - log(c)*log(d))*a^2*b - 3*(2*n^2*log(d) - 2*n*log(c)*log(d) + log(c)^2*log(d))*
a*b^2 + (6*n^3*log(d) - 6*n^2*log(c)*log(d) + 3*n*log(c)^2*log(d) - log(c)^3*log(d))*b^3 - a^3*log(d))*x*e + 2
7*(b^3*x*e*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*e*log(x^n)^2 + 3*((2*n^2 - 2*n*log(c) + log(c)^2)*b^3 -
2*a*b^2*(n - log(c)) + a^2*b)*x*e*log(x^n) + (3*(2*n^2 - 2*n*log(c) + log(c)^2)*a*b^2 - (6*n^3 - 6*n^2*log(c)
+ 3*n*log(c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x*e)*log(f*sqrt(x) + e) - (9*b^3*f*x^2*log(x^n)^
3 - 9*((5*f*n - 3*f*log(c))*b^3 - 3*a*b^2*f)*x^2*log(x^n)^2 - 3*(6*(5*f*n - 3*f*log(c))*a*b^2 - (38*f*n^2 - 30
*f*n*log(c) + 9*f*log(c)^2)*b^3 - 9*a^2*b*f)*x^2*log(x^n) - (9*(5*f*n - 3*f*log(c))*a^2*b - 3*(38*f*n^2 - 30*f
*n*log(c) + 9*f*log(c)^2)*a*b^2 + (130*f*n^3 - 114*f*n^2*log(c) + 45*f*n*log(c)^2 - 9*f*log(c)^3)*b^3 - 9*a^3*
f)*x^2)/sqrt(x))*e^(-1) + integrate(1/2*(b^3*f^2*x*log(x^n)^3 + 3*(a*b^2*f^2 - (f^2*n - f^2*log(c))*b^3)*x*log
(x^n)^2 + 3*(a^2*b*f^2 - 2*(f^2*n - f^2*log(c))*a*b^2 + (2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*b^3)*x*log
(x^n) + (a^3*f^2 - 3*(f^2*n - f^2*log(c))*a^2*b + 3*(2*f^2*n^2 - 2*f^2*n*log(c) + f^2*log(c)^2)*a*b^2 - (6*f^2
*n^3 - 6*f^2*n^2*log(c) + 3*f^2*n*log(c)^2 - f^2*log(c)^3)*b^3)*x)/(f*e^(1/2*log(x) + 1) + e^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*(e+f*x^(1/2))),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) + d*e), 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*(e+f*x**(1/2))),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*(e+f*x^(1/2))),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \ln \left (d\,\left (e+f\,\sqrt {x}\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(log(d*(e + f*x^(1/2)))*(a + b*log(c*x^n))^3,x)

[Out]

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

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