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

Optimal. Leaf size=604 \[ \frac{12 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{24 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac{6 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac{12 b^3 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{24 b^3 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{48 b^3 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{6 b^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+6 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{42 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x+\frac{3 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-3 b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{9 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )+\frac{6 b^3 n^3 \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}-\frac{90 b^3 n^3 \sqrt{x}}{d f}-6 b^3 n^3 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+12 b^3 n^3 x \]

[Out]

(-90*b^3*n^3*Sqrt[x])/(d*f) - 6*a*b^2*n^2*x + 12*b^3*n^3*x - 6*b^3*n^3*x*Log[d*(d^(-1) + f*Sqrt[x])] + (6*b^3*
n^3*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) - 6*b^3*n^2*x*Log[c*x^n] + (42*b^2*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) -
3*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b^2*n^2*x*Log[d*(d^(-1) + 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]))/(d^2*f^2) - (9*b*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) + 3*b*n*x*(a + b*L
og[c*x^n])^2 - 3*b*n*x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*L
og[c*x^n])^2)/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^3)/(d*f) - (x*(a + b*Log[c*x^n])^3)/2 + x*Log[d*(d^(-1)
+ f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(d^2*f^2) - (12*b^3*n^3*PolyL
og[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (12*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (6*b
*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (24*b^3*n^3*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f
^2) + (24*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) - (48*b^3*n^3*PolyLog[4, -(d*f*Sqrt
[x])])/(d^2*f^2)

________________________________________________________________________________________

Rubi [A]  time = 0.519995, antiderivative size = 604, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 12, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {2448, 266, 43, 2370, 2296, 2295, 2305, 2304, 2391, 2374, 6589, 2383} \[ \frac{12 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{24 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac{6 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac{12 b^3 n^3 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{24 b^3 n^3 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{48 b^3 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right )}{d^2 f^2}-\frac{6 b^2 n^2 \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+6 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{42 b^2 n^2 \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}-3 b^2 n^2 x \left (a+b \log \left (c x^n\right )\right )-6 a b^2 n^2 x+\frac{3 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d^2 f^2}-3 b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac{9 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^3}{d f}+3 b n x \left (a+b \log \left (c x^n\right )\right )^2-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^3-6 b^3 n^2 x \log \left (c x^n\right )+\frac{6 b^3 n^3 \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}-\frac{90 b^3 n^3 \sqrt{x}}{d f}-6 b^3 n^3 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+12 b^3 n^3 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-90*b^3*n^3*Sqrt[x])/(d*f) - 6*a*b^2*n^2*x + 12*b^3*n^3*x - 6*b^3*n^3*x*Log[d*(d^(-1) + f*Sqrt[x])] + (6*b^3*
n^3*Log[1 + d*f*Sqrt[x]])/(d^2*f^2) - 6*b^3*n^2*x*Log[c*x^n] + (42*b^2*n^2*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) -
3*b^2*n^2*x*(a + b*Log[c*x^n]) + 6*b^2*n^2*x*Log[d*(d^(-1) + 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]))/(d^2*f^2) - (9*b*n*Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) + 3*b*n*x*(a + b*L
og[c*x^n])^2 - 3*b*n*x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n])^2 + (3*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*L
og[c*x^n])^2)/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^3)/(d*f) - (x*(a + b*Log[c*x^n])^3)/2 + x*Log[d*(d^(-1)
+ f*Sqrt[x])]*(a + b*Log[c*x^n])^3 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^3)/(d^2*f^2) - (12*b^3*n^3*PolyL
og[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (12*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (6*b
*n*(a + b*Log[c*x^n])^2*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) - (24*b^3*n^3*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f
^2) + (24*b^2*n^2*(a + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])])/(d^2*f^2) - (48*b^3*n^3*PolyLog[4, -(d*f*Sqrt
[x])])/(d^2*f^2)

Rule 2448

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 266

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 43

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 2370

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 2296

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 2295

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

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2391

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

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2383

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_.)])/(x_), x_Symbol] :> Simp[(PolyL
og[k + 1, e*x^q]*(a + b*Log[c*x^n])^p)/q, x] - Dist[(b*n*p)/q, Int[(PolyLog[k + 1, e*x^q]*(a + b*Log[c*x^n])^(
p - 1))/x, x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.49359, size = 986, normalized size = 1.63 \[ -\frac{d^2 f^2 x a^3-2 d^2 f^2 x \log \left (d \sqrt{x} f+1\right ) a^3+2 \log \left (d \sqrt{x} f+1\right ) a^3-2 d f \sqrt{x} a^3-6 b d^2 f^2 n x a^2-6 b n \log \left (d \sqrt{x} f+1\right ) a^2+6 b d^2 f^2 n x \log \left (d \sqrt{x} f+1\right ) a^2+3 b d^2 f^2 x \log \left (c x^n\right ) a^2+6 b \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) a^2-6 b d^2 f^2 x \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) a^2-6 b d f \sqrt{x} \log \left (c x^n\right ) a^2+18 b d f n \sqrt{x} a^2+3 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) a+6 b^2 \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right ) a-6 b^2 d^2 f^2 x \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right ) a-6 b^2 d f \sqrt{x} \log ^2\left (c x^n\right ) a+18 b^2 d^2 f^2 n^2 x a+12 b^2 n^2 \log \left (d \sqrt{x} f+1\right ) a-12 b^2 d^2 f^2 n^2 x \log \left (d \sqrt{x} f+1\right ) a-12 b^2 d^2 f^2 n x \log \left (c x^n\right ) a-12 b^2 n \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) a+12 b^2 d^2 f^2 n x \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right ) a+36 b^2 d f n \sqrt{x} \log \left (c x^n\right ) a-84 b^2 d f n^2 \sqrt{x} a+b^3 d^2 f^2 x \log ^3\left (c x^n\right )+2 b^3 \log \left (d \sqrt{x} f+1\right ) \log ^3\left (c x^n\right )-2 b^3 d^2 f^2 x \log \left (d \sqrt{x} f+1\right ) \log ^3\left (c x^n\right )-2 b^3 d f \sqrt{x} \log ^3\left (c x^n\right )-6 b^3 d^2 f^2 n x \log ^2\left (c x^n\right )-6 b^3 n \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right )+6 b^3 d^2 f^2 n x \log \left (d \sqrt{x} f+1\right ) \log ^2\left (c x^n\right )+18 b^3 d f n \sqrt{x} \log ^2\left (c x^n\right )-24 b^3 d^2 f^2 n^3 x-12 b^3 n^3 \log \left (d \sqrt{x} f+1\right )+12 b^3 d^2 f^2 n^3 x \log \left (d \sqrt{x} f+1\right )+18 b^3 d^2 f^2 n^2 x \log \left (c x^n\right )+12 b^3 n^2 \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )-12 b^3 d^2 f^2 n^2 x \log \left (d \sqrt{x} f+1\right ) \log \left (c x^n\right )-84 b^3 d f n^2 \sqrt{x} \log \left (c x^n\right )+12 b n \left (a^2-2 b n a+2 b^2 n^2+b^2 \log ^2\left (c x^n\right )+2 b (a-b n) \log \left (c x^n\right )\right ) \text{PolyLog}\left (2,-d f \sqrt{x}\right )-48 b^2 n^2 \left (a-b n+b \log \left (c x^n\right )\right ) \text{PolyLog}\left (3,-d f \sqrt{x}\right )+96 b^3 n^3 \text{PolyLog}\left (4,-d f \sqrt{x}\right )+180 b^3 d f n^3 \sqrt{x}}{2 d^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*a^3*d*f*Sqrt[x] + 18*a^2*b*d*f*n*Sqrt[x] - 84*a*b^2*d*f*n^2*Sqrt[x] + 180*b^3*d*f*n^3*Sqrt[x] + a^3*d^2*f
^2*x - 6*a^2*b*d^2*f^2*n*x + 18*a*b^2*d^2*f^2*n^2*x - 24*b^3*d^2*f^2*n^3*x + 2*a^3*Log[1 + d*f*Sqrt[x]] - 6*a^
2*b*n*Log[1 + d*f*Sqrt[x]] + 12*a*b^2*n^2*Log[1 + d*f*Sqrt[x]] - 12*b^3*n^3*Log[1 + d*f*Sqrt[x]] - 2*a^3*d^2*f
^2*x*Log[1 + d*f*Sqrt[x]] + 6*a^2*b*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]] - 12*a*b^2*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt
[x]] + 12*b^3*d^2*f^2*n^3*x*Log[1 + d*f*Sqrt[x]] - 6*a^2*b*d*f*Sqrt[x]*Log[c*x^n] + 36*a*b^2*d*f*n*Sqrt[x]*Log
[c*x^n] - 84*b^3*d*f*n^2*Sqrt[x]*Log[c*x^n] + 3*a^2*b*d^2*f^2*x*Log[c*x^n] - 12*a*b^2*d^2*f^2*n*x*Log[c*x^n] +
18*b^3*d^2*f^2*n^2*x*Log[c*x^n] + 6*a^2*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 12*a*b^2*n*Log[1 + d*f*Sqrt[x]]*L
og[c*x^n] + 12*b^3*n^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 6*a^2*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 1
2*a*b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 12*b^3*d^2*f^2*n^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 6
*a*b^2*d*f*Sqrt[x]*Log[c*x^n]^2 + 18*b^3*d*f*n*Sqrt[x]*Log[c*x^n]^2 + 3*a*b^2*d^2*f^2*x*Log[c*x^n]^2 - 6*b^3*d
^2*f^2*n*x*Log[c*x^n]^2 + 6*a*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 6*b^3*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^
2 - 6*a*b^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 6*b^3*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2
- 2*b^3*d*f*Sqrt[x]*Log[c*x^n]^3 + b^3*d^2*f^2*x*Log[c*x^n]^3 + 2*b^3*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3 - 2*b^
3*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^3 + 12*b*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)*PolyLog[2, -(d*f*Sqrt[x])] - 48*b^2*n^2*(a - b*n + b*Log[c*x^n])*PolyLog[3, -(d*f*Sqrt[x])]
+ 96*b^3*n^3*PolyLog[4, -(d*f*Sqrt[x])])/(2*d^2*f^2)

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{3}\ln \left ( d \left ({d}^{-1}+f\sqrt{x} \right ) \right ) \, dx \end{align*}

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)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="maxima")

[Out]

(b^3*x*log(x^n)^3 - 3*(b^3*(n - log(c)) - a*b^2)*x*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*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*l
og(c)^2 - log(c)^3)*b^3 - 3*a^2*b*(n - log(c)) + a^3)*x)*log(d*f*sqrt(x) + 1) - 1/27*(9*b^3*d*f*x^2*log(x^n)^3
+ 9*(3*a*b^2*d*f - (5*d*f*n - 3*d*f*log(c))*b^3)*x^2*log(x^n)^2 + 3*(9*a^2*b*d*f - 6*(5*d*f*n - 3*d*f*log(c))
*a*b^2 + (38*d*f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*b^3)*x^2*log(x^n) + (9*a^3*d*f - 9*(5*d*f*n - 3*d*f*l
og(c))*a^2*b + 3*(38*d*f*n^2 - 30*d*f*n*log(c) + 9*d*f*log(c)^2)*a*b^2 - (130*d*f*n^3 - 114*d*f*n^2*log(c) + 4
5*d*f*n*log(c)^2 - 9*d*f*log(c)^3)*b^3)*x^2)/sqrt(x) + integrate(1/2*(b^3*d^2*f^2*x*log(x^n)^3 + 3*(a*b^2*d^2*
f^2 - (d^2*f^2*n - d^2*f^2*log(c))*b^3)*x*log(x^n)^2 + 3*(a^2*b*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b^2
+ (2*d^2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*b^3)*x*log(x^n) + (a^3*d^2*f^2 - 3*(d^2*f^2*n - d^2
*f^2*log(c))*a^2*b + 3*(2*d^2*f^2*n^2 - 2*d^2*f^2*n*log(c) + d^2*f^2*log(c)^2)*a*b^2 - (6*d^2*f^2*n^3 - 6*d^2*
f^2*n^2*log(c) + 3*d^2*f^2*n*log(c)^2 - d^2*f^2*log(c)^3)*b^3)*x)/(d*f*sqrt(x) + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt{x} + 1\right ), x\right ) \end{align*}

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, 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)

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

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)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt{x} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

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, algorithm="giac")

[Out]

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