3.55 \(\int \log (d (\frac{1}{d}+f \sqrt{x})) (a+b \log (c x^n))^2 \, dx\)

Optimal. Leaf size=374 \[ -\frac{4 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{4 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^2 f^2}+\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}+\frac{14 b^2 n^2 \sqrt{x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )-3 b^2 n^2 x \]

[Out]

(14*b^2*n^2*Sqrt[x])/(d*f) + a*b*n*x - 3*b^2*n^2*x + 2*b^2*n^2*x*Log[d*(d^(-1) + f*Sqrt[x])] - (2*b^2*n^2*Log[
1 + d*f*Sqrt[x]])/(d^2*f^2) + b^2*n*x*Log[c*x^n] - (6*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) + b*n*x*(a + b*Log
[c*x^n]) - 2*b*n*x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x
^n]))/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - (x*(a + b*Log[c*x^n])^2)/2 + x*Log[d*(d^(-1) + f*Sqrt
[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (4*b^2*n^2*PolyLog[2, -(d
*f*Sqrt[x])])/(d^2*f^2) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (8*b^2*n^2*PolyLog
[3, -(d*f*Sqrt[x])])/(d^2*f^2)

________________________________________________________________________________________

Rubi [A]  time = 0.268821, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2448, 266, 43, 2370, 2295, 2304, 2391, 2374, 6589} \[ -\frac{4 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{4 b^2 n^2 \text{PolyLog}\left (2,-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )}{d^2 f^2}+\frac{2 b n \log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}-\frac{\log \left (d f \sqrt{x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 f^2}-2 b n x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+a b n x+b^2 n x \log \left (c x^n\right )-\frac{2 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{d^2 f^2}+\frac{14 b^2 n^2 \sqrt{x}}{d f}+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )-3 b^2 n^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*b^2*n^2*Sqrt[x])/(d*f) + a*b*n*x - 3*b^2*n^2*x + 2*b^2*n^2*x*Log[d*(d^(-1) + f*Sqrt[x])] - (2*b^2*n^2*Log[
1 + d*f*Sqrt[x]])/(d^2*f^2) + b^2*n*x*Log[c*x^n] - (6*b*n*Sqrt[x]*(a + b*Log[c*x^n]))/(d*f) + b*n*x*(a + b*Log
[c*x^n]) - 2*b*n*x*Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]) + (2*b*n*Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x
^n]))/(d^2*f^2) + (Sqrt[x]*(a + b*Log[c*x^n])^2)/(d*f) - (x*(a + b*Log[c*x^n])^2)/2 + x*Log[d*(d^(-1) + f*Sqrt
[x])]*(a + b*Log[c*x^n])^2 - (Log[1 + d*f*Sqrt[x]]*(a + b*Log[c*x^n])^2)/(d^2*f^2) + (4*b^2*n^2*PolyLog[2, -(d
*f*Sqrt[x])])/(d^2*f^2) - (4*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])])/(d^2*f^2) + (8*b^2*n^2*PolyLog
[3, -(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 2295

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

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]

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 )^2 \, dx &=\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}-(2 b n) \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{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}-(b n) \int \left (-a-b \log \left (c x^n\right )\right ) \, dx-(2 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 ) \, dx+\frac{(2 b n) \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{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{x}} \, dx}{d f}\\ &=\frac{8 b^2 n^2 \sqrt{x}}{d f}+a b n x-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\left (b^2 n\right ) \int \log \left (c x^n\right ) \, dx+\left (2 b^2 n^2\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 (4 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-d f \sqrt{x}\right )}{x} \, dx}{d^2 f^2}\\ &=\frac{12 b^2 n^2 \sqrt{x}}{d f}+a b n x-2 b^2 n^2 x+b^2 n x \log \left (c x^n\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^2 f^2}+\left (2 b^2 n^2\right ) \int \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right ) \, dx-\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+d f \sqrt{x}\right )}{x} \, dx}{d^2 f^2}\\ &=\frac{12 b^2 n^2 \sqrt{x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^2 f^2}-\left (b^2 f n^2\right ) \int \frac{\sqrt{x}}{\frac{1}{d}+f \sqrt{x}} \, dx\\ &=\frac{12 b^2 n^2 \sqrt{x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^2 f^2}-\left (2 b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{\frac{1}{d}+f x} \, dx,x,\sqrt{x}\right )\\ &=\frac{12 b^2 n^2 \sqrt{x}}{d f}+a b n x-2 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )+b^2 n x \log \left (c x^n\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^2 f^2}-\left (2 b^2 f n^2\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{14 b^2 n^2 \sqrt{x}}{d f}+a b n x-3 b^2 n^2 x+2 b^2 n^2 x \log \left (d \left (\frac{1}{d}+f \sqrt{x}\right )\right )-\frac{2 b^2 n^2 \log \left (1+d f \sqrt{x}\right )}{d^2 f^2}+b^2 n x \log \left (c x^n\right )-\frac{6 b n \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{d f}+b n x \left (a+b \log \left (c x^n\right )\right )-2 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 )+\frac{2 b n \log \left (1+d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 f^2}+\frac{\sqrt{x} \left (a+b \log \left (c x^n\right )\right )^2}{d f}-\frac{1}{2} x \left (a+b \log \left (c x^n\right )\right )^2+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}+\frac{4 b^2 n^2 \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}-\frac{4 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-d f \sqrt{x}\right )}{d^2 f^2}+\frac{8 b^2 n^2 \text{Li}_3\left (-d f \sqrt{x}\right )}{d^2 f^2}\\ \end{align*}

Mathematica [A]  time = 0.31961, size = 527, normalized size = 1.41 \[ -\frac{8 b n \text{PolyLog}\left (2,-d f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )-b n\right )-16 b^2 n^2 \text{PolyLog}\left (3,-d f \sqrt{x}\right )+a^2 d^2 f^2 x-2 a^2 d^2 f^2 x \log \left (d f \sqrt{x}+1\right )-2 a^2 d f \sqrt{x}+2 a^2 \log \left (d f \sqrt{x}+1\right )+2 a b d^2 f^2 x \log \left (c x^n\right )-4 a b d^2 f^2 x \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+4 a b \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-4 a b d f \sqrt{x} \log \left (c x^n\right )-4 a b d^2 f^2 n x+4 a b d^2 f^2 n x \log \left (d f \sqrt{x}+1\right )+12 a b d f n \sqrt{x}-4 a b n \log \left (d f \sqrt{x}+1\right )+b^2 d^2 f^2 x \log ^2\left (c x^n\right )-2 b^2 d^2 f^2 x \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-4 b^2 d^2 f^2 n x \log \left (c x^n\right )+4 b^2 d^2 f^2 n x \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+2 b^2 \log ^2\left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )-2 b^2 d f \sqrt{x} \log ^2\left (c x^n\right )-4 b^2 n \log \left (c x^n\right ) \log \left (d f \sqrt{x}+1\right )+12 b^2 d f n \sqrt{x} \log \left (c x^n\right )+6 b^2 d^2 f^2 n^2 x-4 b^2 d^2 f^2 n^2 x \log \left (d f \sqrt{x}+1\right )-28 b^2 d f n^2 \sqrt{x}+4 b^2 n^2 \log \left (d f \sqrt{x}+1\right )}{2 d^2 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2*a^2*d*f*Sqrt[x] + 12*a*b*d*f*n*Sqrt[x] - 28*b^2*d*f*n^2*Sqrt[x] + a^2*d^2*f^2*x - 4*a*b*d^2*f^2*n*x + 6*b
^2*d^2*f^2*n^2*x + 2*a^2*Log[1 + d*f*Sqrt[x]] - 4*a*b*n*Log[1 + d*f*Sqrt[x]] + 4*b^2*n^2*Log[1 + d*f*Sqrt[x]]
- 2*a^2*d^2*f^2*x*Log[1 + d*f*Sqrt[x]] + 4*a*b*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]] - 4*b^2*d^2*f^2*n^2*x*Log[1 +
d*f*Sqrt[x]] - 4*a*b*d*f*Sqrt[x]*Log[c*x^n] + 12*b^2*d*f*n*Sqrt[x]*Log[c*x^n] + 2*a*b*d^2*f^2*x*Log[c*x^n] - 4
*b^2*d^2*f^2*n*x*Log[c*x^n] + 4*a*b*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 4*b^2*n*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]
- 4*a*b*d^2*f^2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] + 4*b^2*d^2*f^2*n*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n] - 2*b^2*
d*f*Sqrt[x]*Log[c*x^n]^2 + b^2*d^2*f^2*x*Log[c*x^n]^2 + 2*b^2*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 - 2*b^2*d^2*f^
2*x*Log[1 + d*f*Sqrt[x]]*Log[c*x^n]^2 + 8*b*n*(a - b*n + b*Log[c*x^n])*PolyLog[2, -(d*f*Sqrt[x])] - 16*b^2*n^2
*PolyLog[3, -(d*f*Sqrt[x])])/(2*d^2*f^2)

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Maple [F]  time = 0.019, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}\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))^2*ln(d*(1/d+f*x^(1/2))),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\left (b^{2} x \log \left (x^{n}\right )^{2} - 2 \,{\left (b^{2}{\left (n - \log \left (c\right )\right )} - a b\right )} x \log \left (x^{n}\right ) +{\left ({\left (2 \, n^{2} - 2 \, n \log \left (c\right ) + \log \left (c\right )^{2}\right )} b^{2} - 2 \, a b{\left (n - \log \left (c\right )\right )} + a^{2}\right )} x\right )} \log \left (d f \sqrt{x} + 1\right ) - \frac{9 \, b^{2} d f x^{2} \log \left (x^{n}\right )^{2} + 6 \,{\left (3 \, a b d f -{\left (5 \, d f n - 3 \, d f \log \left (c\right )\right )} b^{2}\right )} x^{2} \log \left (x^{n}\right ) +{\left (9 \, a^{2} d f - 6 \,{\left (5 \, d f n - 3 \, d f \log \left (c\right )\right )} a b +{\left (38 \, d f n^{2} - 30 \, d f n \log \left (c\right ) + 9 \, d f \log \left (c\right )^{2}\right )} b^{2}\right )} x^{2}}{27 \, \sqrt{x}} + \int \frac{b^{2} d^{2} f^{2} x \log \left (x^{n}\right )^{2} + 2 \,{\left (a b d^{2} f^{2} -{\left (d^{2} f^{2} n - d^{2} f^{2} \log \left (c\right )\right )} b^{2}\right )} x \log \left (x^{n}\right ) +{\left (a^{2} d^{2} f^{2} - 2 \,{\left (d^{2} f^{2} n - d^{2} f^{2} \log \left (c\right )\right )} a b +{\left (2 \, d^{2} f^{2} n^{2} - 2 \, d^{2} f^{2} n \log \left (c\right ) + d^{2} f^{2} \log \left (c\right )^{2}\right )} b^{2}\right )} x}{2 \,{\left (d f \sqrt{x} + 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^2*x*log(x^n)^2 - 2*(b^2*(n - log(c)) - a*b)*x*log(x^n) + ((2*n^2 - 2*n*log(c) + log(c)^2)*b^2 - 2*a*b*(n -
log(c)) + a^2)*x)*log(d*f*sqrt(x) + 1) - 1/27*(9*b^2*d*f*x^2*log(x^n)^2 + 6*(3*a*b*d*f - (5*d*f*n - 3*d*f*log(
c))*b^2)*x^2*log(x^n) + (9*a^2*d*f - 6*(5*d*f*n - 3*d*f*log(c))*a*b + (38*d*f*n^2 - 30*d*f*n*log(c) + 9*d*f*lo
g(c)^2)*b^2)*x^2)/sqrt(x) + integrate(1/2*(b^2*d^2*f^2*x*log(x^n)^2 + 2*(a*b*d^2*f^2 - (d^2*f^2*n - d^2*f^2*lo
g(c))*b^2)*x*log(x^n) + (a^2*d^2*f^2 - 2*(d^2*f^2*n - d^2*f^2*log(c))*a*b + (2*d^2*f^2*n^2 - 2*d^2*f^2*n*log(c
) + d^2*f^2*log(c)^2)*b^2)*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^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\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))^2*log(d*(1/d+f*x^(1/2))),x, algorithm="fricas")

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)*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))**2*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 )}^{2} \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))^2*log(d*(1/d+f*x^(1/2))),x, algorithm="giac")

[Out]

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