3.115 \(\int x^2 \log (d (e+f \sqrt{x})^k) (a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=403 \[ \frac{2 b e^6 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 f^6}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{b e^3 k n x^{3/2}}{9 f^3}+\frac{5 b e^2 k n x^2}{72 f^2}-\frac{7 b e^5 k n \sqrt{x}}{9 f^5}+\frac{2 b e^4 k n x}{9 f^4}+\frac{b e^6 k n \log \left (e+f \sqrt{x}\right )}{9 f^6}+\frac{2 b e^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^6}-\frac{11 b e k n x^{5/2}}{225 f}+\frac{1}{27} b k n x^3 \]

[Out]

(-7*b*e^5*k*n*Sqrt[x])/(9*f^5) + (2*b*e^4*k*n*x)/(9*f^4) - (b*e^3*k*n*x^(3/2))/(9*f^3) + (5*b*e^2*k*n*x^2)/(72
*f^2) - (11*b*e*k*n*x^(5/2))/(225*f) + (b*k*n*x^3)/27 + (b*e^6*k*n*Log[e + f*Sqrt[x]])/(9*f^6) - (b*n*x^3*Log[
d*(e + f*Sqrt[x])^k])/9 + (2*b*e^6*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*f^6) + (e^5*k*Sqrt[x]*(a +
b*Log[c*x^n]))/(3*f^5) - (e^4*k*x*(a + b*Log[c*x^n]))/(6*f^4) + (e^3*k*x^(3/2)*(a + b*Log[c*x^n]))/(9*f^3) -
(e^2*k*x^2*(a + b*Log[c*x^n]))/(12*f^2) + (e*k*x^(5/2)*(a + b*Log[c*x^n]))/(15*f) - (k*x^3*(a + b*Log[c*x^n]))
/18 - (e^6*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^6) + (x^3*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])
)/3 + (2*b*e^6*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^6)

________________________________________________________________________________________

Rubi [A]  time = 0.344635, antiderivative size = 403, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2454, 2395, 43, 2376, 2394, 2315} \[ \frac{2 b e^6 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 f^6}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-\frac{b e^3 k n x^{3/2}}{9 f^3}+\frac{5 b e^2 k n x^2}{72 f^2}-\frac{7 b e^5 k n \sqrt{x}}{9 f^5}+\frac{2 b e^4 k n x}{9 f^4}+\frac{b e^6 k n \log \left (e+f \sqrt{x}\right )}{9 f^6}+\frac{2 b e^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^6}-\frac{11 b e k n x^{5/2}}{225 f}+\frac{1}{27} b k n x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*b*e^5*k*n*Sqrt[x])/(9*f^5) + (2*b*e^4*k*n*x)/(9*f^4) - (b*e^3*k*n*x^(3/2))/(9*f^3) + (5*b*e^2*k*n*x^2)/(72
*f^2) - (11*b*e*k*n*x^(5/2))/(225*f) + (b*k*n*x^3)/27 + (b*e^6*k*n*Log[e + f*Sqrt[x]])/(9*f^6) - (b*n*x^3*Log[
d*(e + f*Sqrt[x])^k])/9 + (2*b*e^6*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*f^6) + (e^5*k*Sqrt[x]*(a +
b*Log[c*x^n]))/(3*f^5) - (e^4*k*x*(a + b*Log[c*x^n]))/(6*f^4) + (e^3*k*x^(3/2)*(a + b*Log[c*x^n]))/(9*f^3) -
(e^2*k*x^2*(a + b*Log[c*x^n]))/(12*f^2) + (e*k*x^(5/2)*(a + b*Log[c*x^n]))/(15*f) - (k*x^3*(a + b*Log[c*x^n]))
/18 - (e^6*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*f^6) + (x^3*Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n])
)/3 + (2*b*e^6*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])/(3*f^6)

Rule 2454

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 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 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 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2394

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 2315

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

Rubi steps

\begin{align*} \int x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac{e^4 k}{6 f^4}+\frac{e^5 k}{3 f^5 \sqrt{x}}+\frac{e^3 k \sqrt{x}}{9 f^3}-\frac{e^2 k x}{12 f^2}+\frac{e k x^{3/2}}{15 f}-\frac{k x^2}{18}-\frac{e^6 k \log \left (e+f \sqrt{x}\right )}{3 f^6 x}+\frac{1}{3} x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )\right ) \, dx\\ &=-\frac{2 b e^5 k n \sqrt{x}}{3 f^5}+\frac{b e^4 k n x}{6 f^4}-\frac{2 b e^3 k n x^{3/2}}{27 f^3}+\frac{b e^2 k n x^2}{24 f^2}-\frac{2 b e k n x^{5/2}}{75 f}+\frac{1}{54} b k n x^3+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (b n) \int x^2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \, dx+\frac{\left (b e^6 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{3 f^6}\\ &=-\frac{2 b e^5 k n \sqrt{x}}{3 f^5}+\frac{b e^4 k n x}{6 f^4}-\frac{2 b e^3 k n x^{3/2}}{27 f^3}+\frac{b e^2 k n x^2}{24 f^2}-\frac{2 b e k n x^{5/2}}{75 f}+\frac{1}{54} b k n x^3+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{1}{3} (2 b n) \operatorname{Subst}\left (\int x^5 \log \left (d (e+f x)^k\right ) \, dx,x,\sqrt{x}\right )+\frac{\left (2 b e^6 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{3 f^6}\\ &=-\frac{2 b e^5 k n \sqrt{x}}{3 f^5}+\frac{b e^4 k n x}{6 f^4}-\frac{2 b e^3 k n x^{3/2}}{27 f^3}+\frac{b e^2 k n x^2}{24 f^2}-\frac{2 b e k n x^{5/2}}{75 f}+\frac{1}{54} b k n x^3-\frac{1}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^6}+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{\left (2 b e^6 k n\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx,x,\sqrt{x}\right )}{3 f^5}+\frac{1}{9} (b f k n) \operatorname{Subst}\left (\int \frac{x^6}{e+f x} \, dx,x,\sqrt{x}\right )\\ &=-\frac{2 b e^5 k n \sqrt{x}}{3 f^5}+\frac{b e^4 k n x}{6 f^4}-\frac{2 b e^3 k n x^{3/2}}{27 f^3}+\frac{b e^2 k n x^2}{24 f^2}-\frac{2 b e k n x^{5/2}}{75 f}+\frac{1}{54} b k n x^3-\frac{1}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^6}+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^6 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 f^6}+\frac{1}{9} (b f k n) \operatorname{Subst}\left (\int \left (-\frac{e^5}{f^6}+\frac{e^4 x}{f^5}-\frac{e^3 x^2}{f^4}+\frac{e^2 x^3}{f^3}-\frac{e x^4}{f^2}+\frac{x^5}{f}+\frac{e^6}{f^6 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{7 b e^5 k n \sqrt{x}}{9 f^5}+\frac{2 b e^4 k n x}{9 f^4}-\frac{b e^3 k n x^{3/2}}{9 f^3}+\frac{5 b e^2 k n x^2}{72 f^2}-\frac{11 b e k n x^{5/2}}{225 f}+\frac{1}{27} b k n x^3+\frac{b e^6 k n \log \left (e+f \sqrt{x}\right )}{9 f^6}-\frac{1}{9} b n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+\frac{2 b e^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 f^6}+\frac{e^5 k \sqrt{x} \left (a+b \log \left (c x^n\right )\right )}{3 f^5}-\frac{e^4 k x \left (a+b \log \left (c x^n\right )\right )}{6 f^4}+\frac{e^3 k x^{3/2} \left (a+b \log \left (c x^n\right )\right )}{9 f^3}-\frac{e^2 k x^2 \left (a+b \log \left (c x^n\right )\right )}{12 f^2}+\frac{e k x^{5/2} \left (a+b \log \left (c x^n\right )\right )}{15 f}-\frac{1}{18} k x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{e^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 f^6}+\frac{1}{3} x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 b e^6 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 f^6}\\ \end{align*}

Mathematica [A]  time = 0.463241, size = 434, normalized size = 1.08 \[ -\frac{3600 b e^6 k n \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )+600 e^6 k \log \left (e+f \sqrt{x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)-b n\right )-1800 a f^6 x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-600 a e^3 f^3 k x^{3/2}+450 a e^2 f^4 k x^2+900 a e^4 f^2 k x-1800 a e^5 f k \sqrt{x}-360 a e f^5 k x^{5/2}+300 a f^6 k x^3-1800 b f^6 x^3 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )+900 b e^4 f^2 k x \log \left (c x^n\right )-600 b e^3 f^3 k x^{3/2} \log \left (c x^n\right )+450 b e^2 f^4 k x^2 \log \left (c x^n\right )-1800 b e^5 f k \sqrt{x} \log \left (c x^n\right )-360 b e f^5 k x^{5/2} \log \left (c x^n\right )+300 b f^6 k x^3 \log \left (c x^n\right )+600 b f^6 n x^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+600 b e^3 f^3 k n x^{3/2}-375 b e^2 f^4 k n x^2-1200 b e^4 f^2 k n x+4200 b e^5 f k n \sqrt{x}+1800 b e^6 k n \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+264 b e f^5 k n x^{5/2}-200 b f^6 k n x^3}{5400 f^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-1800*a*e^5*f*k*Sqrt[x] + 4200*b*e^5*f*k*n*Sqrt[x] + 900*a*e^4*f^2*k*x - 1200*b*e^4*f^2*k*n*x - 600*a*e^3*f^
3*k*x^(3/2) + 600*b*e^3*f^3*k*n*x^(3/2) + 450*a*e^2*f^4*k*x^2 - 375*b*e^2*f^4*k*n*x^2 - 360*a*e*f^5*k*x^(5/2)
+ 264*b*e*f^5*k*n*x^(5/2) + 300*a*f^6*k*x^3 - 200*b*f^6*k*n*x^3 - 1800*a*f^6*x^3*Log[d*(e + f*Sqrt[x])^k] + 60
0*b*f^6*n*x^3*Log[d*(e + f*Sqrt[x])^k] + 1800*b*e^6*k*n*Log[1 + (f*Sqrt[x])/e]*Log[x] - 1800*b*e^5*f*k*Sqrt[x]
*Log[c*x^n] + 900*b*e^4*f^2*k*x*Log[c*x^n] - 600*b*e^3*f^3*k*x^(3/2)*Log[c*x^n] + 450*b*e^2*f^4*k*x^2*Log[c*x^
n] - 360*b*e*f^5*k*x^(5/2)*Log[c*x^n] + 300*b*f^6*k*x^3*Log[c*x^n] - 1800*b*f^6*x^3*Log[d*(e + f*Sqrt[x])^k]*L
og[c*x^n] + 600*e^6*k*Log[e + f*Sqrt[x]]*(3*a - b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) + 3600*b*e^6*k*n*PolyLog[
2, -((f*Sqrt[x])/e)])/(5400*f^6)

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{147 \, b e x^{3} \log \left (d\right ) \log \left (x^{n}\right ) + 49 \,{\left (3 \, a e \log \left (d\right ) -{\left (e n \log \left (d\right ) - 3 \, e \log \left (c\right ) \log \left (d\right )\right )} b\right )} x^{3} + 49 \,{\left (3 \, b e x^{3} \log \left (x^{n}\right ) -{\left ({\left (e n - 3 \, e \log \left (c\right )\right )} b - 3 \, a e\right )} x^{3}\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) - \frac{21 \, b f k x^{4} \log \left (x^{n}\right ) +{\left (21 \, a f k -{\left (13 \, f k n - 21 \, f k \log \left (c\right )\right )} b\right )} x^{4}}{\sqrt{x}}}{441 \, e} + \int \frac{3 \, b f^{2} k x^{3} \log \left (x^{n}\right ) +{\left (3 \, a f^{2} k -{\left (f^{2} k n - 3 \, f^{2} k \log \left (c\right )\right )} b\right )} x^{3}}{18 \,{\left (e f \sqrt{x} + e^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*(147*b*e*x^3*log(d)*log(x^n) + 49*(3*a*e*log(d) - (e*n*log(d) - 3*e*log(c)*log(d))*b)*x^3 + 49*(3*b*e*x^
3*log(x^n) - ((e*n - 3*e*log(c))*b - 3*a*e)*x^3)*log((f*sqrt(x) + e)^k) - (21*b*f*k*x^4*log(x^n) + (21*a*f*k -
(13*f*k*n - 21*f*k*log(c))*b)*x^4)/sqrt(x))/e + integrate(1/18*(3*b*f^2*k*x^3*log(x^n) + (3*a*f^2*k - (f^2*k*
n - 3*f^2*k*log(c))*b)*x^3)/(e*f*sqrt(x) + e^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*log(c*x^n) + a*x^2)*log((f*sqrt(x) + e)^k*d), 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(x**2*(a+b*ln(c*x**n))*ln(d*(e+f*x**(1/2))**k),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 )} x^{2} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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