3.121 \(\int \frac{\log (d (e+f \sqrt{x})^k) (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=434 \[ -\frac{2 b f^6 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 e^6}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{3 x^3}+\frac{f^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^6}-\frac{f^6 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{6 e^6}-\frac{f^5 k \left (a+b \log \left (c x^n\right )\right )}{3 e^5 \sqrt{x}}+\frac{f^4 k \left (a+b \log \left (c x^n\right )\right )}{6 e^4 x}-\frac{f^3 k \left (a+b \log \left (c x^n\right )\right )}{9 e^3 x^{3/2}}+\frac{f^2 k \left (a+b \log \left (c x^n\right )\right )}{12 e^2 x^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{15 e x^{5/2}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^3}-\frac{b f^3 k n}{9 e^3 x^{3/2}}+\frac{5 b f^2 k n}{72 e^2 x^2}-\frac{7 b f^5 k n}{9 e^5 \sqrt{x}}+\frac{2 b f^4 k n}{9 e^4 x}+\frac{b f^6 k n \log ^2(x)}{12 e^6}+\frac{b f^6 k n \log \left (e+f \sqrt{x}\right )}{9 e^6}-\frac{2 b f^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^6}-\frac{b f^6 k n \log (x)}{18 e^6}-\frac{11 b f k n}{225 e x^{5/2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.34986, antiderivative size = 434, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2454, 2395, 44, 2376, 2394, 2315, 2301} \[ -\frac{2 b f^6 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 e^6}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{3 x^3}+\frac{f^6 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^6}-\frac{f^6 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{6 e^6}-\frac{f^5 k \left (a+b \log \left (c x^n\right )\right )}{3 e^5 \sqrt{x}}+\frac{f^4 k \left (a+b \log \left (c x^n\right )\right )}{6 e^4 x}-\frac{f^3 k \left (a+b \log \left (c x^n\right )\right )}{9 e^3 x^{3/2}}+\frac{f^2 k \left (a+b \log \left (c x^n\right )\right )}{12 e^2 x^2}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{15 e x^{5/2}}-\frac{b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^3}-\frac{b f^3 k n}{9 e^3 x^{3/2}}+\frac{5 b f^2 k n}{72 e^2 x^2}-\frac{7 b f^5 k n}{9 e^5 \sqrt{x}}+\frac{2 b f^4 k n}{9 e^4 x}+\frac{b f^6 k n \log ^2(x)}{12 e^6}+\frac{b f^6 k n \log \left (e+f \sqrt{x}\right )}{9 e^6}-\frac{2 b f^6 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^6}-\frac{b f^6 k n \log (x)}{18 e^6}-\frac{11 b f k n}{225 e x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

(-11*b*f*k*n)/(225*e*x^(5/2)) + (5*b*f^2*k*n)/(72*e^2*x^2) - (b*f^3*k*n)/(9*e^3*x^(3/2)) + (2*b*f^4*k*n)/(9*e^
4*x) - (7*b*f^5*k*n)/(9*e^5*Sqrt[x]) + (b*f^6*k*n*Log[e + f*Sqrt[x]])/(9*e^6) - (b*n*Log[d*(e + f*Sqrt[x])^k])
/(9*x^3) - (2*b*f^6*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^6) - (b*f^6*k*n*Log[x])/(18*e^6) + (b*f
^6*k*n*Log[x]^2)/(12*e^6) - (f*k*(a + b*Log[c*x^n]))/(15*e*x^(5/2)) + (f^2*k*(a + b*Log[c*x^n]))/(12*e^2*x^2)
- (f^3*k*(a + b*Log[c*x^n]))/(9*e^3*x^(3/2)) + (f^4*k*(a + b*Log[c*x^n]))/(6*e^4*x) - (f^5*k*(a + b*Log[c*x^n]
))/(3*e^5*Sqrt[x]) + (f^6*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^6) - (Log[d*(e + f*Sqrt[x])^k]*(a + b*
Log[c*x^n]))/(3*x^3) - (f^6*k*Log[x]*(a + b*Log[c*x^n]))/(6*e^6) - (2*b*f^6*k*n*PolyLog[2, 1 + (f*Sqrt[x])/e])
/(3*e^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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[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]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rubi steps

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

Mathematica [A]  time = 0.496986, size = 457, normalized size = 1.05 \[ -\frac{-1200 b f^6 k n x^3 \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )-200 f^6 k x^3 \log \left (e+f \sqrt{x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)+b n\right )+600 a e^6 \log \left (d \left (e+f \sqrt{x}\right )^k\right )+200 a e^3 f^3 k x^{3/2}-300 a e^2 f^4 k x^2-150 a e^4 f^2 k x+120 a e^5 f k \sqrt{x}+600 a e f^5 k x^{5/2}+300 a f^6 k x^3 \log (x)+600 b e^6 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-150 b e^4 f^2 k x \log \left (c x^n\right )+200 b e^3 f^3 k x^{3/2} \log \left (c x^n\right )-300 b e^2 f^4 k x^2 \log \left (c x^n\right )+120 b e^5 f k \sqrt{x} \log \left (c x^n\right )+600 b e f^5 k x^{5/2} \log \left (c x^n\right )+300 b f^6 k x^3 \log (x) \log \left (c x^n\right )+200 b e^6 n \log \left (d \left (e+f \sqrt{x}\right )^k\right )+200 b e^3 f^3 k n x^{3/2}-400 b e^2 f^4 k n x^2-125 b e^4 f^2 k n x+88 b e^5 f k n \sqrt{x}+1400 b e f^5 k n x^{5/2}-600 b f^6 k n x^3 \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )-150 b f^6 k n x^3 \log ^2(x)+100 b f^6 k n x^3 \log (x)}{1800 e^6 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

-(120*a*e^5*f*k*Sqrt[x] + 88*b*e^5*f*k*n*Sqrt[x] - 150*a*e^4*f^2*k*x - 125*b*e^4*f^2*k*n*x + 200*a*e^3*f^3*k*x
^(3/2) + 200*b*e^3*f^3*k*n*x^(3/2) - 300*a*e^2*f^4*k*x^2 - 400*b*e^2*f^4*k*n*x^2 + 600*a*e*f^5*k*x^(5/2) + 140
0*b*e*f^5*k*n*x^(5/2) + 600*a*e^6*Log[d*(e + f*Sqrt[x])^k] + 200*b*e^6*n*Log[d*(e + f*Sqrt[x])^k] + 300*a*f^6*
k*x^3*Log[x] + 100*b*f^6*k*n*x^3*Log[x] - 600*b*f^6*k*n*x^3*Log[1 + (f*Sqrt[x])/e]*Log[x] - 150*b*f^6*k*n*x^3*
Log[x]^2 + 120*b*e^5*f*k*Sqrt[x]*Log[c*x^n] - 150*b*e^4*f^2*k*x*Log[c*x^n] + 200*b*e^3*f^3*k*x^(3/2)*Log[c*x^n
] - 300*b*e^2*f^4*k*x^2*Log[c*x^n] + 600*b*e*f^5*k*x^(5/2)*Log[c*x^n] + 600*b*e^6*Log[d*(e + f*Sqrt[x])^k]*Log
[c*x^n] + 300*b*f^6*k*x^3*Log[x]*Log[c*x^n] - 200*f^6*k*x^3*Log[e + f*Sqrt[x]]*(3*a + b*n - 3*b*n*Log[x] + 3*b
*Log[c*x^n]) - 1200*b*f^6*k*n*x^3*PolyLog[2, -((f*Sqrt[x])/e)])/(1800*e^6*x^3)

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{4}}\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((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^4,x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^4,x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{75 \, b e \log \left (d\right ) \log \left (x^{n}\right ) + 75 \, a e \log \left (d\right ) + 25 \,{\left (e n \log \left (d\right ) + 3 \, e \log \left (c\right ) \log \left (d\right )\right )} b + 25 \,{\left (3 \, b e \log \left (x^{n}\right ) +{\left (e n + 3 \, e \log \left (c\right )\right )} b + 3 \, a e\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right ) + \frac{15 \, b f k x \log \left (x^{n}\right ) +{\left (15 \, a f k +{\left (11 \, f k n + 15 \, f k \log \left (c\right )\right )} b\right )} x}{\sqrt{x}}}{225 \, e x^{3}} - \int \frac{3 \, b f^{2} k \log \left (x^{n}\right ) + 3 \, a f^{2} k +{\left (f^{2} k n + 3 \, f^{2} k \log \left (c\right )\right )} b}{18 \,{\left (e f x^{\frac{7}{2}} + e^{2} x^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^4,x, algorithm="maxima")

[Out]

-1/225*(75*b*e*log(d)*log(x^n) + 75*a*e*log(d) + 25*(e*n*log(d) + 3*e*log(c)*log(d))*b + 25*(3*b*e*log(x^n) +
(e*n + 3*e*log(c))*b + 3*a*e)*log((f*sqrt(x) + e)^k) + (15*b*f*k*x*log(x^n) + (15*a*f*k + (11*f*k*n + 15*f*k*l
og(c))*b)*x)/sqrt(x))/(e*x^3) - integrate(1/18*(3*b*f^2*k*log(x^n) + 3*a*f^2*k + (f^2*k*n + 3*f^2*k*log(c))*b)
/(e*f*x^(7/2) + e^2*x^3), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^4,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^4, x)

________________________________________________________________________________________

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))*ln(d*(e+f*x**(1/2))**k)/x**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))*log(d*(e+f*x^(1/2))^k)/x^4,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^4, x)