3.136 \(\int \frac{\log (d (e+f \sqrt{x})^k) (a+b \log (c x^n))}{x^{5/2}} \, dx\)

Optimal. Leaf size=310 \[ \frac{4 b f^3 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 e^3}-\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{3 x^{3/2}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^{3/2}}+\frac{16 b f^2 k n}{9 e^2 \sqrt{x}}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right )}{9 e^3}+\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^3}+\frac{2 b f^3 k n \log (x)}{9 e^3}-\frac{5 b f k n}{9 e x} \]

[Out]

(-5*b*f*k*n)/(9*e*x) + (16*b*f^2*k*n)/(9*e^2*Sqrt[x]) - (4*b*f^3*k*n*Log[e + f*Sqrt[x]])/(9*e^3) - (4*b*n*Log[
d*(e + f*Sqrt[x])^k])/(9*x^(3/2)) + (4*b*f^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^3) + (2*b*f^3*
k*n*Log[x])/(9*e^3) - (b*f^3*k*n*Log[x]^2)/(6*e^3) - (f*k*(a + b*Log[c*x^n]))/(3*e*x) + (2*f^2*k*(a + b*Log[c*
x^n]))/(3*e^2*Sqrt[x]) - (2*f^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^3) - (2*Log[d*(e + f*Sqrt[x])^k]
*(a + b*Log[c*x^n]))/(3*x^(3/2)) + (f^3*k*Log[x]*(a + b*Log[c*x^n]))/(3*e^3) + (4*b*f^3*k*n*PolyLog[2, 1 + (f*
Sqrt[x])/e])/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.24198, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233, Rules used = {2454, 2395, 44, 2376, 2394, 2315, 2301} \[ \frac{4 b f^3 k n \text{PolyLog}\left (2,\frac{f \sqrt{x}}{e}+1\right )}{3 e^3}-\frac{2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{3 x^{3/2}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^{3/2}}+\frac{16 b f^2 k n}{9 e^2 \sqrt{x}}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right )}{9 e^3}+\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^3}+\frac{2 b f^3 k n \log (x)}{9 e^3}-\frac{5 b f k n}{9 e x} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(5/2),x]

[Out]

(-5*b*f*k*n)/(9*e*x) + (16*b*f^2*k*n)/(9*e^2*Sqrt[x]) - (4*b*f^3*k*n*Log[e + f*Sqrt[x]])/(9*e^3) - (4*b*n*Log[
d*(e + f*Sqrt[x])^k])/(9*x^(3/2)) + (4*b*f^3*k*n*Log[e + f*Sqrt[x]]*Log[-((f*Sqrt[x])/e)])/(3*e^3) + (2*b*f^3*
k*n*Log[x])/(9*e^3) - (b*f^3*k*n*Log[x]^2)/(6*e^3) - (f*k*(a + b*Log[c*x^n]))/(3*e*x) + (2*f^2*k*(a + b*Log[c*
x^n]))/(3*e^2*Sqrt[x]) - (2*f^3*k*Log[e + f*Sqrt[x]]*(a + b*Log[c*x^n]))/(3*e^3) - (2*Log[d*(e + f*Sqrt[x])^k]
*(a + b*Log[c*x^n]))/(3*x^(3/2)) + (f^3*k*Log[x]*(a + b*Log[c*x^n]))/(3*e^3) + (4*b*f^3*k*n*PolyLog[2, 1 + (f*
Sqrt[x])/e])/(3*e^3)

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^{5/2}} \, dx &=-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-(b n) \int \left (-\frac{f k}{3 e x^2}+\frac{2 f^2 k}{3 e^2 x^{3/2}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right )}{3 e^3 x}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{3 x^{5/2}}+\frac{f^3 k \log (x)}{3 e^3 x}\right ) \, dx\\ &=-\frac{b f k n}{3 e x}+\frac{4 b f^2 k n}{3 e^2 \sqrt{x}}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{1}{3} (2 b n) \int \frac{\log \left (d \left (e+f \sqrt{x}\right )^k\right )}{x^{5/2}} \, dx-\frac{\left (b f^3 k n\right ) \int \frac{\log (x)}{x} \, dx}{3 e^3}+\frac{\left (2 b f^3 k n\right ) \int \frac{\log \left (e+f \sqrt{x}\right )}{x} \, dx}{3 e^3}\\ &=-\frac{b f k n}{3 e x}+\frac{4 b f^2 k n}{3 e^2 \sqrt{x}}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{1}{3} (4 b n) \operatorname{Subst}\left (\int \frac{\log \left (d (e+f x)^k\right )}{x^4} \, dx,x,\sqrt{x}\right )+\frac{\left (4 b f^3 k n\right ) \operatorname{Subst}\left (\int \frac{\log (e+f x)}{x} \, dx,x,\sqrt{x}\right )}{3 e^3}\\ &=-\frac{b f k n}{3 e x}+\frac{4 b f^2 k n}{3 e^2 \sqrt{x}}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^{3/2}}+\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^3}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{1}{9} (4 b f k n) \operatorname{Subst}\left (\int \frac{1}{x^3 (e+f x)} \, dx,x,\sqrt{x}\right )-\frac{\left (4 b f^4 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^3}\\ &=-\frac{b f k n}{3 e x}+\frac{4 b f^2 k n}{3 e^2 \sqrt{x}}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^{3/2}}+\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^3}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{4 b f^3 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 e^3}+\frac{1}{9} (4 b f k n) \operatorname{Subst}\left (\int \left (\frac{1}{e x^3}-\frac{f}{e^2 x^2}+\frac{f^2}{e^3 x}-\frac{f^3}{e^3 (e+f x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{5 b f k n}{9 e x}+\frac{16 b f^2 k n}{9 e^2 \sqrt{x}}-\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right )}{9 e^3}-\frac{4 b n \log \left (d \left (e+f \sqrt{x}\right )^k\right )}{9 x^{3/2}}+\frac{4 b f^3 k n \log \left (e+f \sqrt{x}\right ) \log \left (-\frac{f \sqrt{x}}{e}\right )}{3 e^3}+\frac{2 b f^3 k n \log (x)}{9 e^3}-\frac{b f^3 k n \log ^2(x)}{6 e^3}-\frac{f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac{2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt{x}}-\frac{2 f^3 k \log \left (e+f \sqrt{x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{2 \log \left (d \left (e+f \sqrt{x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac{f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{4 b f^3 k n \text{Li}_2\left (1+\frac{f \sqrt{x}}{e}\right )}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.395856, size = 326, normalized size = 1.05 \[ \frac{-12 b f^3 k n x^{3/2} \text{PolyLog}\left (2,-\frac{f \sqrt{x}}{e}\right )-2 f^3 k x^{3/2} \log \left (e+f \sqrt{x}\right ) \left (3 a+3 b \log \left (c x^n\right )-3 b n \log (x)+2 b n\right )-6 a e^3 \log \left (d \left (e+f \sqrt{x}\right )^k\right )-3 a e^2 f k \sqrt{x}+6 a e f^2 k x+3 a f^3 k x^{3/2} \log (x)-6 b e^3 \log \left (c x^n\right ) \log \left (d \left (e+f \sqrt{x}\right )^k\right )-3 b e^2 f k \sqrt{x} \log \left (c x^n\right )+6 b e f^2 k x \log \left (c x^n\right )+3 b f^3 k x^{3/2} \log (x) \log \left (c x^n\right )-4 b e^3 n \log \left (d \left (e+f \sqrt{x}\right )^k\right )-5 b e^2 f k n \sqrt{x}-6 b f^3 k n x^{3/2} \log (x) \log \left (\frac{f \sqrt{x}}{e}+1\right )+16 b e f^2 k n x-\frac{3}{2} b f^3 k n x^{3/2} \log ^2(x)+2 b f^3 k n x^{3/2} \log (x)}{9 e^3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(e + f*Sqrt[x])^k]*(a + b*Log[c*x^n]))/x^(5/2),x]

[Out]

(-3*a*e^2*f*k*Sqrt[x] - 5*b*e^2*f*k*n*Sqrt[x] + 6*a*e*f^2*k*x + 16*b*e*f^2*k*n*x - 6*a*e^3*Log[d*(e + f*Sqrt[x
])^k] - 4*b*e^3*n*Log[d*(e + f*Sqrt[x])^k] + 3*a*f^3*k*x^(3/2)*Log[x] + 2*b*f^3*k*n*x^(3/2)*Log[x] - 6*b*f^3*k
*n*x^(3/2)*Log[1 + (f*Sqrt[x])/e]*Log[x] - (3*b*f^3*k*n*x^(3/2)*Log[x]^2)/2 - 3*b*e^2*f*k*Sqrt[x]*Log[c*x^n] +
6*b*e*f^2*k*x*Log[c*x^n] - 6*b*e^3*Log[d*(e + f*Sqrt[x])^k]*Log[c*x^n] + 3*b*f^3*k*x^(3/2)*Log[x]*Log[c*x^n]
- 2*f^3*k*x^(3/2)*Log[e + f*Sqrt[x]]*(3*a + 2*b*n - 3*b*n*Log[x] + 3*b*Log[c*x^n]) - 12*b*f^3*k*n*x^(3/2)*Poly
Log[2, -((f*Sqrt[x])/e)])/(9*e^3*x^(3/2))

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{(a+b\ln \left ( c{x}^{n} \right ) )\ln \left ( d \left ( e+f\sqrt{x} \right ) ^{k} \right ){x}^{-{\frac{5}{2}}}}\, 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^(5/2),x)

[Out]

int((a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^(5/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{5 \, b f k n}{x} - \frac{3 \, b f k \log \left (c\right )}{x} - \frac{3 \, b f k \log \left (x^{n}\right )}{x} - \frac{3 \, a f k}{x}}{9 \, e} + \frac{2 \, b f^{3} k n \log \left (x\right ) + 3 \, b f^{3} k \log \left (c\right ) \log \left (x\right ) + 3 \, a f^{3} k \log \left (x\right ) + \frac{3 \, b f^{3} k \log \left (x^{n}\right )^{2}}{2 \, n}}{9 \, e^{3}} - \frac{\frac{2 \,{\left (b f^{6} k x^{2} \log \left (x^{n}\right ) +{\left (b f^{6} k \log \left (c\right ) + a f^{6} k\right )} x^{2}\right )}}{\sqrt{x}} - \frac{3 \, b e f^{5} k x^{2} \log \left (x^{n}\right ) +{\left (3 \, a e f^{5} k -{\left (e f^{5} k n - 3 \, e f^{5} k \log \left (c\right )\right )} b\right )} x^{2}}{x} + \frac{2 \,{\left (3 \, b e^{2} f^{4} k x^{2} \log \left (x^{n}\right ) +{\left (3 \, a e^{2} f^{4} k -{\left (4 \, e^{2} f^{4} k n - 3 \, e^{2} f^{4} k \log \left (c\right )\right )} b\right )} x^{2}\right )}}{x^{\frac{3}{2}}} + \frac{2 \,{\left (3 \, b e^{6} x \log \left (x^{n}\right ) +{\left (3 \, a e^{6} +{\left (2 \, e^{6} n + 3 \, e^{6} \log \left (c\right )\right )} b\right )} x\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k}\right )}{x^{\frac{5}{2}}} - \frac{2 \,{\left ({\left (3 \, a e^{4} f^{2} k +{\left (8 \, e^{4} f^{2} k n + 3 \, e^{4} f^{2} k \log \left (c\right )\right )} b\right )} x^{2} -{\left (3 \, a e^{6} \log \left (d\right ) +{\left (2 \, e^{6} n \log \left (d\right ) + 3 \, e^{6} \log \left (c\right ) \log \left (d\right )\right )} b\right )} x + 3 \,{\left (b e^{4} f^{2} k x^{2} - b e^{6} x \log \left (d\right )\right )} \log \left (x^{n}\right )\right )}}{x^{\frac{5}{2}}}}{9 \, e^{6}} + \int \frac{3 \, b f^{7} k x \log \left (x^{n}\right ) +{\left (3 \, a f^{7} k +{\left (2 \, f^{7} k n + 3 \, f^{7} k \log \left (c\right )\right )} b\right )} x}{9 \,{\left (e^{6} f \sqrt{x} + e^{7}\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^(5/2),x, algorithm="maxima")

[Out]

1/9*integrate((3*b*f*k*x*log(x^n) + (3*a*f*k + (2*f*k*n + 3*f*k*log(c))*b)*x)/x^3, x)/e + 1/9*integrate((3*b*f
^3*k*x*log(x^n) + (3*a*f^3*k + (2*f^3*k*n + 3*f^3*k*log(c))*b)*x)/x^2, x)/e^3 - 1/9*(2*(b*f^6*k*x^2*log(x^n) +
(b*f^6*k*log(c) + a*f^6*k)*x^2)/sqrt(x) - (3*b*e*f^5*k*x^2*log(x^n) + (3*a*e*f^5*k - (e*f^5*k*n - 3*e*f^5*k*l
og(c))*b)*x^2)/x + 2*(3*b*e^2*f^4*k*x^2*log(x^n) + (3*a*e^2*f^4*k - (4*e^2*f^4*k*n - 3*e^2*f^4*k*log(c))*b)*x^
2)/x^(3/2) + 2*(3*b*e^6*x*log(x^n) + (3*a*e^6 + (2*e^6*n + 3*e^6*log(c))*b)*x)*log((f*sqrt(x) + e)^k)/x^(5/2)
- 2*((3*a*e^4*f^2*k + (8*e^4*f^2*k*n + 3*e^4*f^2*k*log(c))*b)*x^2 - (3*a*e^6*log(d) + (2*e^6*n*log(d) + 3*e^6*
log(c)*log(d))*b)*x + 3*(b*e^4*f^2*k*x^2 - b*e^6*x*log(d))*log(x^n))/x^(5/2))/e^6 + integrate(1/9*(3*b*f^7*k*x
*log(x^n) + (3*a*f^7*k + (2*f^7*k*n + 3*f^7*k*log(c))*b)*x)/(e^6*f*sqrt(x) + e^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b \sqrt{x} \log \left (c x^{n}\right ) + a \sqrt{x}\right )} \log \left ({\left (f \sqrt{x} + e\right )}^{k} d\right )}{x^{3}}, 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^(5/2),x, algorithm="fricas")

[Out]

integral((b*sqrt(x)*log(c*x^n) + a*sqrt(x))*log((f*sqrt(x) + e)^k*d)/x^3, 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))*ln(d*(e+f*x**(1/2))**k)/x**(5/2),x)

[Out]

Timed out

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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^{\frac{5}{2}}}\,{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^(5/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + e)^k*d)/x^(5/2), x)