3.2.36 \(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^{5/2}} \, dx\) [136]

Optimal. Leaf size=310 \[ -\frac {5 b f k n}{9 e x}+\frac {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right )}{9 e^3}-\frac {4 b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^{3/2}}+\frac {4 b f^3 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^3}+\frac {2 b f^3 k n \log (x)}{9 e^3}-\frac {b f^3 k n \log ^2(x)}{6 e^3}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{3 e x}+\frac {2 f^2 k \left (a+b \log \left (c x^n\right )\right )}{3 e^2 \sqrt {x}}-\frac {2 f^3 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac {2 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^{3/2}}+\frac {f^3 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac {4 b f^3 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 e^3} \]

[Out]

-5/9*b*f*k*n/e/x+2/9*b*f^3*k*n*ln(x)/e^3-1/6*b*f^3*k*n*ln(x)^2/e^3-1/3*f*k*(a+b*ln(c*x^n))/e/x+1/3*f^3*k*ln(x)
*(a+b*ln(c*x^n))/e^3-4/9*b*f^3*k*n*ln(e+f*x^(1/2))/e^3-2/3*f^3*k*(a+b*ln(c*x^n))*ln(e+f*x^(1/2))/e^3+4/3*b*f^3
*k*n*ln(-f*x^(1/2)/e)*ln(e+f*x^(1/2))/e^3-4/9*b*n*ln(d*(e+f*x^(1/2))^k)/x^(3/2)-2/3*(a+b*ln(c*x^n))*ln(d*(e+f*
x^(1/2))^k)/x^(3/2)+4/3*b*f^3*k*n*polylog(2,1+f*x^(1/2)/e)/e^3+16/9*b*f^2*k*n/e^2/x^(1/2)+2/3*f^2*k*(a+b*ln(c*
x^n))/e^2/x^(1/2)

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Rubi [A]
time = 0.16, antiderivative size = 310, normalized size of antiderivative = 1.00, 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 = {2504, 2442, 46, 2423, 2441, 2352, 2338} \begin {gather*} \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 {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 {16 b f^2 k n}{9 e^2 \sqrt {x}}-\frac {5 b f k n}{9 e x} \end {gather*}

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 46

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] && Lt
Q[m + n + 2, 0])

Rule 2338

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]

Rule 2352

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

Rule 2423

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 2441

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 2442

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 2504

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])

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) \text {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 ) \text {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) \text {Subst}\left (\int \frac {1}{x^3 (e+f x)} \, dx,x,\sqrt {x}\right )-\frac {\left (4 b f^4 k n\right ) \text {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) \text {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*}

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Mathematica [A]
time = 0.25, size = 326, normalized size = 1.05 \begin {gather*} \frac {-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 \left (d \left (e+f \sqrt {x}\right )^k\right )-4 b e^3 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+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 \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-\frac {3}{2} b f^3 k n x^{3/2} \log ^2(x)-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 )-6 b e^3 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+3 b f^3 k x^{3/2} \log (x) \log \left (c x^n\right )-2 f^3 k x^{3/2} \log \left (e+f \sqrt {x}\right ) \left (3 a+2 b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )-12 b f^3 k n x^{3/2} \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{9 e^3 x^{3/2}} \end {gather*}

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))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )}{x^{\frac {5}{2}}}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 8010 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (d\,{\left (e+f\,\sqrt {x}\right )}^k\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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