3.2.21 \(\int \frac {\log (d (e+f \sqrt {x})^k) (a+b \log (c x^n))}{x^4} \, dx\) [121]

Optimal. Leaf size=434 \[ -\frac {11 b f k n}{225 e x^{5/2}}+\frac {5 b f^2 k n}{72 e^2 x^2}-\frac {b f^3 k n}{9 e^3 x^{3/2}}+\frac {2 b f^4 k n}{9 e^4 x}-\frac {7 b f^5 k n}{9 e^5 \sqrt {x}}+\frac {b f^6 k n \log \left (e+f \sqrt {x}\right )}{9 e^6}-\frac {b n \log \left (d \left (e+f \sqrt {x}\right )^k\right )}{9 x^3}-\frac {2 b f^6 k n \log \left (e+f \sqrt {x}\right ) \log \left (-\frac {f \sqrt {x}}{e}\right )}{3 e^6}-\frac {b f^6 k n \log (x)}{18 e^6}+\frac {b f^6 k n \log ^2(x)}{12 e^6}-\frac {f k \left (a+b \log \left (c x^n\right )\right )}{15 e x^{5/2}}+\frac {f^2 k \left (a+b \log \left (c x^n\right )\right )}{12 e^2 x^2}-\frac {f^3 k \left (a+b \log \left (c x^n\right )\right )}{9 e^3 x^{3/2}}+\frac {f^4 k \left (a+b \log \left (c x^n\right )\right )}{6 e^4 x}-\frac {f^5 k \left (a+b \log \left (c x^n\right )\right )}{3 e^5 \sqrt {x}}+\frac {f^6 k \log \left (e+f \sqrt {x}\right ) \left (a+b \log \left (c x^n\right )\right )}{3 e^6}-\frac {\log \left (d \left (e+f \sqrt {x}\right )^k\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {f^6 k \log (x) \left (a+b \log \left (c x^n\right )\right )}{6 e^6}-\frac {2 b f^6 k n \text {Li}_2\left (1+\frac {f \sqrt {x}}{e}\right )}{3 e^6} \]

[Out]

-11/225*b*f*k*n/e/x^(5/2)+5/72*b*f^2*k*n/e^2/x^2-1/9*b*f^3*k*n/e^3/x^(3/2)+2/9*b*f^4*k*n/e^4/x-1/18*b*f^6*k*n*
ln(x)/e^6+1/12*b*f^6*k*n*ln(x)^2/e^6-1/15*f*k*(a+b*ln(c*x^n))/e/x^(5/2)+1/12*f^2*k*(a+b*ln(c*x^n))/e^2/x^2-1/9
*f^3*k*(a+b*ln(c*x^n))/e^3/x^(3/2)+1/6*f^4*k*(a+b*ln(c*x^n))/e^4/x-1/6*f^6*k*ln(x)*(a+b*ln(c*x^n))/e^6+1/9*b*f
^6*k*n*ln(e+f*x^(1/2))/e^6+1/3*f^6*k*(a+b*ln(c*x^n))*ln(e+f*x^(1/2))/e^6-2/3*b*f^6*k*n*ln(-f*x^(1/2)/e)*ln(e+f
*x^(1/2))/e^6-1/9*b*n*ln(d*(e+f*x^(1/2))^k)/x^3-1/3*(a+b*ln(c*x^n))*ln(d*(e+f*x^(1/2))^k)/x^3-2/3*b*f^6*k*n*po
lylog(2,1+f*x^(1/2)/e)/e^6-7/9*b*f^5*k*n/e^5/x^(1/2)-1/3*f^5*k*(a+b*ln(c*x^n))/e^5/x^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 434, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2442, 46, 2423, 2441, 2352, 2338} \begin {gather*} -\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^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 {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^3 k n}{9 e^3 x^{3/2}}+\frac {5 b f^2 k n}{72 e^2 x^2}-\frac {11 b f k n}{225 e x^{5/2}} \end {gather*}

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 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^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) \text {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 ) \text {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) \text {Subst}\left (\int \frac {1}{x^6 (e+f x)} \, dx,x,\sqrt {x}\right )+\frac {\left (2 b f^7 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^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) \text {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*}

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Mathematica [A]
time = 0.34, size = 457, normalized size = 1.05 \begin {gather*} -\frac {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}+1400 b e f^5 k n x^{5/2}+600 a e^6 \log \left (d \left (e+f \sqrt {x}\right )^k\right )+200 b e^6 n \log \left (d \left (e+f \sqrt {x}\right )^k\right )+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 \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)-150 b f^6 k n x^3 \log ^2(x)+120 b e^5 f k \sqrt {x} \log \left (c x^n\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 )+600 b e f^5 k x^{5/2} \log \left (c x^n\right )+600 b e^6 \log \left (d \left (e+f \sqrt {x}\right )^k\right ) \log \left (c x^n\right )+300 b f^6 k x^3 \log (x) \log \left (c x^n\right )-200 f^6 k x^3 \log \left (e+f \sqrt {x}\right ) \left (3 a+b n-3 b n \log (x)+3 b \log \left (c x^n\right )\right )-1200 b f^6 k n x^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{1800 e^6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1800*(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
) + 1400*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)*Lo
g[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)])/(e^6*x^3)

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \ln \left (d \left (e +f \sqrt {x}\right )^{k}\right )}{x^{4}}\, dx\]

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^4,x)

[Out]

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

[Out]

-1/225*(75*b*e*log(d)*log(x^n) + 25*(3*b*e*log(x^n) + (b*(n + 3*log(c)) + 3*a)*e)*k*log(f*sqrt(x) + e) + 25*((
n*log(d) + 3*log(c)*log(d))*b + 3*a*log(d))*e + (15*b*f*k*x*log(x^n) + (15*a*f*k + (11*f*k*n + 15*f*k*log(c))*
b)*x)/sqrt(x))*e^(-1)/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)/(f*
x^3*e^(1/2*log(x) + 1) + x^3*e^2), 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^4,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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**4,x)

[Out]

Timed out

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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^4,x, algorithm="giac")

[Out]

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

[Out]

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

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