3.52 \(\int \frac {\log (d (\frac {1}{d}+f \sqrt {x})) (a+b \log (c x^n))}{x^4} \, dx\)

Optimal. Leaf size=372 \[ \frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right )+\frac {1}{12} b d^6 f^6 n \log ^2(x)+\frac {1}{9} b d^6 f^6 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{18} b d^6 f^6 n \log (x)-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {2 b d^4 f^4 n}{9 x}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {11 b d f n}{225 x^{5/2}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 x^3} \]

[Out]

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

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Rubi [A]  time = 0.25, antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2454, 2395, 44, 2376, 2391, 2301} \[ \frac {2}{3} b d^6 f^6 n \text {PolyLog}\left (2,-d f \sqrt {x}\right )+\frac {1}{3} d^6 f^6 \log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {1}{6} d^6 f^6 \log (x) \left (a+b \log \left (c x^n\right )\right )-\frac {d^5 f^5 \left (a+b \log \left (c x^n\right )\right )}{3 \sqrt {x}}+\frac {d^4 f^4 \left (a+b \log \left (c x^n\right )\right )}{6 x}-\frac {d^3 f^3 \left (a+b \log \left (c x^n\right )\right )}{9 x^{3/2}}+\frac {d^2 f^2 \left (a+b \log \left (c x^n\right )\right )}{12 x^2}-\frac {d f \left (a+b \log \left (c x^n\right )\right )}{15 x^{5/2}}-\frac {\log \left (d f \sqrt {x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{3 x^3}-\frac {b d^3 f^3 n}{9 x^{3/2}}+\frac {5 b d^2 f^2 n}{72 x^2}-\frac {7 b d^5 f^5 n}{9 \sqrt {x}}+\frac {2 b d^4 f^4 n}{9 x}+\frac {1}{12} b d^6 f^6 n \log ^2(x)+\frac {1}{9} b d^6 f^6 n \log \left (d f \sqrt {x}+1\right )-\frac {1}{18} b d^6 f^6 n \log (x)-\frac {11 b d f n}{225 x^{5/2}}-\frac {b n \log \left (d f \sqrt {x}+1\right )}{9 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

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

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 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]

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 2391

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

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 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])

Rubi steps

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

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Mathematica [A]  time = 0.36, size = 288, normalized size = 0.77 \[ \frac {\left (d^6 f^6 x^3-1\right ) \log \left (d f \sqrt {x}+1\right ) \left (3 a+3 b \log \left (c x^n\right )+b n\right )}{9 x^3}-\frac {d f \left (100 d^5 f^5 x^{5/2} \log (x) \left (3 a+3 b \log \left (c x^n\right )+b n\right )+600 a d^4 f^4 x^2-300 a d^3 f^3 x^{3/2}+200 a d^2 f^2 x-150 a d f \sqrt {x}+120 a+10 b \left (60 d^4 f^4 x^2-30 d^3 f^3 x^{3/2}+20 d^2 f^2 x-15 d f \sqrt {x}+12\right ) \log \left (c x^n\right )-150 b d^5 f^5 n x^{5/2} \log ^2(x)+1400 b d^4 f^4 n x^2-400 b d^3 f^3 n x^{3/2}+200 b d^2 f^2 n x-125 b d f n \sqrt {x}+88 b n\right )}{1800 x^{5/2}}+\frac {2}{3} b d^6 f^6 n \text {Li}_2\left (-d f \sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(Log[d*(d^(-1) + f*Sqrt[x])]*(a + b*Log[c*x^n]))/x^4,x]

[Out]

((-1 + d^6*f^6*x^3)*Log[1 + d*f*Sqrt[x]]*(3*a + b*n + 3*b*Log[c*x^n]))/(9*x^3) - (d*f*(120*a + 88*b*n - 150*a*
d*f*Sqrt[x] - 125*b*d*f*n*Sqrt[x] + 200*a*d^2*f^2*x + 200*b*d^2*f^2*n*x - 300*a*d^3*f^3*x^(3/2) - 400*b*d^3*f^
3*n*x^(3/2) + 600*a*d^4*f^4*x^2 + 1400*b*d^4*f^4*n*x^2 - 150*b*d^5*f^5*n*x^(5/2)*Log[x]^2 + 10*b*(12 - 15*d*f*
Sqrt[x] + 20*d^2*f^2*x - 30*d^3*f^3*x^(3/2) + 60*d^4*f^4*x^2)*Log[c*x^n] + 100*d^5*f^5*x^(5/2)*Log[x]*(3*a + b
*n + 3*b*Log[c*x^n])))/(1800*x^(5/2)) + (2*b*d^6*f^6*n*PolyLog[2, -(d*f*Sqrt[x])])/3

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fricas [F]  time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left (d f \sqrt {x} + 1\right )}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*log(c*x^n) + a)*log(d*f*sqrt(x) + 1)/x^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^4, x)

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maple [F]  time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (\left (f \sqrt {x}+\frac {1}{d}\right ) d \right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f \sqrt {x} + \frac {1}{d}\right )} d\right )}{x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*x^n) + a)*log((f*sqrt(x) + 1/d)*d)/x^4, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\ln \left (d\,\left (f\,\sqrt {x}+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))*ln(d*(1/d+f*x**(1/2)))/x**4,x)

[Out]

Timed out

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