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

Optimal. Leaf size=274 \[ \frac{b f^3 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{f^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}-\frac{b n \log \left (d (e+f x)^m\right )}{9 x^3}+\frac{4 b f^2 m n}{9 e^2 x}-\frac{b f^3 m n \log ^2(x)}{6 e^3}+\frac{b f^3 m n \log (x)}{9 e^3}-\frac{b f^3 m n \log (e+f x)}{9 e^3}+\frac{b f^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac{5 b f m n}{36 e x^2} \]

[Out]

(-5*b*f*m*n)/(36*e*x^2) + (4*b*f^2*m*n)/(9*e^2*x) + (b*f^3*m*n*Log[x])/(9*e^3) - (b*f^3*m*n*Log[x]^2)/(6*e^3)
- (f*m*(a + b*Log[c*x^n]))/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n]))/(3*e^2*x) + (f^3*m*Log[x]*(a + b*Log[c*x^n])
)/(3*e^3) - (b*f^3*m*n*Log[e + f*x])/(9*e^3) + (b*f^3*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(3*e^3) - (f^3*m*(a +
b*Log[c*x^n])*Log[e + f*x])/(3*e^3) - (b*n*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m
])/(3*x^3) + (b*f^3*m*n*PolyLog[2, 1 + (f*x)/e])/(3*e^3)

________________________________________________________________________________________

Rubi [A]  time = 0.183956, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2395, 44, 2376, 2301, 2394, 2315} \[ \frac{b f^3 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{f^3 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}-\frac{b n \log \left (d (e+f x)^m\right )}{9 x^3}+\frac{4 b f^2 m n}{9 e^2 x}-\frac{b f^3 m n \log ^2(x)}{6 e^3}+\frac{b f^3 m n \log (x)}{9 e^3}-\frac{b f^3 m n \log (e+f x)}{9 e^3}+\frac{b f^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac{5 b f m n}{36 e x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b*f*m*n)/(36*e*x^2) + (4*b*f^2*m*n)/(9*e^2*x) + (b*f^3*m*n*Log[x])/(9*e^3) - (b*f^3*m*n*Log[x]^2)/(6*e^3)
- (f*m*(a + b*Log[c*x^n]))/(6*e*x^2) + (f^2*m*(a + b*Log[c*x^n]))/(3*e^2*x) + (f^3*m*Log[x]*(a + b*Log[c*x^n])
)/(3*e^3) - (b*f^3*m*n*Log[e + f*x])/(9*e^3) + (b*f^3*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(3*e^3) - (f^3*m*(a +
b*Log[c*x^n])*Log[e + f*x])/(3*e^3) - (b*n*Log[d*(e + f*x)^m])/(9*x^3) - ((a + b*Log[c*x^n])*Log[d*(e + f*x)^m
])/(3*x^3) + (b*f^3*m*n*PolyLog[2, 1 + (f*x)/e])/(3*e^3)

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 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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{x^4} \, dx &=-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}-(b n) \int \left (-\frac{f m}{6 e x^3}+\frac{f^2 m}{3 e^2 x^2}+\frac{f^3 m \log (x)}{3 e^3 x}-\frac{f^3 m \log (e+f x)}{3 e^3 x}-\frac{\log \left (d (e+f x)^m\right )}{3 x^4}\right ) \, dx\\ &=-\frac{b f m n}{12 e x^2}+\frac{b f^2 m n}{3 e^2 x}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{1}{3} (b n) \int \frac{\log \left (d (e+f x)^m\right )}{x^4} \, dx-\frac{\left (b f^3 m n\right ) \int \frac{\log (x)}{x} \, dx}{3 e^3}+\frac{\left (b f^3 m n\right ) \int \frac{\log (e+f x)}{x} \, dx}{3 e^3}\\ &=-\frac{b f m n}{12 e x^2}+\frac{b f^2 m n}{3 e^2 x}-\frac{b f^3 m n \log ^2(x)}{6 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{b f^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac{b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{1}{9} (b f m n) \int \frac{1}{x^3 (e+f x)} \, dx-\frac{\left (b f^4 m n\right ) \int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx}{3 e^3}\\ &=-\frac{b f m n}{12 e x^2}+\frac{b f^2 m n}{3 e^2 x}-\frac{b f^3 m n \log ^2(x)}{6 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}+\frac{b f^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac{b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{b f^3 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{3 e^3}+\frac{1}{9} (b f m n) \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\\ &=-\frac{5 b f m n}{36 e x^2}+\frac{4 b f^2 m n}{9 e^2 x}+\frac{b f^3 m n \log (x)}{9 e^3}-\frac{b f^3 m n \log ^2(x)}{6 e^3}-\frac{f m \left (a+b \log \left (c x^n\right )\right )}{6 e x^2}+\frac{f^2 m \left (a+b \log \left (c x^n\right )\right )}{3 e^2 x}+\frac{f^3 m \log (x) \left (a+b \log \left (c x^n\right )\right )}{3 e^3}-\frac{b f^3 m n \log (e+f x)}{9 e^3}+\frac{b f^3 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{3 e^3}-\frac{f^3 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{3 e^3}-\frac{b n \log \left (d (e+f x)^m\right )}{9 x^3}-\frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{3 x^3}+\frac{b f^3 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{3 e^3}\\ \end{align*}

Mathematica [A]  time = 0.175692, size = 280, normalized size = 1.02 \[ -\frac{12 b f^3 m n x^3 \text{PolyLog}\left (2,-\frac{f x}{e}\right )-4 f^3 m x^3 \log (x) \left (3 a+3 b \log \left (c x^n\right )+3 b n \log (e+f x)-3 b n \log \left (\frac{f x}{e}+1\right )+b n\right )+12 a e^3 \log \left (d (e+f x)^m\right )+6 a e^2 f m x-12 a e f^2 m x^2+12 a f^3 m x^3 \log (e+f x)+12 b e^3 \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+6 b e^2 f m x \log \left (c x^n\right )-12 b e f^2 m x^2 \log \left (c x^n\right )+12 b f^3 m x^3 \log \left (c x^n\right ) \log (e+f x)+4 b e^3 n \log \left (d (e+f x)^m\right )+5 b e^2 f m n x-16 b e f^2 m n x^2+4 b f^3 m n x^3 \log (e+f x)+6 b f^3 m n x^3 \log ^2(x)}{36 e^3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*a*e^2*f*m*x + 5*b*e^2*f*m*n*x - 12*a*e*f^2*m*x^2 - 16*b*e*f^2*m*n*x^2 + 6*b*f^3*m*n*x^3*Log[x]^2 + 6*b*e^2
*f*m*x*Log[c*x^n] - 12*b*e*f^2*m*x^2*Log[c*x^n] + 12*a*f^3*m*x^3*Log[e + f*x] + 4*b*f^3*m*n*x^3*Log[e + f*x] +
12*b*f^3*m*x^3*Log[c*x^n]*Log[e + f*x] + 12*a*e^3*Log[d*(e + f*x)^m] + 4*b*e^3*n*Log[d*(e + f*x)^m] + 12*b*e^
3*Log[c*x^n]*Log[d*(e + f*x)^m] - 4*f^3*m*x^3*Log[x]*(3*a + b*n + 3*b*Log[c*x^n] + 3*b*n*Log[e + f*x] - 3*b*n*
Log[1 + (f*x)/e]) + 12*b*f^3*m*n*x^3*PolyLog[2, -((f*x)/e)])/(36*e^3*x^3)

________________________________________________________________________________________

Maple [C]  time = 0.384, size = 2282, normalized size = 8.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*f^3*m*n*ln(-f*x/e)*ln(f*x+e)/e^3+1/9*b*f^3*m*n*ln(x)/e^3-1/6*b*f^3*m*n*ln(x)^2/e^3-1/9*b*f^3*m*n*ln(f*x+
e)/e^3-5/36*b*f*m*n/e/x^2+4/9*b*f^2*m*n/e^2/x-1/6*I/x^3*Pi*a*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/6*I*Pi*
csgn(I*d*(f*x+e)^m)^3*b/x^3*ln(x^n)-1/6*I/e^3*f^3*m*ln(f*x+e)*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I/e^2*f^2*m
/x*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/12*I/e*f*m/x^2*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2-1/12*I/e*f*m/x^2*b*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2+1/12*I/e*f*m/x^2*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I/e^3*f^3*m*ln(x)*b*Pi*c
sgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3/x^3*b*csgn(I*c)*csgn(I*c*x^n)^2-1/12*Pi^2*c
sgn(I*d*(f*x+e)^m)^3/x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/6*I/x^3*ln(c)*Pi*b*csgn(I*d*(f*x+e)^m)^3+1/6*I/x^3*Pi
*ln(d)*b*csgn(I*c*x^n)^3+1/18*I/x^3*Pi*b*n*csgn(I*d*(f*x+e)^m)^3-1/6*I/x^3*Pi*a*csgn(I*d)*csgn(I*d*(f*x+e)^m)^
2+1/6*I/e^3*f^3*m*ln(f*x+e)*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/6*I/e^2*f^2*m/x*b*Pi*csgn(I*c)*csgn(I*x
^n)*csgn(I*c*x^n)+1/6*I/e^3*f^3*m*ln(x)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/6*I/e^3*f^3*m*ln(x)*b*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2-1/6*I/e^3*f^3*m*ln(f*x+e)*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)
*csgn(I*d*(f*x+e)^m)/x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/6*I/e^2*f^2*m/x*b*Pi*csgn(I*c)*csgn(I*c*x^n)^
2+(-1/3*b/x^3*ln(x^n)-1/18*(-3*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+3*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+3
*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-3*I*b*Pi*csgn(I*c*x^n)^3+6*b*ln(c)+2*b*n+6*a)/x^3)*ln((f*x+e)^m)+1/12*Pi^2
*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/12*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e
)^m)^2/x^3*b*csgn(I*c)*csgn(I*c*x^n)^2+1/12*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(I*x^n)*csg
n(I*c*x^n)^2+1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3/x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/12*Pi^2*csgn(I*d)*csg
n(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)/x^3*b*csgn(I*c*x^n)^3+1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(
I*c)*csgn(I*c*x^n)^2-1/6*I/x^3*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2-1/6*I/x^3*Pi*ln(d)*b*csgn(I*x^n)*csgn(I*c*
x^n)^2-1/18*I/x^3*Pi*b*n*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/18*I/x^3*Pi*b*n*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^
m)^2+1/6*I/x^3*Pi*a*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/6*I/x^3*ln(c)*Pi*b*csgn(I*d)*csgn(I*d*(f
*x+e)^m)^2-1/6*I*Pi*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*b/x^3*ln(x^n)-1/6*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^
m)^2*b/x^3*ln(x^n)-1/6*I/x^3*ln(c)*Pi*b*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/6/e*f*m/x^2*b*ln(c)+1/3/e^2*
f^2*m/x*b*ln(c)-1/3/e^3*f^3*m*ln(f*x+e)*b*ln(c)+1/3/e^3*f^3*m*ln(x)*b*ln(c)+1/12*Pi^2*csgn(I*d*(f*x+e)^m)^3/x^
3*b*csgn(I*c*x^n)^3+1/6*I/x^3*Pi*a*csgn(I*d*(f*x+e)^m)^3+1/6*I/x^3*Pi*ln(d)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x
^n)-1/6*I/e^3*f^3*m*ln(x)*b*Pi*csgn(I*c*x^n)^3+1/6*I/e^3*f^3*m*ln(f*x+e)*b*Pi*csgn(I*c*x^n)^3+1/6*I/x^3*ln(c)*
Pi*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/3*m*f^3*b*ln(x^n)/e^3*ln(x)+1/3*m*f^2*b*ln(x^n)/e^2/x-1
/3*m*f^3*b*ln(x^n)/e^3*ln(f*x+e)-1/6*m*f*b*ln(x^n)/e/x^2-1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(
I*c*x^n)^3-1/12*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(I*c*x^n)^3+1/3/e^2*f^2*m/x*a-1/6/e*f*m
/x^2*a+1/3/e^3*f^3*m*ln(x)*a-1/3/e^3*f^3*m*ln(f*x+e)*a-1/3/x^3*ln(c)*ln(d)*b-1/9/x^3*ln(d)*b*n+1/3*n*f^3*b*m/e
^3*dilog(-f*x/e)-1/3/x^3*ln(d)*a-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)/x^3*b*csgn(I*c)*csg
n(I*c*x^n)^2+1/6*I*Pi*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*b/x^3*ln(x^n)-1/6*I/e^2*f^2*m/x*b*Pi*csg
n(I*c*x^n)^3+1/12*I/e*f*m/x^2*b*Pi*csgn(I*c*x^n)^3+1/18*I/x^3*Pi*b*n*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x
+e)^m)-1/12*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/12*Pi^2*csgn(I*(f
*x+e)^m)*csgn(I*d*(f*x+e)^m)^2/x^3*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/12*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)
*csgn(I*d*(f*x+e)^m)/x^3*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/3*ln(d)*b/x^3*ln(x^n)

________________________________________________________________________________________

Maxima [A]  time = 1.66343, size = 462, normalized size = 1.69 \begin{align*} -\frac{{\left (\log \left (\frac{f x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{f x}{e}\right )\right )} b f^{3} m n}{3 \, e^{3}} - \frac{{\left (3 \, a f^{3} m +{\left (f^{3} m n + 3 \, f^{3} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{9 \, e^{3}} + \frac{12 \, b f^{3} m n x^{3} \log \left (f x + e\right ) \log \left (x\right ) - 6 \, b f^{3} m n x^{3} \log \left (x\right )^{2} - 12 \, a e^{3} \log \left (d\right ) + 4 \,{\left (3 \, a f^{3} m +{\left (f^{3} m n + 3 \, f^{3} m \log \left (c\right )\right )} b\right )} x^{3} \log \left (x\right ) + 4 \,{\left (3 \, a e f^{2} m +{\left (4 \, e f^{2} m n + 3 \, e f^{2} m \log \left (c\right )\right )} b\right )} x^{2} - 4 \,{\left (e^{3} n \log \left (d\right ) + 3 \, e^{3} \log \left (c\right ) \log \left (d\right )\right )} b -{\left (6 \, a e^{2} f m +{\left (5 \, e^{2} f m n + 6 \, e^{2} f m \log \left (c\right )\right )} b\right )} x - 4 \,{\left (3 \, b e^{3} \log \left (x^{n}\right ) + 3 \, a e^{3} +{\left (e^{3} n + 3 \, e^{3} \log \left (c\right )\right )} b\right )} \log \left ({\left (f x + e\right )}^{m}\right ) - 6 \,{\left (2 \, b f^{3} m x^{3} \log \left (f x + e\right ) - 2 \, b f^{3} m x^{3} \log \left (x\right ) - 2 \, b e f^{2} m x^{2} + b e^{2} f m x + 2 \, b e^{3} \log \left (d\right )\right )} \log \left (x^{n}\right )}{36 \, e^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*f^3*m*n/e^3 - 1/9*(3*a*f^3*m + (f^3*m*n + 3*f^3*m*log(c))*b)*lo
g(f*x + e)/e^3 + 1/36*(12*b*f^3*m*n*x^3*log(f*x + e)*log(x) - 6*b*f^3*m*n*x^3*log(x)^2 - 12*a*e^3*log(d) + 4*(
3*a*f^3*m + (f^3*m*n + 3*f^3*m*log(c))*b)*x^3*log(x) + 4*(3*a*e*f^2*m + (4*e*f^2*m*n + 3*e*f^2*m*log(c))*b)*x^
2 - 4*(e^3*n*log(d) + 3*e^3*log(c)*log(d))*b - (6*a*e^2*f*m + (5*e^2*f*m*n + 6*e^2*f*m*log(c))*b)*x - 4*(3*b*e
^3*log(x^n) + 3*a*e^3 + (e^3*n + 3*e^3*log(c))*b)*log((f*x + e)^m) - 6*(2*b*f^3*m*x^3*log(f*x + e) - 2*b*f^3*m
*x^3*log(x) - 2*b*e*f^2*m*x^2 + b*e^2*f*m*x + 2*b*e^3*log(d))*log(x^n))/(e^3*x^3)

________________________________________________________________________________________

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 x + e\right )}^{m} 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*(f*x+e)^m)/x^4,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)*log((f*x + e)^m*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*(f*x+e)**m)/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 x + e\right )}^{m} 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*(f*x+e)^m)/x^4,x, algorithm="giac")

[Out]

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