3.366 \(\int \frac{\log (f x^m) (a+b \log (c (d+e x)^n))}{x^5} \, dx\)
Optimal. Leaf size=230 \[ -\frac{b e^4 m n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{4 d^4}-\frac{1}{16} \left (\frac{4 \log \left (f x^m\right )}{x^4}+\frac{m}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e^4 n \log \left (\frac{d}{e x}+1\right ) \log \left (f x^m\right )}{4 d^4}-\frac{b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac{b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}+\frac{3 b e^2 m n}{32 d^2 x^2}-\frac{5 b e^3 m n}{16 d^3 x}-\frac{b e^4 m n \log (x)}{16 d^4}+\frac{b e^4 m n \log (d+e x)}{16 d^4}-\frac{b e n \log \left (f x^m\right )}{12 d x^3}-\frac{7 b e m n}{144 d x^3} \]
[Out]
(-7*b*e*m*n)/(144*d*x^3) + (3*b*e^2*m*n)/(32*d^2*x^2) - (5*b*e^3*m*n)/(16*d^3*x) - (b*e^4*m*n*Log[x])/(16*d^4)
- (b*e*n*Log[f*x^m])/(12*d*x^3) + (b*e^2*n*Log[f*x^m])/(8*d^2*x^2) - (b*e^3*n*Log[f*x^m])/(4*d^3*x) + (b*e^4*
n*Log[1 + d/(e*x)]*Log[f*x^m])/(4*d^4) + (b*e^4*m*n*Log[d + e*x])/(16*d^4) - ((m/x^4 + (4*Log[f*x^m])/x^4)*(a
+ b*Log[c*(d + e*x)^n]))/16 - (b*e^4*m*n*PolyLog[2, -(d/(e*x))])/(4*d^4)
________________________________________________________________________________________
Rubi [A] time = 0.218946, antiderivative size = 249, normalized size of antiderivative =
1.08, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules
used = {2426, 44, 2351, 2304, 2301, 2317, 2391} \[ \frac{b e^4 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{4 d^4}-\frac{1}{16} \left (\frac{4 \log \left (f x^m\right )}{x^4}+\frac{m}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e^4 n \log ^2\left (f x^m\right )}{8 d^4 m}+\frac{b e^4 n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{4 d^4}-\frac{b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac{b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}+\frac{3 b e^2 m n}{32 d^2 x^2}-\frac{5 b e^3 m n}{16 d^3 x}-\frac{b e^4 m n \log (x)}{16 d^4}+\frac{b e^4 m n \log (d+e x)}{16 d^4}-\frac{b e n \log \left (f x^m\right )}{12 d x^3}-\frac{7 b e m n}{144 d x^3} \]
Antiderivative was successfully verified.
[In]
Int[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/x^5,x]
[Out]
(-7*b*e*m*n)/(144*d*x^3) + (3*b*e^2*m*n)/(32*d^2*x^2) - (5*b*e^3*m*n)/(16*d^3*x) - (b*e^4*m*n*Log[x])/(16*d^4)
- (b*e*n*Log[f*x^m])/(12*d*x^3) + (b*e^2*n*Log[f*x^m])/(8*d^2*x^2) - (b*e^3*n*Log[f*x^m])/(4*d^3*x) - (b*e^4*
n*Log[f*x^m]^2)/(8*d^4*m) + (b*e^4*m*n*Log[d + e*x])/(16*d^4) - ((m/x^4 + (4*Log[f*x^m])/x^4)*(a + b*Log[c*(d
+ e*x)^n]))/16 + (b*e^4*n*Log[f*x^m]*Log[1 + (e*x)/d])/(4*d^4) + (b*e^4*m*n*PolyLog[2, -((e*x)/d)])/(4*d^4)
Rule 2426
Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :
> -Simp[(((m*(g*x)^(q + 1))/(q + 1) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] +
(-Dist[(b*e*n)/(g*(q + 1)), Int[((g*x)^(q + 1)*Log[f*x^m])/(d + e*x), x], x] + Dist[(b*e*m*n)/(g*(q + 1)^2), I
nt[(g*x)^(q + 1)/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && 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 2351
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))
Rule 2304
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -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 2317
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]
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]
Rubi steps
\begin{align*} \int \frac{\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^5} \, dx &=-\frac{1}{16} \left (\frac{m}{x^4}+\frac{4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{4} (b e n) \int \frac{\log \left (f x^m\right )}{x^4 (d+e x)} \, dx+\frac{1}{16} (b e m n) \int \frac{1}{x^4 (d+e x)} \, dx\\ &=-\frac{1}{16} \left (\frac{m}{x^4}+\frac{4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{4} (b e n) \int \left (\frac{\log \left (f x^m\right )}{d x^4}-\frac{e \log \left (f x^m\right )}{d^2 x^3}+\frac{e^2 \log \left (f x^m\right )}{d^3 x^2}-\frac{e^3 \log \left (f x^m\right )}{d^4 x}+\frac{e^4 \log \left (f x^m\right )}{d^4 (d+e x)}\right ) \, dx+\frac{1}{16} (b e m n) \int \left (\frac{1}{d x^4}-\frac{e}{d^2 x^3}+\frac{e^2}{d^3 x^2}-\frac{e^3}{d^4 x}+\frac{e^4}{d^4 (d+e x)}\right ) \, dx\\ &=-\frac{b e m n}{48 d x^3}+\frac{b e^2 m n}{32 d^2 x^2}-\frac{b e^3 m n}{16 d^3 x}-\frac{b e^4 m n \log (x)}{16 d^4}+\frac{b e^4 m n \log (d+e x)}{16 d^4}-\frac{1}{16} \left (\frac{m}{x^4}+\frac{4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{(b e n) \int \frac{\log \left (f x^m\right )}{x^4} \, dx}{4 d}-\frac{\left (b e^2 n\right ) \int \frac{\log \left (f x^m\right )}{x^3} \, dx}{4 d^2}+\frac{\left (b e^3 n\right ) \int \frac{\log \left (f x^m\right )}{x^2} \, dx}{4 d^3}-\frac{\left (b e^4 n\right ) \int \frac{\log \left (f x^m\right )}{x} \, dx}{4 d^4}+\frac{\left (b e^5 n\right ) \int \frac{\log \left (f x^m\right )}{d+e x} \, dx}{4 d^4}\\ &=-\frac{7 b e m n}{144 d x^3}+\frac{3 b e^2 m n}{32 d^2 x^2}-\frac{5 b e^3 m n}{16 d^3 x}-\frac{b e^4 m n \log (x)}{16 d^4}-\frac{b e n \log \left (f x^m\right )}{12 d x^3}+\frac{b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac{b e^3 n \log \left (f x^m\right )}{4 d^3 x}-\frac{b e^4 n \log ^2\left (f x^m\right )}{8 d^4 m}+\frac{b e^4 m n \log (d+e x)}{16 d^4}-\frac{1}{16} \left (\frac{m}{x^4}+\frac{4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e^4 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{4 d^4}-\frac{\left (b e^4 m n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{4 d^4}\\ &=-\frac{7 b e m n}{144 d x^3}+\frac{3 b e^2 m n}{32 d^2 x^2}-\frac{5 b e^3 m n}{16 d^3 x}-\frac{b e^4 m n \log (x)}{16 d^4}-\frac{b e n \log \left (f x^m\right )}{12 d x^3}+\frac{b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac{b e^3 n \log \left (f x^m\right )}{4 d^3 x}-\frac{b e^4 n \log ^2\left (f x^m\right )}{8 d^4 m}+\frac{b e^4 m n \log (d+e x)}{16 d^4}-\frac{1}{16} \left (\frac{m}{x^4}+\frac{4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e^4 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{4 d^4}+\frac{b e^4 m n \text{Li}_2\left (-\frac{e x}{d}\right )}{4 d^4}\\ \end{align*}
Mathematica [A] time = 0.150132, size = 273, normalized size = 1.19 \[ -\frac{-72 b e^4 m n x^4 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+72 a d^4 \log \left (f x^m\right )+18 a d^4 m+72 b d^4 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+18 b d^4 m \log \left (c (d+e x)^n\right )-36 b d^2 e^2 n x^2 \log \left (f x^m\right )-27 b d^2 e^2 m n x^2+24 b d^3 e n x \log \left (f x^m\right )+14 b d^3 e m n x+72 b d e^3 n x^3 \log \left (f x^m\right )-72 b e^4 n x^4 \log (d+e x) \log \left (f x^m\right )+18 b e^4 n x^4 \log (x) \left (4 m \log (d+e x)-4 m \log \left (\frac{e x}{d}+1\right )+4 \log \left (f x^m\right )+m\right )+90 b d e^3 m n x^3-18 b e^4 m n x^4 \log (d+e x)-36 b e^4 m n x^4 \log ^2(x)}{288 d^4 x^4} \]
Antiderivative was successfully verified.
[In]
Integrate[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/x^5,x]
[Out]
-(18*a*d^4*m + 14*b*d^3*e*m*n*x - 27*b*d^2*e^2*m*n*x^2 + 90*b*d*e^3*m*n*x^3 - 36*b*e^4*m*n*x^4*Log[x]^2 + 72*a
*d^4*Log[f*x^m] + 24*b*d^3*e*n*x*Log[f*x^m] - 36*b*d^2*e^2*n*x^2*Log[f*x^m] + 72*b*d*e^3*n*x^3*Log[f*x^m] - 18
*b*e^4*m*n*x^4*Log[d + e*x] - 72*b*e^4*n*x^4*Log[f*x^m]*Log[d + e*x] + 18*b*d^4*m*Log[c*(d + e*x)^n] + 72*b*d^
4*Log[f*x^m]*Log[c*(d + e*x)^n] + 18*b*e^4*n*x^4*Log[x]*(m + 4*Log[f*x^m] + 4*m*Log[d + e*x] - 4*m*Log[1 + (e*
x)/d]) - 72*b*e^4*m*n*x^4*PolyLog[2, -((e*x)/d)])/(288*d^4*x^4)
________________________________________________________________________________________
Maple [C] time = 0.89, size = 2387, normalized size = 10.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(f*x^m)*(a+b*ln(c*(e*x+d)^n))/x^5,x)
[Out]
-1/8*I/x^4*Pi*ln(c)*b*csgn(I*x^m)*csgn(I*f*x^m)^2+1/4*e^4*n*b*ln(x^m)/d^4*ln(e*x+d)-1/4*e^3*n*b*ln(x^m)/d^3/x-
1/4*b*ln(c)/x^4*ln(x^m)-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^4*csgn(I*f)*csgn(I*f*x^m)^2-1/4/x^4*ln(f)*ln(c)*b-
1/16/x^4*ln(c)*b*m+1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/x^4*csgn(I*f)*csgn(I*f*x^m)^2+1/16*b*Pi^2*csgn(
I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^4*csgn(I*f*x^m)^3+1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/x^4
*csgn(I*x^m)*csgn(I*f*x^m)^2+1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^4*csgn(I*f)*csgn(I*f*x^m)^2
-1/4*a/x^4*ln(x^m)+1/16*I/d^2*b*e^2*n/x^2*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/24*I/d*e*b*n/x^3*Pi*csgn(I*f)*csgn(
I*f*x^m)^2-1/4/x^4*ln(f)*a+(-1/4*b/x^4*ln(x^m)-1/16*(-2*I*Pi*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+2*I*Pi*b*cs
gn(I*f)*csgn(I*f*x^m)^2+2*I*Pi*b*csgn(I*x^m)*csgn(I*f*x^m)^2-2*I*Pi*b*csgn(I*f*x^m)^3+4*b*ln(f)+b*m)/x^4)*ln((
e*x+d)^n)-1/4*m*e^4*b*n/d^4*ln(e*x+d)*ln(-e*x/d)-1/16/x^4*a*m+1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)^2/x^4*csgn(I*x^m)*csgn(I*f*x^m)^2+1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^4*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+
1/32*I/x^4*Pi*b*m*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)/x^4*csgn(I*f)*csgn(I*f*x^m)^2-1/16*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^4*
csgn(I*x^m)*csgn(I*f*x^m)^2-1/16*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/x^4*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m
)+1/8*I/x^4*ln(f)*Pi*b*csgn(I*c*(e*x+d)^n)^3+1/8*I/x^4*Pi*ln(c)*b*csgn(I*f*x^m)^3-1/16*b*Pi^2*csgn(I*c)*csgn(I
*c*(e*x+d)^n)^2/x^4*csgn(I*f*x^m)^3+1/24*I/d*e*b*n/x^3*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/16*I/d^2*b*e^2
*n/x^2*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/8*I/d^3*b*e^3*n/x*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/8*I
/d^4*e^4*b*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/8*I/x^4*ln(f)*Pi*b*csgn(I*c)*csgn(I*c*(e*x+d)^
n)^2-1/8*I/x^4*ln(f)*Pi*b*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/8*I/x^4*Pi*ln(c)*b*csgn(I*f)*csgn(I*f*x^m)
^2-1/32*I/x^4*Pi*b*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/32*I/x^4*Pi*b*m*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
-1/8*I/d^3*b*e^3*n/x*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/8*I/x^4*Pi*a*csgn(I*f)*csgn(I*f*x^m)^2-1/8*I/x^4*Pi*a*csgn
(I*x^m)*csgn(I*f*x^m)^2+1/8*I/d^3*b*e^3*n/x*Pi*csgn(I*f*x^m)^3-1/8*I/d^4*e^4*b*n*ln(e*x+d)*Pi*csgn(I*f*x^m)^3-
1/16*I/d^2*b*e^2*n/x^2*Pi*csgn(I*f*x^m)^3+1/24*I/d*e*b*n/x^3*Pi*csgn(I*f*x^m)^3+1/8*I/d^4*e^4*b*n*ln(x)*Pi*csg
n(I*f*x^m)^3+1/8*I/x^4*Pi*a*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)
^n)^2/x^4*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/8*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^4*l
n(x^m)+1/16*b*e^4*m*n*ln(e*x+d)/d^4+1/8/d^4*n*m*e^4*b*ln(x)^2+1/8*I/x^4*ln(f)*Pi*b*csgn(I*c)*csgn(I*(e*x+d)^n)
*csgn(I*c*(e*x+d)^n)+1/8*I/x^4*Pi*ln(c)*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*m*e^4*b*n/d^4*dilog(-e*x/d)+
1/8*I/d^4*e^4*b*n*ln(x)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4/d^4*e^4*b*n*ln(x)*ln(f)+1/4/d^4*e^4*b*n*ln(
e*x+d)*ln(f)-1/8*I/d^3*b*e^3*n/x*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/16*I/d^2*b*e^2*n/x^2*Pi*csgn(I*f)*csgn(I*f*x
^m)^2+1/8/d^2*b*e^2*n/x^2*ln(f)-1/12/d*e*b*n/x^3*ln(f)-1/4/d^3*b*e^3*n/x*ln(f)-1/8*I*b*Pi*csgn(I*c)*csgn(I*c*(
e*x+d)^n)^2/x^4*ln(x^m)+1/8*I/d^4*e^4*b*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/8*I/d^4*e^4*b*n*ln(e*x+d)*P
i*csgn(I*x^m)*csgn(I*f*x^m)^2+1/8*e^2*n*b*ln(x^m)/d^2/x^2-1/4*e^4*n*b*ln(x^m)/d^4*ln(x)-1/12*e*n*b*ln(x^m)/d/x
^3-1/16*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^4*csgn(I*f*x^m)^3+1/8*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/x^
4*ln(x^m)-1/16*b*e^4*m*n*ln(x)/d^4-1/24*I/d*e*b*n/x^3*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/8*I/d^4*e^4*b*n*ln(x)*P
i*csgn(I*f)*csgn(I*f*x^m)^2-1/8*I/d^4*e^4*b*n*ln(x)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/8*I*b*Pi*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)^2/x^4*ln(x^m)-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^4*csgn(I*x^m)*csgn(I*f*x^m)^2+1/16*b*P
i^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^4*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-7/144*b*e*m*n/d/x^
3+3/32*b*e^2*m*n/d^2/x^2-5/16*b*e^3*m*n/d^3/x+1/32*I/x^4*Pi*b*m*csgn(I*c*(e*x+d)^n)^3+1/16*b*Pi^2*csgn(I*c*(e*
x+d)^n)^3/x^4*csgn(I*f*x^m)^3+1/8*I/x^4*Pi*a*csgn(I*f*x^m)^3
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Maxima [A] time = 1.25757, size = 342, normalized size = 1.49 \begin{align*} \frac{1}{288} \,{\left (\frac{72 \,{\left (\log \left (\frac{e x}{d} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{e x}{d}\right )\right )} b e^{4} n}{d^{4}} + \frac{18 \, b e^{4} n \log \left (e x + d\right )}{d^{4}} - \frac{72 \, b e^{4} n x^{4} \log \left (e x + d\right ) \log \left (x\right ) - 36 \, b e^{4} n x^{4} \log \left (x\right )^{2} + 18 \, b e^{4} n x^{4} \log \left (x\right ) + 90 \, b d e^{3} n x^{3} - 27 \, b d^{2} e^{2} n x^{2} + 14 \, b d^{3} e n x + 18 \, b d^{4} \log \left ({\left (e x + d\right )}^{n}\right ) + 18 \, b d^{4} \log \left (c\right ) + 18 \, a d^{4}}{d^{4} x^{4}}\right )} m + \frac{1}{24} \,{\left (b e n{\left (\frac{6 \, e^{3} \log \left (e x + d\right )}{d^{4}} - \frac{6 \, e^{3} \log \left (x\right )}{d^{4}} - \frac{6 \, e^{2} x^{2} - 3 \, d e x + 2 \, d^{2}}{d^{3} x^{3}}\right )} - \frac{6 \, b \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{4}} - \frac{6 \, a}{x^{4}}\right )} \log \left (f x^{m}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))/x^5,x, algorithm="maxima")
[Out]
1/288*(72*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))*b*e^4*n/d^4 + 18*b*e^4*n*log(e*x + d)/d^4 - (72*b*e^4*n*x^4*
log(e*x + d)*log(x) - 36*b*e^4*n*x^4*log(x)^2 + 18*b*e^4*n*x^4*log(x) + 90*b*d*e^3*n*x^3 - 27*b*d^2*e^2*n*x^2
+ 14*b*d^3*e*n*x + 18*b*d^4*log((e*x + d)^n) + 18*b*d^4*log(c) + 18*a*d^4)/(d^4*x^4))*m + 1/24*(b*e*n*(6*e^3*l
og(e*x + d)/d^4 - 6*e^3*log(x)/d^4 - (6*e^2*x^2 - 3*d*e*x + 2*d^2)/(d^3*x^3)) - 6*b*log((e*x + d)^n*c)/x^4 - 6
*a/x^4)*log(f*x^m)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left ({\left (e x + d\right )}^{n} c\right ) \log \left (f x^{m}\right ) + a \log \left (f x^{m}\right )}{x^{5}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))/x^5,x, algorithm="fricas")
[Out]
integral((b*log((e*x + d)^n*c)*log(f*x^m) + a*log(f*x^m))/x^5, 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(ln(f*x**m)*(a+b*ln(c*(e*x+d)**n))/x**5,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 ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(f*x^m)*(a+b*log(c*(e*x+d)^n))/x^5,x, algorithm="giac")
[Out]
integrate((b*log((e*x + d)^n*c) + a)*log(f*x^m)/x^5, x)