3.365 \(\int \frac{\log (f x^m) (a+b \log (c (d+e x)^n))}{x^4} \, dx\)
Optimal. Leaf size=193 \[ \frac{b e^3 m n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{3 d^3}-\frac{1}{9} \left (\frac{3 \log \left (f x^m\right )}{x^3}+\frac{m}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e^3 n \log \left (\frac{d}{e x}+1\right ) \log \left (f x^m\right )}{3 d^3}+\frac{b e^2 n \log \left (f x^m\right )}{3 d^2 x}+\frac{4 b e^2 m n}{9 d^2 x}+\frac{b e^3 m n \log (x)}{9 d^3}-\frac{b e^3 m n \log (d+e x)}{9 d^3}-\frac{b e n \log \left (f x^m\right )}{6 d x^2}-\frac{5 b e m n}{36 d x^2} \]
[Out]
(-5*b*e*m*n)/(36*d*x^2) + (4*b*e^2*m*n)/(9*d^2*x) + (b*e^3*m*n*Log[x])/(9*d^3) - (b*e*n*Log[f*x^m])/(6*d*x^2)
+ (b*e^2*n*Log[f*x^m])/(3*d^2*x) - (b*e^3*n*Log[1 + d/(e*x)]*Log[f*x^m])/(3*d^3) - (b*e^3*m*n*Log[d + e*x])/(9
*d^3) - ((m/x^3 + (3*Log[f*x^m])/x^3)*(a + b*Log[c*(d + e*x)^n]))/9 + (b*e^3*m*n*PolyLog[2, -(d/(e*x))])/(3*d^
3)
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Rubi [A] time = 0.181487, antiderivative size = 212, normalized size of antiderivative = 1.1,
number of steps used = 10, 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} \[ -\frac{b e^3 m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{3 d^3}-\frac{1}{9} \left (\frac{3 \log \left (f x^m\right )}{x^3}+\frac{m}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e^3 n \log ^2\left (f x^m\right )}{6 d^3 m}-\frac{b e^3 n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{3 d^3}+\frac{b e^2 n \log \left (f x^m\right )}{3 d^2 x}+\frac{4 b e^2 m n}{9 d^2 x}+\frac{b e^3 m n \log (x)}{9 d^3}-\frac{b e^3 m n \log (d+e x)}{9 d^3}-\frac{b e n \log \left (f x^m\right )}{6 d x^2}-\frac{5 b e m n}{36 d x^2} \]
Antiderivative was successfully verified.
[In]
Int[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/x^4,x]
[Out]
(-5*b*e*m*n)/(36*d*x^2) + (4*b*e^2*m*n)/(9*d^2*x) + (b*e^3*m*n*Log[x])/(9*d^3) - (b*e*n*Log[f*x^m])/(6*d*x^2)
+ (b*e^2*n*Log[f*x^m])/(3*d^2*x) + (b*e^3*n*Log[f*x^m]^2)/(6*d^3*m) - (b*e^3*m*n*Log[d + e*x])/(9*d^3) - ((m/x
^3 + (3*Log[f*x^m])/x^3)*(a + b*Log[c*(d + e*x)^n]))/9 - (b*e^3*n*Log[f*x^m]*Log[1 + (e*x)/d])/(3*d^3) - (b*e^
3*m*n*PolyLog[2, -((e*x)/d)])/(3*d^3)
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^4} \, dx &=-\frac{1}{9} \left (\frac{m}{x^3}+\frac{3 \log \left (f x^m\right )}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{3} (b e n) \int \frac{\log \left (f x^m\right )}{x^3 (d+e x)} \, dx+\frac{1}{9} (b e m n) \int \frac{1}{x^3 (d+e x)} \, dx\\ &=-\frac{1}{9} \left (\frac{m}{x^3}+\frac{3 \log \left (f x^m\right )}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{1}{3} (b e n) \int \left (\frac{\log \left (f x^m\right )}{d x^3}-\frac{e \log \left (f x^m\right )}{d^2 x^2}+\frac{e^2 \log \left (f x^m\right )}{d^3 x}-\frac{e^3 \log \left (f x^m\right )}{d^3 (d+e x)}\right ) \, dx+\frac{1}{9} (b e m n) \int \left (\frac{1}{d x^3}-\frac{e}{d^2 x^2}+\frac{e^2}{d^3 x}-\frac{e^3}{d^3 (d+e x)}\right ) \, dx\\ &=-\frac{b e m n}{18 d x^2}+\frac{b e^2 m n}{9 d^2 x}+\frac{b e^3 m n \log (x)}{9 d^3}-\frac{b e^3 m n \log (d+e x)}{9 d^3}-\frac{1}{9} \left (\frac{m}{x^3}+\frac{3 \log \left (f x^m\right )}{x^3}\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^3} \, dx}{3 d}-\frac{\left (b e^2 n\right ) \int \frac{\log \left (f x^m\right )}{x^2} \, dx}{3 d^2}+\frac{\left (b e^3 n\right ) \int \frac{\log \left (f x^m\right )}{x} \, dx}{3 d^3}-\frac{\left (b e^4 n\right ) \int \frac{\log \left (f x^m\right )}{d+e x} \, dx}{3 d^3}\\ &=-\frac{5 b e m n}{36 d x^2}+\frac{4 b e^2 m n}{9 d^2 x}+\frac{b e^3 m n \log (x)}{9 d^3}-\frac{b e n \log \left (f x^m\right )}{6 d x^2}+\frac{b e^2 n \log \left (f x^m\right )}{3 d^2 x}+\frac{b e^3 n \log ^2\left (f x^m\right )}{6 d^3 m}-\frac{b e^3 m n \log (d+e x)}{9 d^3}-\frac{1}{9} \left (\frac{m}{x^3}+\frac{3 \log \left (f x^m\right )}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e^3 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^3}+\frac{\left (b e^3 m n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{3 d^3}\\ &=-\frac{5 b e m n}{36 d x^2}+\frac{4 b e^2 m n}{9 d^2 x}+\frac{b e^3 m n \log (x)}{9 d^3}-\frac{b e n \log \left (f x^m\right )}{6 d x^2}+\frac{b e^2 n \log \left (f x^m\right )}{3 d^2 x}+\frac{b e^3 n \log ^2\left (f x^m\right )}{6 d^3 m}-\frac{b e^3 m n \log (d+e x)}{9 d^3}-\frac{1}{9} \left (\frac{m}{x^3}+\frac{3 \log \left (f x^m\right )}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e^3 n \log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{3 d^3}-\frac{b e^3 m n \text{Li}_2\left (-\frac{e x}{d}\right )}{3 d^3}\\ \end{align*}
Mathematica [A] time = 0.143647, size = 240, normalized size = 1.24 \[ -\frac{12 b e^3 m n x^3 \text{PolyLog}\left (2,-\frac{e x}{d}\right )+12 a d^3 \log \left (f x^m\right )+4 a d^3 m+12 b d^3 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+4 b d^3 m \log \left (c (d+e x)^n\right )+6 b d^2 e n x \log \left (f x^m\right )+5 b d^2 e m n x-12 b d e^2 n x^2 \log \left (f x^m\right )+12 b e^3 n x^3 \log (d+e x) \log \left (f x^m\right )-4 b e^3 n x^3 \log (x) \left (3 m \log (d+e x)-3 m \log \left (\frac{e x}{d}+1\right )+3 \log \left (f x^m\right )+m\right )-16 b d e^2 m n x^2+4 b e^3 m n x^3 \log (d+e x)+6 b e^3 m n x^3 \log ^2(x)}{36 d^3 x^3} \]
Antiderivative was successfully verified.
[In]
Integrate[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/x^4,x]
[Out]
-(4*a*d^3*m + 5*b*d^2*e*m*n*x - 16*b*d*e^2*m*n*x^2 + 6*b*e^3*m*n*x^3*Log[x]^2 + 12*a*d^3*Log[f*x^m] + 6*b*d^2*
e*n*x*Log[f*x^m] - 12*b*d*e^2*n*x^2*Log[f*x^m] + 4*b*e^3*m*n*x^3*Log[d + e*x] + 12*b*e^3*n*x^3*Log[f*x^m]*Log[
d + e*x] + 4*b*d^3*m*Log[c*(d + e*x)^n] + 12*b*d^3*Log[f*x^m]*Log[c*(d + e*x)^n] - 4*b*e^3*n*x^3*Log[x]*(m + 3
*Log[f*x^m] + 3*m*Log[d + e*x] - 3*m*Log[1 + (e*x)/d]) + 12*b*e^3*m*n*x^3*PolyLog[2, -((e*x)/d)])/(36*d^3*x^3)
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Maple [C] time = 0.874, size = 2220, normalized size = 11.5 \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^4,x)
[Out]
-1/6*I/d^2*b*e^2*n/x*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/3/x^3*ln(f)*a+(-1/3*b/x^3*ln(x^m)-1/18*(-3*I*Pi*
b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+3*I*Pi*b*csgn(I*f)*csgn(I*f*x^m)^2+3*I*Pi*b*csgn(I*x^m)*csgn(I*f*x^m)^2-
3*I*Pi*b*csgn(I*f*x^m)^3+6*b*ln(f)+2*b*m)/x^3)*ln((e*x+d)^n)+1/6*I/x^3*ln(f)*Pi*b*csgn(I*c)*csgn(I*(e*x+d)^n)*
csgn(I*c*(e*x+d)^n)-1/3*a/x^3*ln(x^m)-1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^3*csgn(I*f)*csgn(I
*x^m)*csgn(I*f*x^m)-1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^3*csgn(I*f)*csgn(I*f*x^m)^2-
1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^3*csgn(I*x^m)*csgn(I*f*x^m)^2-1/9/x^3*a*m-1/6/d^
3*b*e^3*m*n*ln(x)^2-1/18*I/x^3*Pi*b*m*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/18*I/x^3*Pi*b*m*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^2+1/3*m*b*e^3*n/d^3*dilog(-e*x/d)-1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/x^3*csgn(I*f)*cs
gn(I*x^m)*csgn(I*f*x^m)+1/6*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^3*ln(x^m)+1/6*I/d^3*b*e^3
*n*ln(e*x+d)*Pi*csgn(I*f*x^m)^3-1/3*b*ln(c)/x^3*ln(x^m)-1/6*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^3
*ln(x^m)-1/6*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/x^3*ln(x^m)+1/6*I/d^3*b*e^3*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(
I*x^m)*csgn(I*f*x^m)-1/6*I/d^3*b*e^3*n*ln(x)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/12*b*Pi^2*csgn(I*c*(e*x+
d)^n)^3/x^3*csgn(I*f*x^m)^3+1/6*I/x^3*Pi*a*csgn(I*f*x^m)^3+1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2
/x^3*csgn(I*x^m)*csgn(I*f*x^m)^2+1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^3*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/6
*I/d^3*b*e^3*n*ln(x)*Pi*csgn(I*f*x^m)^3+1/12*I/d*e*b*n/x^2*Pi*csgn(I*f*x^m)^3-1/6*I/d^2*b*e^2*n/x*Pi*csgn(I*f*
x^m)^3+1/12*I/d*e*b*n/x^2*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/6*I/x^3*Pi*a*csgn(I*f)*csgn(I*x^m)*csgn(I*f
*x^m)-1/6*I/x^3*ln(f)*Pi*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/6*I/x^3*ln(f)*Pi*b*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)^2+1/6*I/x^3*Pi*ln(c)*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/18*I/x^3*Pi*b*m*csgn(I*c)*csgn(I*(e*x+d)^
n)*csgn(I*c*(e*x+d)^n)+1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/x^3*ln(x^m)-1/6*I/x^3*Pi*a*csgn(I*f)*csgn(I*f*x^m)^2-1
/6*I/x^3*Pi*a*csgn(I*x^m)*csgn(I*f*x^m)^2+1/6*I/x^3*ln(f)*Pi*b*csgn(I*c*(e*x+d)^n)^3-1/12*b*Pi^2*csgn(I*c)*csg
n(I*c*(e*x+d)^n)^2/x^3*csgn(I*f*x^m)^3-1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^3*csgn(I*f*x^m)^3
-1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^3*csgn(I*f)*csgn(I*f*x^m)^2+1/3*m*b*e^3*n/d^3*ln(e*x+d)*ln(-e*x/d)+1/12*b
*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/x^3*csgn(I*f*x^m)^3+1/12*b*Pi^2*csgn(I*c)*csgn(I*c*(e*x+
d)^n)^2/x^3*csgn(I*f)*csgn(I*f*x^m)^2+1/6*I/d^3*b*e^3*n*ln(x)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/12*I/d*e*b*n/x^
2*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/6*e*n*b*ln(x^m)/d/x^2+1/12*b*Pi^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d
)^n)/x^3*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/6*I/d^2*b*e^2*n/x*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/12*I/d*e*b*n
/x^2*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/6*I/d^2*b*e^2*n/x*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/12*b*Pi^2*csgn(I*c)*csg
n(I*c*(e*x+d)^n)^2/x^3*csgn(I*x^m)*csgn(I*f*x^m)^2+1/12*b*Pi^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^3*csg
n(I*f)*csgn(I*f*x^m)^2+1/3/d^3*b*e^3*n*ln(x)*ln(f)-1/3/d^3*b*e^3*n*ln(e*x+d)*ln(f)+1/3/d^2*b*e^2*n/x*ln(f)-1/6
/d*e*b*n/x^2*ln(f)-1/12*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^3*csgn(I*x^m)*csgn(I*f*x^m)^2+1/6*I/x^3*Pi*ln(c)*b*csgn
(I*f*x^m)^3+1/18*I/x^3*Pi*b*m*csgn(I*c*(e*x+d)^n)^3-1/6*I/d^3*b*e^3*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1
/6*I/d^3*b*e^3*n*ln(e*x+d)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/6*I/x^3*Pi*ln(c)*b*csgn(I*f)*csgn(I*f*x^m)^2-1/6*I
/x^3*Pi*ln(c)*b*csgn(I*x^m)*csgn(I*f*x^m)^2+1/6*I/d^3*b*e^3*n*ln(x)*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/3*e^3*n*b*l
n(x^m)/d^3*ln(x)+1/3*e^2*n*b*ln(x^m)/d^2/x-1/3*e^3*n*b*ln(x^m)/d^3*ln(e*x+d)-1/3/x^3*ln(f)*ln(c)*b-1/9/x^3*ln(
c)*b*m+1/9*b*e^3*m*n*ln(x)/d^3-1/9*b*e^3*m*n*ln(e*x+d)/d^3-5/36*b*e*m*n/d/x^2+4/9*b*e^2*m*n/x/d^2
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Maxima [A] time = 1.23475, size = 309, normalized size = 1.6 \begin{align*} -\frac{1}{36} \,{\left (\frac{12 \,{\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^{3} n}{d^{3}} + \frac{4 \, b e^{3} n \log \left (e x + d\right )}{d^{3}} - \frac{12 \, b e^{3} n x^{3} \log \left (e x + d\right ) \log \left (x\right ) - 6 \, b e^{3} n x^{3} \log \left (x\right )^{2} + 4 \, b e^{3} n x^{3} \log \left (x\right ) + 16 \, b d e^{2} n x^{2} - 5 \, b d^{2} e n x - 4 \, b d^{3} \log \left ({\left (e x + d\right )}^{n}\right ) - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{d^{3} x^{3}}\right )} m - \frac{1}{6} \,{\left (b e n{\left (\frac{2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac{2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac{2 \, e x - d}{d^{2} x^{2}}\right )} + \frac{2 \, b \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{3}} + \frac{2 \, a}{x^{3}}\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^4,x, algorithm="maxima")
[Out]
-1/36*(12*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))*b*e^3*n/d^3 + 4*b*e^3*n*log(e*x + d)/d^3 - (12*b*e^3*n*x^3*l
og(e*x + d)*log(x) - 6*b*e^3*n*x^3*log(x)^2 + 4*b*e^3*n*x^3*log(x) + 16*b*d*e^2*n*x^2 - 5*b*d^2*e*n*x - 4*b*d^
3*log((e*x + d)^n) - 4*b*d^3*log(c) - 4*a*d^3)/(d^3*x^3))*m - 1/6*(b*e*n*(2*e^2*log(e*x + d)/d^3 - 2*e^2*log(x
)/d^3 - (2*e*x - d)/(d^2*x^2)) + 2*b*log((e*x + d)^n*c)/x^3 + 2*a/x^3)*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^{4}}, 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^4,x, algorithm="fricas")
[Out]
integral((b*log((e*x + d)^n*c)*log(f*x^m) + a*log(f*x^m))/x^4, x)
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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**4,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^{4}}\,{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^4,x, algorithm="giac")
[Out]
integrate((b*log((e*x + d)^n*c) + a)*log(f*x^m)/x^4, x)