3.4.66 \(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))}{x^5} \, dx\) [366]

Optimal. Leaf size=230 \[ -\frac {7 b e m n}{144 d x^3}+\frac {3 b e^2 m n}{32 d^2 x^2}-\frac {5 b e^3 m n}{16 d^3 x}-\frac {b e^4 m n \log (x)}{16 d^4}-\frac {b e n \log \left (f x^m\right )}{12 d x^3}+\frac {b e^2 n \log \left (f x^m\right )}{8 d^2 x^2}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}+\frac {b e^4 n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{4 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {1}{16} \left (\frac {m}{x^4}+\frac {4 \log \left (f x^m\right )}{x^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e^4 m n \text {Li}_2\left (-\frac {d}{e x}\right )}{4 d^4} \]

[Out]

-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-1/16*b*e^4*m*n*ln(x)/d^4-1/12*b*e*n*ln(f*x^m)
/d/x^3+1/8*b*e^2*n*ln(f*x^m)/d^2/x^2-1/4*b*e^3*n*ln(f*x^m)/d^3/x+1/4*b*e^4*n*ln(1+d/e/x)*ln(f*x^m)/d^4+1/16*b*
e^4*m*n*ln(e*x+d)/d^4-1/16*(m/x^4+4*ln(f*x^m)/x^4)*(a+b*ln(c*(e*x+d)^n))-1/4*b*e^4*m*n*polylog(2,-d/e/x)/d^4

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Rubi [A]
time = 0.18, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2473, 2380, 2341, 2379, 2438, 46} \begin {gather*} -\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^4 m n \log (x)}{16 d^4}+\frac {b e^4 m n \log (d+e x)}{16 d^4}-\frac {b e^3 n \log \left (f x^m\right )}{4 d^3 x}-\frac {5 b e^3 m n}{16 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 {b e n \log \left (f x^m\right )}{12 d x^3}-\frac {7 b e m n}{144 d x^3} \end {gather*}

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[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)

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 2341

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 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2380

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)*(x_)^(r_.)), x_Symbol] :> Dist[1/d,
Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Dist[e/d, Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /;
FreeQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -1]

Rule 2438

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 2473

Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Symbol] :
> Simp[(-(g*(q + 1))^(-1))*(m*((g*x)^(q + 1)/(q + 1)) - (g*x)^(q + 1)*Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]),
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
)), Int[(g*x)^(q + 1)/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[q, -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*}

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Mathematica [A]
time = 0.10, size = 273, normalized size = 1.19 \begin {gather*} -\frac {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 ^2(x)+72 a d^4 \log \left (f x^m\right )+24 b d^3 e n x \log \left (f x^m\right )-36 b d^2 e^2 n x^2 \log \left (f x^m\right )+72 b d e^3 n x^3 \log \left (f x^m\right )-18 b e^4 m n x^4 \log (d+e x)-72 b e^4 n x^4 \log \left (f x^m\right ) \log (d+e x)+18 b d^4 m \log \left (c (d+e x)^n\right )+72 b d^4 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )+18 b e^4 n x^4 \log (x) \left (m+4 \log \left (f x^m\right )+4 m \log (d+e x)-4 m \log \left (1+\frac {e x}{d}\right )\right )-72 b e^4 m n x^4 \text {Li}_2\left (-\frac {e x}{d}\right )}{288 d^4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Log[f*x^m]*(a + b*Log[c*(d + e*x)^n]))/x^5,x]

[Out]

-1/288*(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] + 7
2*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)])/(d^4*x^4)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.81, size = 2387, normalized size = 10.38

method	result	size
risch	\(\text {Expression too large to display}\)	\(2387\)

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,method=_RETURNVERBOSE)

[Out]

1/8/d^2*e^2*b*n/x^2*ln(f)-1/4/d^3*e^3*b*n/x*ln(f)-1/12/d*e*b*n/x^3*ln(f)-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)-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^4*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*m*e^4*b*n/d^4*dil
og(-e*x/d)-1/4/x^4*ln(f)*a-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)+1/16*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+1/16*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)+1/16*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*(e*x+d)^n)*csgn(I*c*(e*x+d)^
n)^2/x^4*csgn(I*f*x^m)^3-1/16*b*Pi^2*csgn(I*c*(e*x+d)^n)^3/x^4*csgn(I*f)*csgn(I*f*x^m)^2+1/16*I/d^2*e^2*b*n/x^
2*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/8*I/d^4*e^4*b*n*ln(x)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/8*I/d^3*e^3*
b*n/x*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4/x^4*ln(c)*ln(f)*b-1/16/x^4*ln(c)*b*m-1/8*I/x^4*ln(f)*Pi*b*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/8*I/x^4*ln(c)*Pi*b*csgn(I*f)*csgn(I*f*x^m)^2-7/144*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)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/8*I/
x^4*ln(c)*Pi*b*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*c*(e*x+d)^n)^2-1/8*I/x^4
*ln(f)*Pi*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/8*I/x^4*ln(c)*Pi*b*csgn(I*f*x^m)^3+1/8*I/x^4*ln(f)*Pi*b*csgn(I*c
*(e*x+d)^n)^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+1/16*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*c*(e*x+d)^n)^2/x^4*csgn(I*x^m)*csgn(I*f
*x^m)^2+(-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*csgn(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)+1/8*I/d^4
*e^4*b*n*ln(e*x+d)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/16*I/d^2*e^2*b*n/x^2*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1/8*I/d^
3*e^3*b*n/x*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/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/d^4*e^4*b*n*ln(x)*Pi*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*e^4*n*b*ln(x^m)/d^4*ln(x)+1/8*e^2*n*b*ln(x^m)/d^2/x^2+1/8/d^4*n*m*b*e^4*l
n(x)^2+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*e^2*b*n/x^2*Pi*csgn(I*f)*csgn(I*x^
m)*csgn(I*f*x^m)-1/8*I/d^3*e^3*b*n/x*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/24*I/d*e*b*n/x^3*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)^2-1/8*I/d^4*e^4*b*n*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*x^m)*c
sgn(I*f*x^m)-1/4*a/x^4*ln(x^m)+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*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*f*x^m)^2-1/12*e*n*b*ln(x^m)/d/x^3-1/4*m*e
^4*b*n/d^4*ln(e*x+d)*ln(-e*x/d)-1/8*I/x^4*ln(c)*Pi*b*csgn(I*x^m)*csgn(I*f*x^m)^2-1/8*I*b*Pi*csgn(I*c)*csgn(I*c
*(e*x+d)^n)^2/x^4*ln(x^m)-1/8*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/x^4*ln(x^m)-1/32*I/x^4*Pi*b*m*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/8*I/x^4*Pi*a*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/16/x^4*a*m-1/4*b*ln(
c)/x^4*ln(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*ln(x^m)-1/8*I/d^4*e^4*b*n*ln(e*x+d)*Pi*csgn(I*f*x^
m)^3-1/8*I/d^4*e^4*b*n*ln(x)*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/8*I/d^3*e^3*b*n/x*Pi*csgn(I*f)*csgn(I*f*x^m)^2-1/1
6*I/d^2*e^2*b*n/x^2*Pi*csgn(I*f*x^m)^3+1/8*I/d^4*e^4*b*n*ln(x)*Pi*csgn(I*f*x^m)^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+1/32*I/x^4*Pi*b*m*csgn(I*c*(e*x+d)^n)^3-1/8*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*b*Pi*csgn(I*c*(e*x+d)^n)^3/x^4*ln
(x^m)+1/16*b*e^4*m*n*ln(e*x+d)/d^4-1/16*b*e^4*m*n*ln(x)/d^4

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Maxima [A]
time = 0.33, size = 253, normalized size = 1.10 \begin {gather*} \frac {1}{288} \, m {\left (\frac {72 \, {\left (\log \left (x\right ) \log \left (\frac {x e}{d} + 1\right ) + {\rm Li}_2\left (-\frac {x e}{d}\right )\right )} b n e^{4}}{d^{4}} + \frac {18 \, b n e^{4} \log \left (x e + d\right )}{d^{4}} - \frac {72 \, b n x^{4} e^{4} \log \left (x e + d\right ) \log \left (x\right ) - 36 \, b n x^{4} e^{4} \log \left (x\right )^{2} + 18 \, b n x^{4} e^{4} \log \left (x\right ) + 90 \, b d n x^{3} e^{3} - 27 \, b d^{2} n x^{2} e^{2} + 14 \, b d^{3} n x e + 18 \, b d^{4} \log \left ({\left (x e + d\right )}^{n}\right ) + 18 \, b d^{4} \log \left (c\right ) + 18 \, a d^{4}}{d^{4} x^{4}}\right )} + \frac {1}{24} \, {\left (b n {\left (\frac {6 \, e^{3} \log \left (x e + d\right )}{d^{4}} - \frac {6 \, e^{3} \log \left (x\right )}{d^{4}} - \frac {6 \, x^{2} e^{2} - 3 \, d x e + 2 \, d^{2}}{d^{3} x^{3}}\right )} e - \frac {6 \, b \log \left ({\left (x e + d\right )}^{n} c\right )}{x^{4}} - \frac {6 \, a}{x^{4}}\right )} \log \left (f x^{m}\right ) \end {gather*}

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*m*(72*(log(x)*log(x*e/d + 1) + dilog(-x*e/d))*b*n*e^4/d^4 + 18*b*n*e^4*log(x*e + d)/d^4 - (72*b*n*x^4*e^
4*log(x*e + d)*log(x) - 36*b*n*x^4*e^4*log(x)^2 + 18*b*n*x^4*e^4*log(x) + 90*b*d*n*x^3*e^3 - 27*b*d^2*n*x^2*e^
2 + 14*b*d^3*n*x*e + 18*b*d^4*log((x*e + d)^n) + 18*b*d^4*log(c) + 18*a*d^4)/(d^4*x^4)) + 1/24*(b*n*(6*e^3*log
(x*e + d)/d^4 - 6*e^3*log(x)/d^4 - (6*x^2*e^2 - 3*d*x*e + 2*d^2)/(d^3*x^3))*e - 6*b*log((x*e + d)^n*c)/x^4 - 6
*a/x^4)*log(f*x^m)

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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(log(f*x^m)*(a+b*log(c*(e*x+d)^n))/x^5,x, algorithm="fricas")

[Out]

integral((b*log((x*e + d)^n*c)*log(f*x^m) + a*log(f*x^m))/x^5, 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(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 = 0.00, 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(log(f*x^m)*(a+b*log(c*(e*x+d)^n))/x^5,x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)*log(f*x^m)/x^5, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n)))/x^5,x)

[Out]

int((log(f*x^m)*(a + b*log(c*(d + e*x)^n)))/x^5, x)

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