\(\int \frac {\log (f x^m) (a+b \log (c (d+e x)^n))}{x^4} \, dx\) [365]

Optimal result

Integrand size = 24, antiderivative size = 193 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {5 b e m n}{36 d x^2}+\frac {4 b e^2 m n}{9 d^2 x}+\frac {b e^3 m n \log (x)}{9 d^3}-\frac {b e n \log \left (f x^m\right )}{6 d x^2}+\frac {b e^2 n \log \left (f x^m\right )}{3 d^2 x}-\frac {b e^3 n \log \left (1+\frac {d}{e x}\right ) \log \left (f x^m\right )}{3 d^3}-\frac {b e^3 m n \log (d+e x)}{9 d^3}-\frac {1}{9} \left (\frac {m}{x^3}+\frac {3 \log \left (f x^m\right )}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b e^3 m n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2473, 2380, 2341, 2379, 2438, 46} \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {1}{9} \left (\frac {3 \log \left (f x^m\right )}{x^3}+\frac {m}{x^3}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac {b e^3 n \log \left (\frac {d}{e x}+1\right ) \log \left (f x^m\right )}{3 d^3}+\frac {b e^3 m n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{3 d^3}+\frac {b e^3 m n \log (x)}{9 d^3}-\frac {b e^3 m n \log (d+e x)}{9 d^3}+\frac {b e^2 n \log \left (f x^m\right )}{3 d^2 x}+\frac {4 b e^2 m n}{9 d^2 x}-\frac {b e n \log \left (f x^m\right )}{6 d x^2}-\frac {5 b e m n}{36 d x^2} \]

[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[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)

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.24 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {4 a d^3 m+5 b d^2 e m n x-16 b d e^2 m n x^2+6 b e^3 m n x^3 \log ^2(x)+12 a d^3 \log \left (f x^m\right )+6 b d^2 e n x \log \left (f x^m\right )-12 b d e^2 n x^2 \log \left (f x^m\right )+4 b e^3 m n x^3 \log (d+e x)+12 b e^3 n x^3 \log \left (f x^m\right ) \log (d+e x)+4 b d^3 m \log \left (c (d+e x)^n\right )+12 b d^3 \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )-4 b e^3 n x^3 \log (x) \left (m+3 \log \left (f x^m\right )+3 m \log (d+e x)-3 m \log \left (1+\frac {e x}{d}\right )\right )+12 b e^3 m n x^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{36 d^3 x^3} \]

[In]

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

[Out]

-1/36*(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)])/(d^3*x^
3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 13.38 (sec) , antiderivative size = 1070, normalized size of antiderivative = 5.54

method	result	size
risch	\(\text {Expression too large to display}\)	\(1070\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{4}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.19 \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=-\frac {1}{36} \, {\left (\frac {12 \, {\left (\log \left (\frac {e x}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x}{d}\right )\right )} b e^{3} n}{d^{3}} + \frac {4 \, b e^{3} n \log \left (e x + d\right )}{d^{3}} - \frac {12 \, b e^{3} n x^{3} \log \left (e x + d\right ) \log \left (x\right ) - 6 \, b e^{3} n x^{3} \log \left (x\right )^{2} + 4 \, b e^{3} n x^{3} \log \left (x\right ) + 16 \, b d e^{2} n x^{2} - 5 \, b d^{2} e n x - 4 \, b d^{3} \log \left ({\left (e x + d\right )}^{n}\right ) - 4 \, b d^{3} \log \left (c\right ) - 4 \, a d^{3}}{d^{3} x^{3}}\right )} m - \frac {1}{6} \, {\left (b e n {\left (\frac {2 \, e^{2} \log \left (e x + d\right )}{d^{3}} - \frac {2 \, e^{2} \log \left (x\right )}{d^{3}} - \frac {2 \, e x - d}{d^{2} x^{2}}\right )} + \frac {2 \, b \log \left ({\left (e x + d\right )}^{n} c\right )}{x^{3}} + \frac {2 \, a}{x^{3}}\right )} \log \left (f x^{m}\right ) \]

[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)

Giac [F]

\[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\right )}{x^{4}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{x^4} \, dx=\int \frac {\ln \left (f\,x^m\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{x^4} \,d x \]

[In]

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

[Out]

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