\(\int \log (f x^m) (a+b \log (c (d+e x)^n)) \, dx\) [361]

Optimal result

Integrand size = 21, antiderivative size = 99 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=2 b m n x-b n x \log \left (f x^m\right )-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{e}+\frac {b d m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2469, 45, 2393, 2332, 2354, 2438} \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (\frac {e x}{d}+1\right ) \log \left (f x^m\right )}{e}+\frac {b d m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}-\frac {b d m n \log (d+e x)}{e}-b n x \log \left (f x^m\right )+2 b m n x \]

[In]

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

[Out]

2*b*m*n*x - b*n*x*Log[f*x^m] - (b*d*m*n*Log[d + e*x])/e - x*(m - Log[f*x^m])*(a + b*Log[c*(d + e*x)^n]) + (b*d
*n*Log[f*x^m]*Log[1 + (e*x)/d])/e + (b*d*m*n*PolyLog[2, -((e*x)/d)])/e

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2354

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 2393

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 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 2469

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

Rubi steps \begin{align*} \text {integral}& = -x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-(b e n) \int \frac {x \log \left (f x^m\right )}{d+e x} \, dx+(b e m n) \int \frac {x}{d+e x} \, dx \\ & = -x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-(b e n) \int \left (\frac {\log \left (f x^m\right )}{e}-\frac {d \log \left (f x^m\right )}{e (d+e x)}\right ) \, dx+(b e m n) \int \left (\frac {1}{e}-\frac {d}{e (d+e x)}\right ) \, dx \\ & = b m n x-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-(b n) \int \log \left (f x^m\right ) \, dx+(b d n) \int \frac {\log \left (f x^m\right )}{d+e x} \, dx \\ & = 2 b m n x-b n x \log \left (f x^m\right )-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{e}-\frac {(b d m n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e} \\ & = 2 b m n x-b n x \log \left (f x^m\right )-\frac {b d m n \log (d+e x)}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {b d n \log \left (f x^m\right ) \log \left (1+\frac {e x}{d}\right )}{e}+\frac {b d m n \text {Li}_2\left (-\frac {e x}{d}\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.17 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {\log \left (f x^m\right ) \left (b d n \log (d+e x)+e x \left (a-b n+b \log \left (c (d+e x)^n\right )\right )\right )-m \left (a e x-2 b e n x+b d n (1+\log (x)) \log (d+e x)+b e x \log \left (c (d+e x)^n\right )-b d n \log (x) \log \left (1+\frac {e x}{d}\right )\right )+b d m n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e} \]

[In]

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

[Out]

(Log[f*x^m]*(b*d*n*Log[d + e*x] + e*x*(a - b*n + b*Log[c*(d + e*x)^n])) - m*(a*e*x - 2*b*e*n*x + b*d*n*(1 + Lo
g[x])*Log[d + e*x] + b*e*x*Log[c*(d + e*x)^n] - b*d*n*Log[x]*Log[1 + (e*x)/d]) + b*d*m*n*PolyLog[2, -((e*x)/d)
])/e

Maple [C] (warning: unable to verify)

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

Time = 5.25 (sec) , antiderivative size = 657, normalized size of antiderivative = 6.64

method	result	size
risch	\(\left (b x \ln \left (x^{m}\right )+\frac {x b \left (-i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )+i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}+i \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}-i \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}+2 \ln \left (f \right )-2 m \right )}{2}\right ) \ln \left (\left (e x +d \right )^{n}\right )+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right )}{4}+\frac {i \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{4}+\frac {i \pi \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} b}{4}-\frac {i \pi \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} b}{4}+\frac {b \ln \left (c \right )}{2}+\frac {a}{2}\right ) \left (i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2} x +i \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2} x +2 x \ln \left (f \right )-i \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3} x -i \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right ) x +2 \ln \left (x^{m}\right ) x -2 m x \right )+\frac {i n b x \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2}-\frac {i n b x \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2}+\frac {i n b x \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{2}-\frac {i n b d \ln \left (e x +d \right ) \pi \operatorname {csgn}\left (i f \,x^{m}\right )^{3}}{2 e}-n b x \ln \left (f \right )+2 b m n x -\frac {i n b x \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2}+\frac {i n b d \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 e}+\frac {i n b d \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i f \,x^{m}\right )^{2}}{2 e}-\frac {i n b d \ln \left (e x +d \right ) \pi \,\operatorname {csgn}\left (i f \right ) \operatorname {csgn}\left (i x^{m}\right ) \operatorname {csgn}\left (i f \,x^{m}\right )}{2 e}+\frac {n b d \ln \left (e x +d \right ) \ln \left (f \right )}{e}-\frac {b d m n \ln \left (e x +d \right )}{e}-n b \ln \left (x^{m}\right ) x +\frac {n b \ln \left (x^{m}\right ) d \ln \left (e x +d \right )}{e}+\frac {n b m d}{e}-\frac {n b m d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e}-\frac {n b m d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e}\)	\(657\)

[In]

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

[Out]

(b*x*ln(x^m)+1/2*x*b*(-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)-2*m))*ln((e*x+d)^n)+(-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*cs
gn(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)*(I*Pi*csgn(I*f)*csgn(I*f*x^m)^2*x+I*Pi*csgn(I*x^m)*csgn(
I*f*x^m)^2*x+2*x*ln(f)-I*Pi*csgn(I*f*x^m)^3*x-I*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*x+2*ln(x^m)*x-2*m*x)+1/
2*I*n*b*x*Pi*csgn(I*f*x^m)^3-1/2*I*n*b*x*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I*n*b*x*Pi*csgn(I*f)*csgn(I*x^m)*c
sgn(I*f*x^m)-1/2*I/e*n*b*d*ln(e*x+d)*Pi*csgn(I*f*x^m)^3-n*b*x*ln(f)+2*b*m*n*x-1/2*I*n*b*x*Pi*csgn(I*f)*csgn(I*
f*x^m)^2+1/2*I/e*n*b*d*ln(e*x+d)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I/e*n*b*d*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*
x^m)^2-1/2*I/e*n*b*d*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/e*n*b*d*ln(e*x+d)*ln(f)-b*d*m*n*ln(e*x
+d)/e-n*b*ln(x^m)*x+1/e*n*b*ln(x^m)*d*ln(e*x+d)+1/e*n*b*m*d-1/e*n*b*m*d*ln(e*x+d)*ln(-e*x/d)-1/e*n*b*m*d*dilog
(-e*x/d)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.40 \[ \int \log \left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-{\left (\frac {{\left (\log \left (e x + d\right ) \log \left (-\frac {e x + d}{d} + 1\right ) + {\rm Li}_2\left (\frac {e x + d}{d}\right )\right )} b d n}{e} + \frac {b d n \log \left (e x + d\right ) + b e x \log \left ({\left (e x + d\right )}^{n}\right ) - {\left ({\left (2 \, e n - e \log \left (c\right )\right )} b - a e\right )} x}{e}\right )} m - {\left (b e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - b x \log \left ({\left (e x + d\right )}^{n} c\right ) - a x\right )} \log \left (f x^{m}\right ) \]

[In]

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

[Out]

-((log(e*x + d)*log(-(e*x + d)/d + 1) + dilog((e*x + d)/d))*b*d*n/e + (b*d*n*log(e*x + d) + b*e*x*log((e*x + d
)^n) - ((2*e*n - e*log(c))*b - a*e)*x)/e)*m - (b*e*n*(x/e - d*log(e*x + d)/e^2) - b*x*log((e*x + d)^n*c) - a*x
)*log(f*x^m)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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