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

Optimal. Leaf size=88 \[ -b n \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+b m n \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \log ^2\left (f x^m\right )}{2 m} \]

[Out]

(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m) - (b*n*Log[f*x^m]^2*Log[1 + (e*x)/d])/(2*m) - b*n*Log[f*x^m]*P
olyLog[2, -((e*x)/d)] + b*m*n*PolyLog[3, -((e*x)/d)]

________________________________________________________________________________________

Rubi [A]  time = 0.0742769, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2425, 2317, 2374, 6589} \[ -b n \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+b m n \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{b n \log \left (\frac{e x}{d}+1\right ) \log ^2\left (f x^m\right )}{2 m} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Log[f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m) - (b*n*Log[f*x^m]^2*Log[1 + (e*x)/d])/(2*m) - b*n*Log[f*x^m]*P
olyLog[2, -((e*x)/d)] + b*m*n*PolyLog[3, -((e*x)/d)]

Rule 2425

Int[(Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)))/(x_), x_Symbol] :> Simp[(Log[
f*x^m]^2*(a + b*Log[c*(d + e*x)^n]))/(2*m), x] - Dist[(b*e*n)/(2*m), Int[Log[f*x^m]^2/(d + e*x), x], x] /; Fre
eQ[{a, b, c, d, e, f, m, 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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

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

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} \, dx &=\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{(b e n) \int \frac{\log ^2\left (f x^m\right )}{d+e x} \, dx}{2 m}\\ &=\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{b n \log ^2\left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{2 m}+(b n) \int \frac{\log \left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{x} \, dx\\ &=\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{b n \log ^2\left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \text{Li}_2\left (-\frac{e x}{d}\right )+(b m n) \int \frac{\text{Li}_2\left (-\frac{e x}{d}\right )}{x} \, dx\\ &=\frac{\log ^2\left (f x^m\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 m}-\frac{b n \log ^2\left (f x^m\right ) \log \left (1+\frac{e x}{d}\right )}{2 m}-b n \log \left (f x^m\right ) \text{Li}_2\left (-\frac{e x}{d}\right )+b m n \text{Li}_3\left (-\frac{e x}{d}\right )\\ \end{align*}

Mathematica [A]  time = 0.0677735, size = 128, normalized size = 1.45 \[ \frac{1}{2} \left (-2 b n \log \left (f x^m\right ) \text{PolyLog}\left (2,-\frac{e x}{d}\right )+2 b m n \text{PolyLog}\left (3,-\frac{e x}{d}\right )+\frac{a \log ^2\left (f x^m\right )}{m}+2 b \log (x) \log \left (f x^m\right ) \log \left (c (d+e x)^n\right )-b m \log ^2(x) \log \left (c (d+e x)^n\right )-2 b n \log (x) \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )+b m n \log ^2(x) \log \left (\frac{e x}{d}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*Log[f*x^m]^2)/m - b*m*Log[x]^2*Log[c*(d + e*x)^n] + 2*b*Log[x]*Log[f*x^m]*Log[c*(d + e*x)^n] + b*m*n*Log[x
]^2*Log[1 + (e*x)/d] - 2*b*n*Log[x]*Log[f*x^m]*Log[1 + (e*x)/d] - 2*b*n*Log[f*x^m]*PolyLog[2, -((e*x)/d)] + 2*
b*m*n*PolyLog[3, -((e*x)/d)])/2

________________________________________________________________________________________

Maple [C]  time = 0.6, size = 1749, normalized size = 19.9 \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,x)

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (b m \log \left (x\right )^{2} - 2 \, b \log \left (f\right ) \log \left (x\right ) - 2 \, b \log \left (x\right ) \log \left (x^{m}\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) - \int -\frac{b e m n x \log \left (x\right )^{2} - 2 \, b e n x \log \left (f\right ) \log \left (x\right ) + 2 \, b d \log \left (c\right ) \log \left (f\right ) + 2 \, a d \log \left (f\right ) + 2 \,{\left (b e \log \left (c\right ) \log \left (f\right ) + a e \log \left (f\right )\right )} x - 2 \,{\left (b e n x \log \left (x\right ) - b d \log \left (c\right ) - a d -{\left (b e \log \left (c\right ) + a e\right )} x\right )} \log \left (x^{m}\right )}{2 \,{\left (e x^{2} + d x\right )}}\,{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,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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}, 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,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

________________________________________________________________________________________

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}\,{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,x, algorithm="giac")

[Out]

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