3.363 \(\int \frac{\log (f x^m) (a+b \log (c (d+e x)^n))}{x^2} \, dx\)

Optimal. Leaf size=102 \[ \frac{b e m n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d}-\left (\frac{\log \left (f x^m\right )}{x}+\frac{m}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-\frac{b e n \log \left (\frac{d}{e x}+1\right ) \log \left (f x^m\right )}{d}+\frac{b e m n \log (x)}{d}-\frac{b e m n \log (d+e x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.0954819, antiderivative size = 120, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2426, 2344, 2301, 2317, 2391, 36, 29, 31} \[ -\frac{b e m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d}-\left (\frac{\log \left (f x^m\right )}{x}+\frac{m}{x}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b e n \log ^2\left (f x^m\right )}{2 d m}-\frac{b e n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{d}+\frac{b e m n \log (x)}{d}-\frac{b e m n \log (d+e x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2344

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

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]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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

Mathematica [A]  time = 0.108873, size = 111, normalized size = 1.09 \[ -\frac{2 b e m n x \text{PolyLog}\left (2,-\frac{e x}{d}\right )+2 \left (\log \left (f x^m\right )+m\right ) \left (a d+b d \log \left (c (d+e x)^n\right )+b e n x \log (d+e x)\right )-2 b e n x \log (x) \left (m \log (d+e x)-m \log \left (\frac{e x}{d}+1\right )+\log \left (f x^m\right )+m\right )+b e m n x \log ^2(x)}{2 d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.796, size = 1859, normalized size = 18.2 \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^2,x)

[Out]

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

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Maxima [A]  time = 1.241, size = 219, normalized size = 2.15 \begin{align*} -\frac{1}{2} \,{\left (\frac{2 \,{\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 n}{d} + \frac{2 \, b e n \log \left (e x + d\right )}{d} - \frac{2 \, b e n x \log \left (e x + d\right ) \log \left (x\right ) - b e n x \log \left (x\right )^{2} + 2 \, b e n x \log \left (x\right ) - 2 \, b d \log \left ({\left (e x + d\right )}^{n}\right ) - 2 \, b d \log \left (c\right ) - 2 \, a d}{d x}\right )} m -{\left (b e n{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} + \frac{b \log \left ({\left (e x + d\right )}^{n} c\right )}{x} + \frac{a}{x}\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^2,x, algorithm="maxima")

[Out]

-1/2*(2*(log(e*x/d + 1)*log(x) + dilog(-e*x/d))*b*e*n/d + 2*b*e*n*log(e*x + d)/d - (2*b*e*n*x*log(e*x + d)*log
(x) - b*e*n*x*log(x)^2 + 2*b*e*n*x*log(x) - 2*b*d*log((e*x + d)^n) - 2*b*d*log(c) - 2*a*d)/(d*x))*m - (b*e*n*(
log(e*x + d)/d - log(x)/d) + b*log((e*x + d)^n*c)/x + a/x)*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^{2}}, 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^2,x, algorithm="fricas")

[Out]

integral((b*log((e*x + d)^n*c)*log(f*x^m) + a*log(f*x^m))/x^2, 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**2,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^{2}}\,{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^2,x, algorithm="giac")

[Out]

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