3.361 \(\int \log (f x^m) (a+b \log (c (d+e x)^n)) \, dx\)
Optimal. Leaf size=99 \[ \frac{b d m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b d n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{e}-\frac{b d m n \log (d+e x)}{e}-b n x \log \left (f x^m\right )+2 b m n x \]
[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
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Rubi [A] time = 0.0938871, antiderivative size = 99, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used =
{2422, 43, 2351, 2295, 2317, 2391} \[ \frac{b d m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e}-x \left (m-\log \left (f x^m\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac{b d n \log \left (\frac{e x}{d}+1\right ) \log \left (f x^m\right )}{e}-\frac{b d m n \log (d+e x)}{e}-b n x \log \left (f x^m\right )+2 b m n x \]
Antiderivative was successfully verified.
[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 2422
Int[Log[(f_.)*(x_)^(m_.)]*((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> -Simp[x*(m - Log[
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, In
t[x/(d + e*x), x], x]) /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Rule 43
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 2351
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 2295
Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{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]
Rubi steps
\begin{align*} \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 )-(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] time = 0.0782515, size = 116, normalized size = 1.17 \[ \frac{b d m n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\log \left (f x^m\right ) \left (e x \left (a+b \log \left (c (d+e x)^n\right )-b n\right )+b d n \log (d+e x)\right )-m \left (a e x+b e x \log \left (c (d+e x)^n\right )+b d n (\log (x)+1) \log (d+e x)-b d n \log (x) \log \left (\frac{e x}{d}+1\right )-2 b e n x\right )}{e} \]
Antiderivative was successfully verified.
[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
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Maple [C] time = 0.938, size = 1724, normalized size = 17.4 \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)
[Out]
-1/2*I*b*d*n/e*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/4*Pi^2*x*b*csgn(I*x^m)*csgn(I*f*x^m)^2*csgn(
I*c)*csgn(I*c*(e*x+d)^n)^2-1/4*Pi^2*x*b*csgn(I*f*x^m)^3*csgn(I*c*(e*x+d)^n)^3-1/2*I*Pi*a*x*csgn(I*f*x^m)^3-n*b
*ln(x^m)*x+ln(x^m)*ln(c)*x*b+ln(f)*ln(c)*b*x-n*ln(f)*b*x+ln(f)*a*x+(b*x*ln(x^m)+1/2*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)*x)*ln((e*x+d)^n)-m*b*d*n/e*ln(e*x+d)*ln(-e*x/d)-m*ln(c)*b*x-a*m*x+ln(x^m)*x*a+1/e*n*b*ln(x^m)*d*ln(e*x+
d)-1/2*I*b*d*n/e*ln(e*x+d)*Pi*csgn(I*f*x^m)^3+2*b*m*n*x+1/2*I*b*d*n/e*ln(e*x+d)*Pi*csgn(I*f)*csgn(I*f*x^m)^2+1
/2*I*b*d*n/e*ln(e*x+d)*Pi*csgn(I*x^m)*csgn(I*f*x^m)^2-1/4*Pi^2*x*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*csgn(I*
c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*I*m*Pi*b*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/2*I*m*Pi*b*x*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/4*Pi^2*x*b*csgn(I*f*x^m)^3*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-
1/4*Pi^2*x*b*csgn(I*f)*csgn(I*f*x^m)^2*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-m*b*d*n/e*dilog(-e*x/d)+1/4*Pi^2*x*b*cs
gn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*Pi^2*x*b*csgn(I*f)*csgn(I*x^m)*csgn(I*f*
x^m)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/4*Pi^2*x*b*csgn(I*f)*csgn(I*f*x^m)^2*csgn(I*c)*csgn(I*(e*x+d)^n
)*csgn(I*c*(e*x+d)^n)-1/2*I*ln(x^m)*Pi*x*b*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*m*Pi*b*x*csgn
(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*Pi*ln(c)*b*x*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I*n*Pi*b*x*csgn
(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/2*I*Pi*ln(c)*b*x*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)-1/2*I*ln(f)*Pi*b*x*csgn
(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*I*ln(f)*Pi*b*x*csgn(I*c*(e*x+d)^n)^3-1/2*I*n*Pi*b*x*csgn(I*x^m
)*csgn(I*f*x^m)^2+1/2*I*ln(f)*Pi*b*x*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/2*I*ln(f)*Pi*b*x*csgn(I*(e*x+d)^n)*csgn
(I*c*(e*x+d)^n)^2+1/2*I*Pi*a*x*csgn(I*f)*csgn(I*f*x^m)^2+1/2*I*Pi*a*x*csgn(I*x^m)*csgn(I*f*x^m)^2+1/2*I*m*Pi*b
*x*csgn(I*c*(e*x+d)^n)^3-1/2*I*ln(x^m)*Pi*x*b*csgn(I*c*(e*x+d)^n)^3-1/4*Pi^2*x*b*csgn(I*f)*csgn(I*x^m)*csgn(I*
f*x^m)*csgn(I*c*(e*x+d)^n)^3+1/2*I*ln(x^m)*Pi*x*b*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/2*I*ln(x^m)*Pi*x*b*csgn(I*
(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+1/4*Pi^2*x*b*csgn(I*x^m)*csgn(I*f*x^m)^2*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c
*(e*x+d)^n)+b*d*n/e*ln(e*x+d)*ln(f)-1/4*Pi^2*x*b*csgn(I*f)*csgn(I*f*x^m)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^
n)^2-1/4*Pi^2*x*b*csgn(I*x^m)*csgn(I*f*x^m)^2*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*n*Pi*b*x*csgn(I*f)
*csgn(I*f*x^m)^2+1/4*Pi^2*x*b*csgn(I*f*x^m)^3*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/4*Pi^2*x*b*csgn(I*f*x^m)^3*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+b*d*m*n/e+1/4*Pi^2*x*b*csgn(I*f)*csgn(I*f*x^m)^2*csgn(I*c*(e*x+d)^n)^3+1/
4*Pi^2*x*b*csgn(I*x^m)*csgn(I*f*x^m)^2*csgn(I*c*(e*x+d)^n)^3-1/2*I*Pi*ln(c)*b*x*csgn(I*f*x^m)^3+1/2*I*n*Pi*b*x
*csgn(I*f*x^m)^3-1/2*I*Pi*a*x*csgn(I*f)*csgn(I*x^m)*csgn(I*f*x^m)+1/2*I*Pi*ln(c)*b*x*csgn(I*f)*csgn(I*f*x^m)^2
-b*d*m*n*ln(e*x+d)/e
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Maxima [A] time = 1.22359, size = 188, normalized size = 1.9 \begin{align*} -{\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 ) \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, 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)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (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\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, algorithm="fricas")
[Out]
integral(b*log((e*x + d)^n*c)*log(f*x^m) + a*log(f*x^m), 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)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{m}\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, algorithm="giac")
[Out]
integrate((b*log((e*x + d)^n*c) + a)*log(f*x^m), x)