3.73 \(\int (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\)
Optimal. Leaf size=117 \[ -\frac{b e m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac{b e n \log \left (-\frac{f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+2 b m n x \]
[Out]
2*b*m*n*x - m*x*(a + b*Log[c*x^n]) - (b*n*(e + f*x)*Log[d*(e + f*x)^m])/f - (b*e*n*Log[-((f*x)/e)]*Log[d*(e +
f*x)^m])/f + ((e + f*x)*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/f - (b*e*m*n*PolyLog[2, 1 + (f*x)/e])/f
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Rubi [A] time = 0.145004, antiderivative size = 117, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used =
{2389, 2295, 2370, 2411, 43, 2351, 2317, 2391} \[ -\frac{b e m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-m x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac{b e n \log \left (-\frac{f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+2 b m n x \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Log[c*x^n])*Log[d*(e + f*x)^m],x]
[Out]
2*b*m*n*x - m*x*(a + b*Log[c*x^n]) - (b*n*(e + f*x)*Log[d*(e + f*x)^m])/f - (b*e*n*Log[-((f*x)/e)]*Log[d*(e +
f*x)^m])/f + ((e + f*x)*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/f - (b*e*m*n*PolyLog[2, 1 + (f*x)/e])/f
Rule 2389
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
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 2370
Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> With[
{u = IntHide[Log[d*(e + f*x^m)^r], x]}, Dist[(a + b*Log[c*x^n])^p, u, x] - Dist[b*n*p, Int[Dist[(a + b*Log[c*x
^n])^(p - 1)/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p, 0] && RationalQ[m] && (EqQ[
p, 1] || (FractionQ[m] && IntegerQ[1/m]) || (EqQ[r, 1] && EqQ[m, 1] && EqQ[d*e, 1]))
Rule 2411
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]
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 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 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=-m x \left (a+b \log \left (c x^n\right )\right )+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-(b n) \int \left (-m+\frac{(e+f x) \log \left (d (e+f x)^m\right )}{f x}\right ) \, dx\\ &=b m n x-m x \left (a+b \log \left (c x^n\right )\right )+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac{(b n) \int \frac{(e+f x) \log \left (d (e+f x)^m\right )}{x} \, dx}{f}\\ &=b m n x-m x \left (a+b \log \left (c x^n\right )\right )+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac{(b n) \operatorname{Subst}\left (\int \frac{x \log \left (d x^m\right )}{-\frac{e}{f}+\frac{x}{f}} \, dx,x,e+f x\right )}{f^2}\\ &=b m n x-m x \left (a+b \log \left (c x^n\right )\right )+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac{(b n) \operatorname{Subst}\left (\int \left (f \log \left (d x^m\right )-\frac{e f \log \left (d x^m\right )}{e-x}\right ) \, dx,x,e+f x\right )}{f^2}\\ &=b m n x-m x \left (a+b \log \left (c x^n\right )\right )+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac{(b n) \operatorname{Subst}\left (\int \log \left (d x^m\right ) \, dx,x,e+f x\right )}{f}+\frac{(b e n) \operatorname{Subst}\left (\int \frac{\log \left (d x^m\right )}{e-x} \, dx,x,e+f x\right )}{f}\\ &=2 b m n x-m x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac{b e n \log \left (-\frac{f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}+\frac{(b e m n) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{e}\right )}{x} \, dx,x,e+f x\right )}{f}\\ &=2 b m n x-m x \left (a+b \log \left (c x^n\right )\right )-\frac{b n (e+f x) \log \left (d (e+f x)^m\right )}{f}-\frac{b e n \log \left (-\frac{f x}{e}\right ) \log \left (d (e+f x)^m\right )}{f}+\frac{(e+f x) \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )}{f}-\frac{b e m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{f}\\ \end{align*}
Mathematica [A] time = 0.0685639, size = 152, normalized size = 1.3 \[ \frac{b e m n \text{PolyLog}\left (2,-\frac{f x}{e}\right )+a f x \log \left (d (e+f x)^m\right )+a e \log \left (d (e+f x)^m\right )-a f m x+b \log \left (c x^n\right ) \left (f x \left (\log \left (d (e+f x)^m\right )-m\right )+e m \log (e+f x)\right )-b f n x \log \left (d (e+f x)^m\right )-b e m n \log (e+f x)-b e m n \log (x) \log (e+f x)+b e m n \log (x) \log \left (\frac{f x}{e}+1\right )+2 b f m n x}{f} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Log[c*x^n])*Log[d*(e + f*x)^m],x]
[Out]
(-(a*f*m*x) + 2*b*f*m*n*x - b*e*m*n*Log[e + f*x] - b*e*m*n*Log[x]*Log[e + f*x] + a*e*Log[d*(e + f*x)^m] + a*f*
x*Log[d*(e + f*x)^m] - b*f*n*x*Log[d*(e + f*x)^m] + b*Log[c*x^n]*(e*m*Log[e + f*x] + f*x*(-m + Log[d*(e + f*x)
^m])) + b*e*m*n*Log[x]*Log[1 + (f*x)/e] + b*e*m*n*PolyLog[2, -((f*x)/e)])/f
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Maple [C] time = 0.274, size = 1762, normalized size = 15.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*ln(c*x^n))*ln(d*(f*x+e)^m),x)
[Out]
-1/2*I*m/f*e*ln(f*x+e)*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+a*m/f*e*ln(f*x+e)+b*e*m*n/f+2*b*m*n*x+ln(d)*a*
x-m*b*ln(x^n)*x+ln(x^n)*ln(d)*x*b-m*ln(c)*b*x+ln(c)*ln(d)*b*x-ln(d)*b*n*x-a*m*x+(b*x*ln(x^n)+1/2*(-I*b*Pi*csgn
(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csg
n(I*c*x^n)^3+2*b*ln(c)-2*b*n+2*a)*x)*ln((f*x+e)^m)-n*b*e*m/f*dilog(-f*x/e)+m/f*e*ln(f*x+e)*b*ln(c)-n*b*e*m/f*l
n(f*x+e)*ln(-f*x/e)-1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*d*(f*x+e)^m)^3-1/2*I*Pi*a*x*csgn(I*d*(f*x+e)^m)^3-1/4*
Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/2*I*m/f*e*ln(f*
x+e)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*m/f*e*ln(f*x+e)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/2*I*Pi*b*n*x*csgn
(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)-1/2*I*ln(x^n)*Pi*x*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^
m)+1/2*I*m*Pi*b*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/2*I*Pi*ln(d)*b*x*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1
/2*I*ln(c)*Pi*b*x*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(
I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*(f*x+e)^m)*
csgn(I*d*(f*x+e)^m)^2+1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)
+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/2*I*m/f*e*ln(f*x+e)*Pi*b*c
sgn(I*c*x^n)^3+m/f*b*ln(x^n)*e*ln(f*x+e)+1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^m)^3+1/4*Pi
^2*x*b*csgn(I*c*x^n)^3*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*(f*x+e)^m)*csgn(I*d
*(f*x+e)^m)^2+1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*d*(f*x+e)^m)^3+1/2*I*Pi*a*x*csgn(I*d)*csgn(I*d*(f*
x+e)^m)^2+1/2*I*Pi*a*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/2*I*m*Pi*b*x*csgn(I*c*x^n)^3-1/2*I*ln(x^n)*Pi
*x*b*csgn(I*d*(f*x+e)^m)^3+1/2*I*Pi*b*n*x*csgn(I*d*(f*x+e)^m)^3-1/2*I*ln(c)*Pi*b*x*csgn(I*d*(f*x+e)^m)^3-1/2*I
*Pi*ln(d)*b*x*csgn(I*c*x^n)^3-b*e*m*n/f*ln(f*x+e)-1/2*I*m*Pi*b*x*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*Pi^2*x*b*csgn
(I*c)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/4*Pi^2*x*b*csgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*d)*csgn(
I*d*(f*x+e)^m)^2-1/4*Pi^2*x*b*csgn(I*c)*csgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/4*Pi^2*x*b*c
sgn(I*x^n)*csgn(I*c*x^n)^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/4*Pi^2*x*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c
*x^n)*csgn(I*d*(f*x+e)^m)^3-1/4*Pi^2*x*b*csgn(I*c*x^n)^3*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/2*I
*ln(x^n)*Pi*x*b*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/2*I*ln(x^n)*Pi*x*b*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1
/2*I*Pi*b*n*x*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/2*I*Pi*b*n*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/2*I*Pi*
ln(d)*b*x*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*Pi*a*x*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/2*I*ln(c)
*Pi*b*x*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2+1/2*I*ln(c)*Pi*b*x*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/2*I*Pi*ln
(d)*b*x*csgn(I*c)*csgn(I*c*x^n)^2-1/2*I*m*Pi*b*x*csgn(I*c)*csgn(I*c*x^n)^2
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Maxima [A] time = 1.72017, size = 254, normalized size = 2.17 \begin{align*} \frac{{\left (\log \left (\frac{f x}{e} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-\frac{f x}{e}\right )\right )} b e m n}{f} + \frac{{\left (a e m -{\left (e m n - e m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{f} - \frac{b e m n \log \left (f x + e\right ) \log \left (x\right ) +{\left ({\left (f m - f \log \left (d\right )\right )} a -{\left (2 \, f m n - f n \log \left (d\right ) -{\left (f m - f \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x -{\left (b f x \log \left (x^{n}\right ) -{\left ({\left (f n - f \log \left (c\right )\right )} b - a f\right )} x\right )} \log \left ({\left (f x + e\right )}^{m}\right ) -{\left (b e m \log \left (f x + e\right ) -{\left (f m - f \log \left (d\right )\right )} b x\right )} \log \left (x^{n}\right )}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="maxima")
[Out]
(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e*m*n/f + (a*e*m - (e*m*n - e*m*log(c))*b)*log(f*x + e)/f - (b*e*m*n
*log(f*x + e)*log(x) + ((f*m - f*log(d))*a - (2*f*m*n - f*n*log(d) - (f*m - f*log(d))*log(c))*b)*x - (b*f*x*lo
g(x^n) - ((f*n - f*log(c))*b - a*f)*x)*log((f*x + e)^m) - (b*e*m*log(f*x + e) - (f*m - f*log(d))*b*x)*log(x^n)
)/f
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="fricas")
[Out]
integral((b*log(c*x^n) + a)*log((f*x + e)^m*d), 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((a+b*ln(c*x**n))*ln(d*(f*x+e)**m),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x + e\right )}^{m} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="giac")
[Out]
integrate((b*log(c*x^n) + a)*log((f*x + e)^m*d), x)