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

Optimal. Leaf size=203 \[ \frac{b e^2 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{2 f^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{e^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \log (e+f x)}{4 f^2}+\frac{b e^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac{3 b e m n x}{4 f}+\frac{1}{4} b m n x^2 \]

[Out]

(-3*b*e*m*n*x)/(4*f) + (b*m*n*x^2)/4 + (e*m*x*(a + b*Log[c*x^n]))/(2*f) - (m*x^2*(a + b*Log[c*x^n]))/4 + (b*e^
2*m*n*Log[e + f*x])/(4*f^2) + (b*e^2*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(2*f^2) - (e^2*m*(a + b*Log[c*x^n])*Log
[e + f*x])/(2*f^2) - (b*n*x^2*Log[d*(e + f*x)^m])/4 + (x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/2 + (b*e^2*m
*n*PolyLog[2, 1 + (f*x)/e])/(2*f^2)

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Rubi [A]  time = 0.126177, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {2395, 43, 2376, 2394, 2315} \[ \frac{b e^2 m n \text{PolyLog}\left (2,\frac{f x}{e}+1\right )}{2 f^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{e^2 m \log (e+f x) \left (a+b \log \left (c x^n\right )\right )}{2 f^2}+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \log (e+f x)}{4 f^2}+\frac{b e^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac{3 b e m n x}{4 f}+\frac{1}{4} b m n x^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*e*m*n*x)/(4*f) + (b*m*n*x^2)/4 + (e*m*x*(a + b*Log[c*x^n]))/(2*f) - (m*x^2*(a + b*Log[c*x^n]))/4 + (b*e^
2*m*n*Log[e + f*x])/(4*f^2) + (b*e^2*m*n*Log[-((f*x)/e)]*Log[e + f*x])/(2*f^2) - (e^2*m*(a + b*Log[c*x^n])*Log
[e + f*x])/(2*f^2) - (b*n*x^2*Log[d*(e + f*x)^m])/4 + (x^2*(a + b*Log[c*x^n])*Log[d*(e + f*x)^m])/2 + (b*e^2*m
*n*PolyLog[2, 1 + (f*x)/e])/(2*f^2)

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rubi steps

\begin{align*} \int x \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right ) \, dx &=\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-(b n) \int \left (\frac{e m}{2 f}-\frac{m x}{4}-\frac{e^2 m \log (e+f x)}{2 f^2 x}+\frac{1}{2} x \log \left (d (e+f x)^m\right )\right ) \, dx\\ &=-\frac{b e m n x}{2 f}+\frac{1}{8} b m n x^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{1}{2} (b n) \int x \log \left (d (e+f x)^m\right ) \, dx+\frac{\left (b e^2 m n\right ) \int \frac{\log (e+f x)}{x} \, dx}{2 f^2}\\ &=-\frac{b e m n x}{2 f}+\frac{1}{8} b m n x^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac{1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )-\frac{\left (b e^2 m n\right ) \int \frac{\log \left (-\frac{f x}{e}\right )}{e+f x} \, dx}{2 f}+\frac{1}{4} (b f m n) \int \frac{x^2}{e+f x} \, dx\\ &=-\frac{b e m n x}{2 f}+\frac{1}{8} b m n x^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac{1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{2 f^2}+\frac{1}{4} (b f m n) \int \left (-\frac{e}{f^2}+\frac{x}{f}+\frac{e^2}{f^2 (e+f x)}\right ) \, dx\\ &=-\frac{3 b e m n x}{4 f}+\frac{1}{4} b m n x^2+\frac{e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac{1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac{b e^2 m n \log (e+f x)}{4 f^2}+\frac{b e^2 m n \log \left (-\frac{f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac{e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac{1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac{1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac{b e^2 m n \text{Li}_2\left (1+\frac{f x}{e}\right )}{2 f^2}\\ \end{align*}

Mathematica [A]  time = 0.121693, size = 208, normalized size = 1.02 \[ \frac{-2 b e^2 m n \text{PolyLog}\left (2,-\frac{f x}{e}\right )+2 a f^2 x^2 \log \left (d (e+f x)^m\right )-2 a e^2 m \log (e+f x)+2 a e f m x-a f^2 m x^2+b \log \left (c x^n\right ) \left (f x \left (2 f x \log \left (d (e+f x)^m\right )+2 e m-f m x\right )-2 e^2 m \log (e+f x)\right )-b f^2 n x^2 \log \left (d (e+f x)^m\right )+b e^2 m n \log (e+f x)+2 b e^2 m n \log (x) \log (e+f x)-2 b e^2 m n \log (x) \log \left (\frac{f x}{e}+1\right )-3 b e f m n x+b f^2 m n x^2}{4 f^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*e*f*m*x - 3*b*e*f*m*n*x - a*f^2*m*x^2 + b*f^2*m*n*x^2 - 2*a*e^2*m*Log[e + f*x] + b*e^2*m*n*Log[e + f*x] +
2*b*e^2*m*n*Log[x]*Log[e + f*x] + 2*a*f^2*x^2*Log[d*(e + f*x)^m] - b*f^2*n*x^2*Log[d*(e + f*x)^m] + b*Log[c*x
^n]*(-2*e^2*m*Log[e + f*x] + f*x*(2*e*m - f*m*x + 2*f*x*Log[d*(e + f*x)^m])) - 2*b*e^2*m*n*Log[x]*Log[1 + (f*x
)/e] - 2*b*e^2*m*n*PolyLog[2, -((f*x)/e)])/(4*f^2)

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Maple [C]  time = 0.319, size = 2041, normalized size = 10.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^2*b*csgn(I*c*x^n)^3-1/4*I/f*Pi*b*e*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*x-1
/4*m*b*ln(x^n)*x^2+1/2*ln(d)*b*x^2*ln(x^n)-1/4*ln(d)*b*n*x^2+1/2*x^2*ln(c)*ln(d)*b-1/4*x^2*ln(c)*b*m+1/2*n*b/f
^2*e^2*m*dilog(-f*x/e)-1/2*m*a*e^2/f^2*ln(f*x+e)+1/2*e*a*m/f*x+1/2*x^2*ln(d)*a+1/4*b*e^2*m*n*ln(f*x+e)/f^2-3/4
*b*e*m*n*x/f+1/4*b*m*n*x^2+1/2*b*e^2*m*n*ln(-f*x/e)*ln(f*x+e)/f^2+(1/2*x^2*b*ln(x^n)+1/4*x^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*csgn(I*
c*x^n)^3+2*b*ln(c)-b*n+2*a))*ln((f*x+e)^m)-1/4*I*x^2*Pi*a*csgn(I*d*(f*x+e)^m)^3-1/4*x^2*a*m+1/2/f*ln(c)*b*e*m*
x-1/2/f^2*e^2*m*ln(f*x+e)*b*ln(c)-5/8*b*e^2*m*n/f^2+1/4*I*x^2*Pi*a*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2-1/4
*I*Pi*csgn(I*d*(f*x+e)^m)^3*b*x^2*ln(x^n)+1/8*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/8
*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*c*x^n)^3+1/8*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x
^2*b*csgn(I*c*x^n)^3+1/8*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^2*b*csgn(I*c)*csgn(I*c*x^n)^2-1/4*I*x^2*ln(c)*Pi*b*csgn(
I*d*(f*x+e)^m)^3-1/4*I*x^2*Pi*ln(d)*b*csgn(I*c*x^n)^3+1/8*I*x^2*Pi*b*m*csgn(I*c*x^n)^3+1/8*I*Pi*b*n*x^2*csgn(I
*d*(f*x+e)^m)^3+1/4*I*x^2*Pi*a*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/4*I*Pi*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(
f*x+e)^m)*b*x^2*ln(x^n)-1/4*I/f*Pi*b*e*m*csgn(I*c*x^n)^3*x+1/4*I/f^2*e^2*m*ln(f*x+e)*Pi*b*csgn(I*c*x^n)^3+1/8*
I*Pi*b*n*x^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)+1/8*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*
x+e)^m)*x^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*x^2*ln(c)*Pi*b*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)
-1/4*I*x^2*Pi*ln(d)*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*I*x^2*Pi*b*m*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)
+1/8*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*Pi^2*csgn(I*(f*x+e)^m)
*csgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/8*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d
*(f*x+e)^m)*x^2*b*csgn(I*c)*csgn(I*c*x^n)^2-1/8*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*c)*csgn(I*c*
x^n)^2-1/8*Pi^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*x^n)*csgn(I*c*x^n)^2-1/8*Pi^2*csgn(I*(f*x+e)^m)*c
sgn(I*d*(f*x+e)^m)^2*x^2*b*csgn(I*c)*csgn(I*c*x^n)^2-1/8*Pi^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*x^2*b*cs
gn(I*x^n)*csgn(I*c*x^n)^2-1/8*Pi^2*csgn(I*d*(f*x+e)^m)^3*x^2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/8*Pi^2*cs
gn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^2*b*csgn(I*c*x^n)^3+1/4*I*x^2*ln(c)*Pi*b*csgn(I*(f*x+e)^m)*csg
n(I*d*(f*x+e)^m)^2+1/4*I*x^2*Pi*ln(d)*b*csgn(I*c)*csgn(I*c*x^n)^2+1/4*I*x^2*Pi*ln(d)*b*csgn(I*x^n)*csgn(I*c*x^
n)^2-1/8*I*Pi*b*n*x^2*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2+1/4*I*Pi*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2*b*x^2*l
n(x^n)+1/4*I*Pi*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)^2*b*x^2*ln(x^n)+1/4*I*x^2*ln(c)*Pi*b*csgn(I*d)*csgn(I*d*
(f*x+e)^m)^2-1/8*I*x^2*Pi*b*m*csgn(I*x^n)*csgn(I*c*x^n)^2-1/4*I*x^2*Pi*a*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*
(f*x+e)^m)-1/8*I*Pi*b*n*x^2*csgn(I*d)*csgn(I*d*(f*x+e)^m)^2-1/8*I*x^2*Pi*b*m*csgn(I*c)*csgn(I*c*x^n)^2+1/4*I/f
^2*e^2*m*ln(f*x+e)*Pi*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2/f*m*b*ln(x^n)*x*e-1/2/f^2*m*b*ln(x^n)*e^2*ln(f
*x+e)+1/4*I/f*Pi*b*e*m*csgn(I*x^n)*csgn(I*c*x^n)^2*x-1/4*I/f^2*e^2*m*ln(f*x+e)*Pi*b*csgn(I*c)*csgn(I*c*x^n)^2-
1/8*Pi^2*csgn(I*d)*csgn(I*(f*x+e)^m)*csgn(I*d*(f*x+e)^m)*x^2*b*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)-1/4*I/f^2*e
^2*m*ln(f*x+e)*Pi*b*csgn(I*x^n)*csgn(I*c*x^n)^2+1/4*I/f*Pi*b*e*m*csgn(I*c)*csgn(I*c*x^n)^2*x

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Maxima [A]  time = 1.82408, size = 363, normalized size = 1.79 \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^{2} m n}{2 \, f^{2}} - \frac{{\left (2 \, a e^{2} m -{\left (e^{2} m n - 2 \, e^{2} m \log \left (c\right )\right )} b\right )} \log \left (f x + e\right )}{4 \, f^{2}} + \frac{2 \, b e^{2} m n \log \left (f x + e\right ) \log \left (x\right ) -{\left ({\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} a -{\left (f^{2} m n - f^{2} n \log \left (d\right ) -{\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} \log \left (c\right )\right )} b\right )} x^{2} +{\left (2 \, a e f m -{\left (3 \, e f m n - 2 \, e f m \log \left (c\right )\right )} b\right )} x +{\left (2 \, b f^{2} x^{2} \log \left (x^{n}\right ) +{\left (2 \, a f^{2} -{\left (f^{2} n - 2 \, f^{2} \log \left (c\right )\right )} b\right )} x^{2}\right )} \log \left ({\left (f x + e\right )}^{m}\right ) +{\left (2 \, b e f m x - 2 \, b e^{2} m \log \left (f x + e\right ) -{\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} b x^{2}\right )} \log \left (x^{n}\right )}{4 \, f^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(log(f*x/e + 1)*log(x) + dilog(-f*x/e))*b*e^2*m*n/f^2 - 1/4*(2*a*e^2*m - (e^2*m*n - 2*e^2*m*log(c))*b)*lo
g(f*x + e)/f^2 + 1/4*(2*b*e^2*m*n*log(f*x + e)*log(x) - ((f^2*m - 2*f^2*log(d))*a - (f^2*m*n - f^2*n*log(d) -
(f^2*m - 2*f^2*log(d))*log(c))*b)*x^2 + (2*a*e*f*m - (3*e*f*m*n - 2*e*f*m*log(c))*b)*x + (2*b*f^2*x^2*log(x^n)
+ (2*a*f^2 - (f^2*n - 2*f^2*log(c))*b)*x^2)*log((f*x + e)^m) + (2*b*e*f*m*x - 2*b*e^2*m*log(f*x + e) - (f^2*m
- 2*f^2*log(d))*b*x^2)*log(x^n))/f^2

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b x \log \left (c x^{n}\right ) + a x\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(x*(a+b*log(c*x^n))*log(d*(f*x+e)^m),x, algorithm="fricas")

[Out]

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

[Out]

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