3.1.72 \(\int x (a+b \log (c x^n)) \log (d (e+f x)^m) \, dx\) [72]

Optimal. Leaf size=203 \[ -\frac {3 b e m n x}{4 f}+\frac {1}{4} b m n x^2+\frac {e m x \left (a+b \log \left (c x^n\right )\right )}{2 f}-\frac {1}{4} m x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {b e^2 m n \log (e+f x)}{4 f^2}+\frac {b e^2 m n \log \left (-\frac {f x}{e}\right ) \log (e+f x)}{2 f^2}-\frac {e^2 m \left (a+b \log \left (c x^n\right )\right ) \log (e+f x)}{2 f^2}-\frac {1}{4} b n x^2 \log \left (d (e+f x)^m\right )+\frac {1}{2} x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d (e+f x)^m\right )+\frac {b e^2 m n \text {Li}_2\left (1+\frac {f x}{e}\right )}{2 f^2} \]

[Out]

-3/4*b*e*m*n*x/f+1/4*b*m*n*x^2+1/2*e*m*x*(a+b*ln(c*x^n))/f-1/4*m*x^2*(a+b*ln(c*x^n))+1/4*b*e^2*m*n*ln(f*x+e)/f
^2+1/2*b*e^2*m*n*ln(-f*x/e)*ln(f*x+e)/f^2-1/2*e^2*m*(a+b*ln(c*x^n))*ln(f*x+e)/f^2-1/4*b*n*x^2*ln(d*(f*x+e)^m)+
1/2*x^2*(a+b*ln(c*x^n))*ln(d*(f*x+e)^m)+1/2*b*e^2*m*n*polylog(2,1+f*x/e)/f^2

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Rubi [A]
time = 0.09, antiderivative size = 203, normalized size of antiderivative = 1.00, 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 = {2442, 45, 2423, 2441, 2352} \begin {gather*} \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 \end {gather*}

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 45

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 2352

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

Rule 2423

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 2441

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 2442

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]

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*}

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Mathematica [A]
time = 0.09, size = 208, normalized size = 1.02 \begin {gather*} \frac {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 \left (d (e+f x)^m\right )-b f^2 n x^2 \log \left (d (e+f x)^m\right )+b \log \left (c x^n\right ) \left (-2 e^2 m \log (e+f x)+f x \left (2 e m-f m x+2 f x \log \left (d (e+f x)^m\right )\right )\right )-2 b e^2 m n \log (x) \log \left (1+\frac {f x}{e}\right )-2 b e^2 m n \text {Li}_2\left (-\frac {f x}{e}\right )}{4 f^2} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.38, size = 2041, normalized size = 10.05

method	result	size
risch	\(\text {Expression too large to display}\)	\(2041\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.39, size = 260, normalized size = 1.28 \begin {gather*} -\frac {{\left (\log \left (f x e^{\left (-1\right )} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-f x e^{\left (-1\right )}\right )\right )} b m n e^{2}}{2 \, f^{2}} + \frac {{\left ({\left (m n - 2 \, m \log \left (c\right )\right )} b - 2 \, a m\right )} e^{2} \log \left (f x + e\right )}{4 \, f^{2}} + \frac {2 \, b m n e^{2} \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 f m - {\left (3 \, f m n - 2 \, f m \log \left (c\right )\right )} b\right )} x e + {\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 f m x e - {\left (f^{2} m - 2 \, f^{2} \log \left (d\right )\right )} b x^{2} - 2 \, b m e^{2} \log \left (f x + e\right )\right )} \log \left (x^{n}\right )}{4 \, f^{2}} \end {gather*}

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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