3.33.48 \(\int \frac {e^{\frac {-12+4 x+(-5 x^2-5 e^x x^2-10 x^3) \log (e^{-x} x)}{-3+x}} (15 x+10 x^2-35 x^3+10 x^4+e^x (15 x-20 x^2+5 x^3)+(30 x+85 x^2-20 x^3+e^x (30 x+10 x^2-5 x^3)) \log (e^{-x} x))}{9-6 x+x^2} \, dx\)

Optimal. Leaf size=32 \[ e^{4+\frac {5 x^2 \left (1+e^x+2 x\right ) \log \left (e^{-x} x\right )}{3-x}} \]

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Rubi [F]  time = 24.54, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\exp \left (\frac {-12+4 x+\left (-5 x^2-5 e^x x^2-10 x^3\right ) \log \left (e^{-x} x\right )}{-3+x}\right ) \left (15 x+10 x^2-35 x^3+10 x^4+e^x \left (15 x-20 x^2+5 x^3\right )+\left (30 x+85 x^2-20 x^3+e^x \left (30 x+10 x^2-5 x^3\right )\right ) \log \left (e^{-x} x\right )\right )}{9-6 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-12 + 4*x + (-5*x^2 - 5*E^x*x^2 - 10*x^3)*Log[x/E^x])/(-3 + x))*(15*x + 10*x^2 - 35*x^3 + 10*x^4 + E^
x*(15*x - 20*x^2 + 5*x^3) + (30*x + 85*x^2 - 20*x^3 + E^x*(30*x + 10*x^2 - 5*x^3))*Log[x/E^x]))/(9 - 6*x + x^2
),x]

[Out]

70*E^4*Defer[Int][(x/E^x)^((-5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] - 35*E^4*Log[x/E^x]*Defer[Int][(x/E^x)^((-5*
x^2*(1 + E^x + 2*x))/(-3 + x)), x] + 10*E^4*Defer[Int][E^x/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] - 20
*E^4*Log[x/E^x]*Defer[Int][E^x/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] + 315*E^4*Log[x/E^x]*Defer[Int][
1/((-3 + x)^2*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x] + 45*E^4*Log[x/E^x]*Defer[Int][E^x/((-3 + x)^2*(
x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x] + 210*E^4*Defer[Int][1/((-3 + x)*(x/E^x)^((5*x^2*(1 + E^x + 2*x
))/(-3 + x))), x] + 30*E^4*Defer[Int][E^x/((-3 + x)*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x] - 45*E^4*L
og[x/E^x]*Defer[Int][E^x/((-3 + x)*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x] + 25*E^4*Defer[Int][x/(x/E^
x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] - 20*E^4*Log[x/E^x]*Defer[Int][x/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-
3 + x)), x] + 5*E^4*Defer[Int][(E^x*x)/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] - 5*E^4*Log[x/E^x]*Defer
[Int][(E^x*x)/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x] + 10*E^4*Defer[Int][x^2/(x/E^x)^((5*x^2*(1 + E^x
+ 2*x))/(-3 + x)), x] - 35*E^4*Defer[Int][Defer[Int][(x/E^x)^((-5*x^2*(1 + E^x + 2*x))/(-3 + x)), x], x] + 35*
E^4*Defer[Int][Defer[Int][(x/E^x)^((-5*x^2*(1 + E^x + 2*x))/(-3 + x)), x]/x, x] - 20*E^4*Defer[Int][Defer[Int]
[E^x/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x], x] + 20*E^4*Defer[Int][Defer[Int][E^x/(x/E^x)^((5*x^2*(1
+ E^x + 2*x))/(-3 + x)), x]/x, x] + 315*E^4*Defer[Int][Defer[Int][1/((-3 + x)^2*(x/E^x)^((5*x^2*(1 + E^x + 2*x
))/(-3 + x))), x], x] - 315*E^4*Defer[Int][Defer[Int][1/((-3 + x)^2*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))
), x]/x, x] + 45*E^4*Defer[Int][Defer[Int][E^x/((-3 + x)^2*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x], x]
- 45*E^4*Defer[Int][Defer[Int][E^x/((-3 + x)^2*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x]/x, x] - 45*E^4
*Defer[Int][Defer[Int][E^x/((-3 + x)*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x], x] + 45*E^4*Defer[Int][D
efer[Int][E^x/((-3 + x)*(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x))), x]/x, x] - 20*E^4*Defer[Int][Defer[Int][x
/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x], x] + 20*E^4*Defer[Int][Defer[Int][x/(x/E^x)^((5*x^2*(1 + E^x
+ 2*x))/(-3 + x)), x]/x, x] - 5*E^4*Defer[Int][Defer[Int][(E^x*x)/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)),
x], x] + 5*E^4*Defer[Int][Defer[Int][(E^x*x)/(x/E^x)^((5*x^2*(1 + E^x + 2*x))/(-3 + x)), x]/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-12+4 x+\left (-5 x^2-5 e^x x^2-10 x^3\right ) \log \left (e^{-x} x\right )}{-3+x}\right ) \left (15 x+10 x^2-35 x^3+10 x^4+e^x \left (15 x-20 x^2+5 x^3\right )+\left (30 x+85 x^2-20 x^3+e^x \left (30 x+10 x^2-5 x^3\right )\right ) \log \left (e^{-x} x\right )\right )}{(-3+x)^2} \, dx\\ &=\int \frac {5 e^4 x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \left (\left (1+e^x+2 x\right ) \left (3-4 x+x^2\right )+\left (6+17 x-4 x^2+e^x \left (6+2 x-x^2\right )\right ) \log \left (e^{-x} x\right )\right )}{(3-x)^2} \, dx\\ &=\left (5 e^4\right ) \int \frac {x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \left (\left (1+e^x+2 x\right ) \left (3-4 x+x^2\right )+\left (6+17 x-4 x^2+e^x \left (6+2 x-x^2\right )\right ) \log \left (e^{-x} x\right )\right )}{(3-x)^2} \, dx\\ &=\left (5 e^4\right ) \int \left (\frac {3 x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2}-\frac {4 x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2}+\frac {2 (-1+x) x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{-3+x}+\frac {x^3 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2}+\frac {6 x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2}+\frac {17 x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2}-\frac {4 x^3 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2}-\frac {e^x x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \left (-3+4 x-x^2-6 \log \left (e^{-x} x\right )-2 x \log \left (e^{-x} x\right )+x^2 \log \left (e^{-x} x\right )\right )}{(-3+x)^2}\right ) \, dx\\ &=\left (5 e^4\right ) \int \frac {x^3 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2} \, dx-\left (5 e^4\right ) \int \frac {e^x x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \left (-3+4 x-x^2-6 \log \left (e^{-x} x\right )-2 x \log \left (e^{-x} x\right )+x^2 \log \left (e^{-x} x\right )\right )}{(-3+x)^2} \, dx+\left (10 e^4\right ) \int \frac {(-1+x) x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{-3+x} \, dx+\left (15 e^4\right ) \int \frac {x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2} \, dx-\left (20 e^4\right ) \int \frac {x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}}}{(-3+x)^2} \, dx-\left (20 e^4\right ) \int \frac {x^3 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2} \, dx+\left (30 e^4\right ) \int \frac {x \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2} \, dx+\left (85 e^4\right ) \int \frac {x^2 \left (e^{-x} x\right )^{-\frac {5 x^2 \left (1+e^x+2 x\right )}{-3+x}} \log \left (e^{-x} x\right )}{(-3+x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.50, size = 98, normalized size = 3.06 \begin {gather*} e^{4+\frac {5 x \left (21 (-3+x)+\left (-21+x+e^x x+2 x^2\right ) \log (x)-\left (-21+x+e^x x+2 x^2\right ) \log \left (e^{-x} x\right )\right )}{-3+x}} x^{-\frac {5 \left (-63+\left (1+e^x\right ) x^2+2 x^3\right )}{-3+x}} \left (e^{-x} x\right )^{-\frac {315}{-3+x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(E^((-12 + 4*x + (-5*x^2 - 5*E^x*x^2 - 10*x^3)*Log[x/E^x])/(-3 + x))*(15*x + 10*x^2 - 35*x^3 + 10*x^
4 + E^x*(15*x - 20*x^2 + 5*x^3) + (30*x + 85*x^2 - 20*x^3 + E^x*(30*x + 10*x^2 - 5*x^3))*Log[x/E^x]))/(9 - 6*x
+ x^2),x]

[Out]

E^(4 + (5*x*(21*(-3 + x) + (-21 + x + E^x*x + 2*x^2)*Log[x] - (-21 + x + E^x*x + 2*x^2)*Log[x/E^x]))/(-3 + x))
/(x^((5*(-63 + (1 + E^x)*x^2 + 2*x^3))/(-3 + x))*(x/E^x)^(315/(-3 + x)))

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fricas [A]  time = 0.59, size = 37, normalized size = 1.16 \begin {gather*} e^{\left (-\frac {5 \, {\left (2 \, x^{3} + x^{2} e^{x} + x^{2}\right )} \log \left (x e^{\left (-x\right )}\right ) - 4 \, x + 12}{x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x^3+10*x^2+30*x)*exp(x)-20*x^3+85*x^2+30*x)*log(x/exp(x))+(5*x^3-20*x^2+15*x)*exp(x)+10*x^4-35
*x^3+10*x^2+15*x)*exp(((-5*exp(x)*x^2-10*x^3-5*x^2)*log(x/exp(x))+4*x-12)/(x-3))/(x^2-6*x+9),x, algorithm="fri
cas")

[Out]

e^(-(5*(2*x^3 + x^2*e^x + x^2)*log(x*e^(-x)) - 4*x + 12)/(x - 3))

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giac [B]  time = 1.71, size = 70, normalized size = 2.19 \begin {gather*} e^{\left (-\frac {10 \, x^{3} \log \left (x e^{\left (-x\right )}\right )}{x - 3} - \frac {5 \, x^{2} e^{x} \log \left (x e^{\left (-x\right )}\right )}{x - 3} - \frac {5 \, x^{2} \log \left (x e^{\left (-x\right )}\right )}{x - 3} + \frac {4 \, x}{x - 3} - \frac {12}{x - 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-5*x^3+10*x^2+30*x)*exp(x)-20*x^3+85*x^2+30*x)*log(x/exp(x))+(5*x^3-20*x^2+15*x)*exp(x)+10*x^4-35
*x^3+10*x^2+15*x)*exp(((-5*exp(x)*x^2-10*x^3-5*x^2)*log(x/exp(x))+4*x-12)/(x-3))/(x^2-6*x+9),x, algorithm="gia
c")

[Out]

e^(-10*x^3*log(x*e^(-x))/(x - 3) - 5*x^2*e^x*log(x*e^(-x))/(x - 3) - 5*x^2*log(x*e^(-x))/(x - 3) + 4*x/(x - 3)
- 12/(x - 3))

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maple [C]  time = 0.64, size = 359, normalized size = 11.22

method	result	size
risch	\({\mathrm e}^{-\frac {-10 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3} x^{3}-5 i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3} x^{2}-5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{3} x^{2}+10 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{3}+5 i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{2}-5 i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{2}+5 i \pi \,{\mathrm e}^{x} \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{2}-10 i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{3}+10 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{3}-5 i \pi \,\mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{2}+5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i x \right ) x^{2}+5 i \pi \mathrm {csgn}\left (i x \,{\mathrm e}^{-x}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{-x}\right ) x^{2}+10 x^{2} {\mathrm e}^{x} \ln \relax (x )+20 x^{3} \ln \relax (x )-10 \,{\mathrm e}^{x} \ln \left ({\mathrm e}^{x}\right ) x^{2}-20 \ln \left ({\mathrm e}^{x}\right ) x^{3}+10 x^{2} \ln \relax (x )-10 \ln \left ({\mathrm e}^{x}\right ) x^{2}-8 x +24}{2 \left (x -3\right )}}\)	\(359\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-5*x^3+10*x^2+30*x)*exp(x)-20*x^3+85*x^2+30*x)*ln(x/exp(x))+(5*x^3-20*x^2+15*x)*exp(x)+10*x^4-35*x^3+10
*x^2+15*x)*exp(((-5*exp(x)*x^2-10*x^3-5*x^2)*ln(x/exp(x))+4*x-12)/(x-3))/(x^2-6*x+9),x,method=_RETURNVERBOSE)

[Out]

exp(-1/2*(-10*I*Pi*csgn(I*x*exp(-x))^3*x^3-5*I*Pi*exp(x)*csgn(I*x*exp(-x))^3*x^2-5*I*Pi*csgn(I*x*exp(-x))^3*x^
2+10*I*Pi*csgn(I*x*exp(-x))^2*csgn(I*x)*x^3+5*I*Pi*exp(x)*csgn(I*x*exp(-x))^2*csgn(I*x)*x^2-5*I*Pi*exp(x)*csgn
(I*x*exp(-x))*csgn(I*x)*csgn(I*exp(-x))*x^2+5*I*Pi*exp(x)*csgn(I*x*exp(-x))^2*csgn(I*exp(-x))*x^2-10*I*Pi*csgn
(I*x*exp(-x))*csgn(I*x)*csgn(I*exp(-x))*x^3+10*I*Pi*csgn(I*x*exp(-x))^2*csgn(I*exp(-x))*x^3-5*I*Pi*csgn(I*x*ex
p(-x))*csgn(I*x)*csgn(I*exp(-x))*x^2+5*I*Pi*csgn(I*x*exp(-x))^2*csgn(I*x)*x^2+5*I*Pi*csgn(I*x*exp(-x))^2*csgn(
I*exp(-x))*x^2+10*x^2*exp(x)*ln(x)+20*x^3*ln(x)-10*exp(x)*ln(exp(x))*x^2-20*ln(exp(x))*x^3+10*x^2*ln(x)-10*ln(
exp(x))*x^2-8*x+24)/(x-3))

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maxima [F(-1)]  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((((-5*x^3+10*x^2+30*x)*exp(x)-20*x^3+85*x^2+30*x)*log(x/exp(x))+(5*x^3-20*x^2+15*x)*exp(x)+10*x^4-35
*x^3+10*x^2+15*x)*exp(((-5*exp(x)*x^2-10*x^3-5*x^2)*log(x/exp(x))+4*x-12)/(x-3))/(x^2-6*x+9),x, algorithm="max
ima")

[Out]

Timed out

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mupad [B]  time = 2.16, size = 79, normalized size = 2.47 \begin {gather*} \frac {{\mathrm {e}}^{\frac {4\,x}{x-3}}\,{\mathrm {e}}^{\frac {5\,x^3\,{\mathrm {e}}^x}{x-3}}\,{\mathrm {e}}^{\frac {5\,x^3}{x-3}}\,{\mathrm {e}}^{\frac {10\,x^4}{x-3}}\,{\mathrm {e}}^{-\frac {12}{x-3}}}{x^{\frac {5\,\left (x^2\,{\mathrm {e}}^x+x^2+2\,x^3\right )}{x-3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(log(x*exp(-x))*(5*x^2*exp(x) + 5*x^2 + 10*x^3) - 4*x + 12)/(x - 3))*(15*x + log(x*exp(-x))*(30*x +
85*x^2 - 20*x^3 + exp(x)*(30*x + 10*x^2 - 5*x^3)) + 10*x^2 - 35*x^3 + 10*x^4 + exp(x)*(15*x - 20*x^2 + 5*x^3))
)/(x^2 - 6*x + 9),x)

[Out]

(exp((4*x)/(x - 3))*exp((5*x^3*exp(x))/(x - 3))*exp((5*x^3)/(x - 3))*exp((10*x^4)/(x - 3))*exp(-12/(x - 3)))/x
^((5*(x^2*exp(x) + x^2 + 2*x^3))/(x - 3))

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sympy [F(-1)]  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((((-5*x**3+10*x**2+30*x)*exp(x)-20*x**3+85*x**2+30*x)*ln(x/exp(x))+(5*x**3-20*x**2+15*x)*exp(x)+10*x
**4-35*x**3+10*x**2+15*x)*exp(((-5*exp(x)*x**2-10*x**3-5*x**2)*ln(x/exp(x))+4*x-12)/(x-3))/(x**2-6*x+9),x)

[Out]

Timed out

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