3.42.75 \(\int \frac {e^{\frac {-50 x-2 x^2}{-43-53 x-2 x^2+e^x (50 x+2 x^2)}} (2150+172 x+6 x^2+e^x (2500 x^2+200 x^3+4 x^4))}{1849+4558 x+2981 x^2+212 x^3+4 x^4+e^x (-4300 x-5472 x^2-412 x^3-8 x^4)+e^{2 x} (2500 x^2+200 x^3+4 x^4)} \, dx\)

Optimal. Leaf size=27 \[ e^{\frac {x}{1+x-e^x x+\frac {-7+x}{2 (25+x)}}} \]

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Rubi [F]  time = 9.06, 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 {-50 x-2 x^2}{-43-53 x-2 x^2+e^x \left (50 x+2 x^2\right )}\right ) \left (2150+172 x+6 x^2+e^x \left (2500 x^2+200 x^3+4 x^4\right )\right )}{1849+4558 x+2981 x^2+212 x^3+4 x^4+e^x \left (-4300 x-5472 x^2-412 x^3-8 x^4\right )+e^{2 x} \left (2500 x^2+200 x^3+4 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-50*x - 2*x^2)/(-43 - 53*x - 2*x^2 + E^x*(50*x + 2*x^2)))*(2150 + 172*x + 6*x^2 + E^x*(2500*x^2 + 200
*x^3 + 4*x^4)))/(1849 + 4558*x + 2981*x^2 + 212*x^3 + 4*x^4 + E^x*(-4300*x - 5472*x^2 - 412*x^3 - 8*x^4) + E^(
2*x)*(2500*x^2 + 200*x^3 + 4*x^4)),x]

[Out]

2150*Defer[Int][1/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))*(-43 - 53*x + 50*E^x*x - 2*x
^2 + 2*E^x*x^2)^2), x] + 2322*Defer[Int][x/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))*(-4
3 - 53*x + 50*E^x*x - 2*x^2 + 2*E^x*x^2)^2), x] + 2742*Defer[Int][x^2/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)
*x + 2*(-1 + E^x)*x^2))*(-43 - 53*x + 50*E^x*x - 2*x^2 + 2*E^x*x^2)^2), x] + 206*Defer[Int][x^3/(E^((2*x*(25 +
x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))*(-43 - 53*x + 50*E^x*x - 2*x^2 + 2*E^x*x^2)^2), x] + 4*Defer
[Int][x^4/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))*(-43 - 53*x + 50*E^x*x - 2*x^2 + 2*E
^x*x^2)^2), x] + 50*Defer[Int][x/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))*(-43 - 53*x +
50*E^x*x - 2*x^2 + 2*E^x*x^2)), x] + 2*Defer[Int][x^2/(E^((2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^
x)*x^2))*(-43 - 53*x + 50*E^x*x - 2*x^2 + 2*E^x*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) \left (2150+172 x+\left (6+2500 e^x\right ) x^2+200 e^x x^3+4 e^x x^4\right )}{\left (43-\left (-53+50 e^x\right ) x-2 \left (-1+e^x\right ) x^2\right )^2} \, dx\\ &=\int \left (\frac {2 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x (25+x)}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2}+\frac {2 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) \left (1075+1161 x+1371 x^2+103 x^3+2 x^4\right )}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x (25+x)}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2} \, dx+2 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) \left (1075+1161 x+1371 x^2+103 x^3+2 x^4\right )}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx\\ &=2 \int \left (\frac {1075 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right )}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}+\frac {1161 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}+\frac {1371 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^2}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}+\frac {103 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^3}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}+\frac {2 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^4}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2}\right ) \, dx+2 \int \left (\frac {25 \exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2}+\frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^2}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^2}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2} \, dx+4 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^4}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx+50 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x}{-43-53 x+50 e^x x-2 x^2+2 e^x x^2} \, dx+206 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^3}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx+2150 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right )}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx+2322 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx+2742 \int \frac {\exp \left (-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}\right ) x^2}{\left (-43-53 x+50 e^x x-2 x^2+2 e^x x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 31, normalized size = 1.15 \begin {gather*} e^{-\frac {2 x (25+x)}{-43+\left (-53+50 e^x\right ) x+2 \left (-1+e^x\right ) x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-50*x - 2*x^2)/(-43 - 53*x - 2*x^2 + E^x*(50*x + 2*x^2)))*(2150 + 172*x + 6*x^2 + E^x*(2500*x^2
+ 200*x^3 + 4*x^4)))/(1849 + 4558*x + 2981*x^2 + 212*x^3 + 4*x^4 + E^x*(-4300*x - 5472*x^2 - 412*x^3 - 8*x^4)
+ E^(2*x)*(2500*x^2 + 200*x^3 + 4*x^4)),x]

[Out]

E^((-2*x*(25 + x))/(-43 + (-53 + 50*E^x)*x + 2*(-1 + E^x)*x^2))

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fricas [A]  time = 2.09, size = 33, normalized size = 1.22 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{2} + 25 \, x\right )}}{2 \, x^{2} - 2 \, {\left (x^{2} + 25 \, x\right )} e^{x} + 53 \, x + 43}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+200*x^3+2500*x^2)*exp(x)+6*x^2+172*x+2150)*exp((-2*x^2-50*x)/((2*x^2+50*x)*exp(x)-2*x^2-53*x
-43))/((4*x^4+200*x^3+2500*x^2)*exp(x)^2+(-8*x^4-412*x^3-5472*x^2-4300*x)*exp(x)+4*x^4+212*x^3+2981*x^2+4558*x
+1849),x, algorithm="fricas")

[Out]

e^(2*(x^2 + 25*x)/(2*x^2 - 2*(x^2 + 25*x)*e^x + 53*x + 43))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (3 \, x^{2} + 2 \, {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2}\right )} e^{x} + 86 \, x + 1075\right )} e^{\left (\frac {2 \, {\left (x^{2} + 25 \, x\right )}}{2 \, x^{2} - 2 \, {\left (x^{2} + 25 \, x\right )} e^{x} + 53 \, x + 43}\right )}}{4 \, x^{4} + 212 \, x^{3} + 2981 \, x^{2} + 4 \, {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2}\right )} e^{\left (2 \, x\right )} - 4 \, {\left (2 \, x^{4} + 103 \, x^{3} + 1368 \, x^{2} + 1075 \, x\right )} e^{x} + 4558 \, x + 1849}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+200*x^3+2500*x^2)*exp(x)+6*x^2+172*x+2150)*exp((-2*x^2-50*x)/((2*x^2+50*x)*exp(x)-2*x^2-53*x
-43))/((4*x^4+200*x^3+2500*x^2)*exp(x)^2+(-8*x^4-412*x^3-5472*x^2-4300*x)*exp(x)+4*x^4+212*x^3+2981*x^2+4558*x
+1849),x, algorithm="giac")

[Out]

integrate(2*(3*x^2 + 2*(x^4 + 50*x^3 + 625*x^2)*e^x + 86*x + 1075)*e^(2*(x^2 + 25*x)/(2*x^2 - 2*(x^2 + 25*x)*e
^x + 53*x + 43))/(4*x^4 + 212*x^3 + 2981*x^2 + 4*(x^4 + 50*x^3 + 625*x^2)*e^(2*x) - 4*(2*x^4 + 103*x^3 + 1368*
x^2 + 1075*x)*e^x + 4558*x + 1849), x)

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maple [A]  time = 0.19, size = 32, normalized size = 1.19

method	result	size
risch	\({\mathrm e}^{-\frac {2 x \left (x +25\right )}{2 \,{\mathrm e}^{x} x^{2}+50 \,{\mathrm e}^{x} x -2 x^{2}-53 x -43}}\)	\(32\)
norman	\(\frac {-53 x \,{\mathrm e}^{\frac {-2 x^{2}-50 x}{\left (2 x^{2}+50 x \right ) {\mathrm e}^{x}-2 x^{2}-53 x -43}}-2 x^{2} {\mathrm e}^{\frac {-2 x^{2}-50 x}{\left (2 x^{2}+50 x \right ) {\mathrm e}^{x}-2 x^{2}-53 x -43}}+50 \,{\mathrm e}^{x} x \,{\mathrm e}^{\frac {-2 x^{2}-50 x}{\left (2 x^{2}+50 x \right ) {\mathrm e}^{x}-2 x^{2}-53 x -43}}+2 \,{\mathrm e}^{x} x^{2} {\mathrm e}^{\frac {-2 x^{2}-50 x}{\left (2 x^{2}+50 x \right ) {\mathrm e}^{x}-2 x^{2}-53 x -43}}-43 \,{\mathrm e}^{\frac {-2 x^{2}-50 x}{\left (2 x^{2}+50 x \right ) {\mathrm e}^{x}-2 x^{2}-53 x -43}}}{2 \,{\mathrm e}^{x} x^{2}+50 \,{\mathrm e}^{x} x -2 x^{2}-53 x -43}\)	\(224\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+200*x^3+2500*x^2)*exp(x)+6*x^2+172*x+2150)*exp((-2*x^2-50*x)/((2*x^2+50*x)*exp(x)-2*x^2-53*x-43))/
((4*x^4+200*x^3+2500*x^2)*exp(x)^2+(-8*x^4-412*x^3-5472*x^2-4300*x)*exp(x)+4*x^4+212*x^3+2981*x^2+4558*x+1849)
,x,method=_RETURNVERBOSE)

[Out]

exp(-2*x*(x+25)/(2*exp(x)*x^2+50*exp(x)*x-2*x^2-53*x-43))

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maxima [B]  time = 0.71, size = 163, normalized size = 6.04 \begin {gather*} e^{\left (\frac {50 \, x e^{x}}{2 \, x^{2} + 2 \, {\left (x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - {\left (4 \, x^{2} + 103 \, x + 43\right )} e^{x} + 53 \, x + 43} - \frac {53 \, x}{2 \, x^{2} + 2 \, {\left (x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - {\left (4 \, x^{2} + 103 \, x + 43\right )} e^{x} + 53 \, x + 43} + \frac {50 \, x}{2 \, x^{2} - 2 \, {\left (x^{2} + 25 \, x\right )} e^{x} + 53 \, x + 43} - \frac {43}{2 \, x^{2} + 2 \, {\left (x^{2} + 25 \, x\right )} e^{\left (2 \, x\right )} - {\left (4 \, x^{2} + 103 \, x + 43\right )} e^{x} + 53 \, x + 43} - \frac {1}{e^{x} - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+200*x^3+2500*x^2)*exp(x)+6*x^2+172*x+2150)*exp((-2*x^2-50*x)/((2*x^2+50*x)*exp(x)-2*x^2-53*x
-43))/((4*x^4+200*x^3+2500*x^2)*exp(x)^2+(-8*x^4-412*x^3-5472*x^2-4300*x)*exp(x)+4*x^4+212*x^3+2981*x^2+4558*x
+1849),x, algorithm="maxima")

[Out]

e^(50*x*e^x/(2*x^2 + 2*(x^2 + 25*x)*e^(2*x) - (4*x^2 + 103*x + 43)*e^x + 53*x + 43) - 53*x/(2*x^2 + 2*(x^2 + 2
5*x)*e^(2*x) - (4*x^2 + 103*x + 43)*e^x + 53*x + 43) + 50*x/(2*x^2 - 2*(x^2 + 25*x)*e^x + 53*x + 43) - 43/(2*x
^2 + 2*(x^2 + 25*x)*e^(2*x) - (4*x^2 + 103*x + 43)*e^x + 53*x + 43) - 1/(e^x - 1))

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mupad [B]  time = 3.25, size = 35, normalized size = 1.30 \begin {gather*} {\mathrm {e}}^{\frac {2\,x^2+50\,x}{53\,x-2\,x^2\,{\mathrm {e}}^x-50\,x\,{\mathrm {e}}^x+2\,x^2+43}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((50*x + 2*x^2)/(53*x - exp(x)*(50*x + 2*x^2) + 2*x^2 + 43))*(172*x + exp(x)*(2500*x^2 + 200*x^3 + 4*x
^4) + 6*x^2 + 2150))/(4558*x - exp(x)*(4300*x + 5472*x^2 + 412*x^3 + 8*x^4) + exp(2*x)*(2500*x^2 + 200*x^3 + 4
*x^4) + 2981*x^2 + 212*x^3 + 4*x^4 + 1849),x)

[Out]

exp((50*x + 2*x^2)/(53*x - 2*x^2*exp(x) - 50*x*exp(x) + 2*x^2 + 43))

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sympy [A]  time = 0.77, size = 32, normalized size = 1.19 \begin {gather*} e^{\frac {- 2 x^{2} - 50 x}{- 2 x^{2} - 53 x + \left (2 x^{2} + 50 x\right ) e^{x} - 43}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+200*x**3+2500*x**2)*exp(x)+6*x**2+172*x+2150)*exp((-2*x**2-50*x)/((2*x**2+50*x)*exp(x)-2*x*
*2-53*x-43))/((4*x**4+200*x**3+2500*x**2)*exp(x)**2+(-8*x**4-412*x**3-5472*x**2-4300*x)*exp(x)+4*x**4+212*x**3
+2981*x**2+4558*x+1849),x)

[Out]

exp((-2*x**2 - 50*x)/(-2*x**2 - 53*x + (2*x**2 + 50*x)*exp(x) - 43))

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