3.84.65 \(\int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} (-2+2 x-12 x^2-10 x^3)}{1-2 x-9 x^2+10 x^3+25 x^4} \, dx\)

Optimal. Leaf size=23 \[ e^{\frac {2 e^{1-x} x}{-1+x (1+5 x)}} \]

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Rubi [F]  time = 4.80, 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 {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} \left (-2+2 x-12 x^2-10 x^3\right )}{1-2 x-9 x^2+10 x^3+25 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))*(-2 + 2*x - 12*x^2 - 10*x^3))/(1 - 2*x - 9*x^2 + 10*x^3 + 25
*x^4),x]

[Out]

(-400*Defer[Int][E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(-1 + Sqrt[21] - 10*x)^2, x])/21 - (20*(1 - Sqrt
[21])*Defer[Int][E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(-1 + Sqrt[21] - 10*x)^2, x])/21 - (20*Defer[Int
][E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(-1 + Sqrt[21] - 10*x), x])/Sqrt[21] - (2*(7 + 3*Sqrt[21])*Defe
r[Int][E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(1 - Sqrt[21] + 10*x), x])/7 - (400*Defer[Int][E^(1 - x +
(2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(1 + Sqrt[21] + 10*x)^2, x])/21 - (20*(1 + Sqrt[21])*Defer[Int][E^(1 - x + (
2*E^(1 - x)*x)/(-1 + x + 5*x^2))/(1 + Sqrt[21] + 10*x)^2, x])/21 - (20*Defer[Int][E^(1 - x + (2*E^(1 - x)*x)/(
-1 + x + 5*x^2))/(1 + Sqrt[21] + 10*x), x])/Sqrt[21] - (2*(7 - 3*Sqrt[21])*Defer[Int][E^(1 - x + (2*E^(1 - x)*
x)/(-1 + x + 5*x^2))/(1 + Sqrt[21] + 10*x), x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} (-2+x)}{\left (-1+x+5 x^2\right )^2}-\frac {2 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} (1+x)}{-1+x+5 x^2}\right ) \, dx\\ &=2 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} (-2+x)}{\left (-1+x+5 x^2\right )^2} \, dx-2 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} (1+x)}{-1+x+5 x^2} \, dx\\ &=-\left (2 \int \left (\frac {\left (1+3 \sqrt {\frac {3}{7}}\right ) e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1-\sqrt {21}+10 x}+\frac {\left (1-3 \sqrt {\frac {3}{7}}\right ) e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x}\right ) \, dx\right )+2 \int \left (-\frac {2 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (-1+x+5 x^2\right )^2}+\frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} x}{\left (-1+x+5 x^2\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}} x}{\left (-1+x+5 x^2\right )^2} \, dx-4 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (-1+x+5 x^2\right )^2} \, dx-\frac {1}{7} \left (2 \left (7-3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x} \, dx-\frac {1}{7} \left (2 \left (7+3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1-\sqrt {21}+10 x} \, dx\\ &=2 \int \left (\frac {10 \left (-1+\sqrt {21}\right ) e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \left (-1+\sqrt {21}-10 x\right )^2}-\frac {10 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \sqrt {21} \left (-1+\sqrt {21}-10 x\right )}+\frac {10 \left (-1-\sqrt {21}\right ) e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \left (1+\sqrt {21}+10 x\right )^2}-\frac {10 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \sqrt {21} \left (1+\sqrt {21}+10 x\right )}\right ) \, dx-4 \int \left (\frac {100 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \left (-1+\sqrt {21}-10 x\right )^2}+\frac {100 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \sqrt {21} \left (-1+\sqrt {21}-10 x\right )}+\frac {100 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \left (1+\sqrt {21}+10 x\right )^2}+\frac {100 e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{21 \sqrt {21} \left (1+\sqrt {21}+10 x\right )}\right ) \, dx-\frac {1}{7} \left (2 \left (7-3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x} \, dx-\frac {1}{7} \left (2 \left (7+3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1-\sqrt {21}+10 x} \, dx\\ &=-\left (\frac {400}{21} \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (-1+\sqrt {21}-10 x\right )^2} \, dx\right )-\frac {400}{21} \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (1+\sqrt {21}+10 x\right )^2} \, dx-\frac {20 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{-1+\sqrt {21}-10 x} \, dx}{21 \sqrt {21}}-\frac {20 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x} \, dx}{21 \sqrt {21}}-\frac {400 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{-1+\sqrt {21}-10 x} \, dx}{21 \sqrt {21}}-\frac {400 \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x} \, dx}{21 \sqrt {21}}-\frac {1}{7} \left (2 \left (7-3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1+\sqrt {21}+10 x} \, dx-\frac {1}{21} \left (20 \left (1-\sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (-1+\sqrt {21}-10 x\right )^2} \, dx-\frac {1}{21} \left (20 \left (1+\sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{\left (1+\sqrt {21}+10 x\right )^2} \, dx-\frac {1}{7} \left (2 \left (7+3 \sqrt {21}\right )\right ) \int \frac {e^{1-x+\frac {2 e^{1-x} x}{-1+x+5 x^2}}}{1-\sqrt {21}+10 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.28, size = 22, normalized size = 0.96 \begin {gather*} e^{\frac {2 e^{1-x} x}{-1+x+5 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(1 - x + (2*E^(1 - x)*x)/(-1 + x + 5*x^2))*(-2 + 2*x - 12*x^2 - 10*x^3))/(1 - 2*x - 9*x^2 + 10*x^
3 + 25*x^4),x]

[Out]

E^((2*E^(1 - x)*x)/(-1 + x + 5*x^2))

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fricas [A]  time = 1.33, size = 40, normalized size = 1.74 \begin {gather*} e^{\left (x - \frac {5 \, x^{3} - 4 \, x^{2} - 2 \, x e^{\left (-x + 1\right )} - 2 \, x + 1}{5 \, x^{2} + x - 1} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-12*x^2+2*x-2)*exp(-x+1)*exp(2*x*exp(-x+1)/(5*x^2+x-1))/(25*x^4+10*x^3-9*x^2-2*x+1),x, algor
ithm="fricas")

[Out]

e^(x - (5*x^3 - 4*x^2 - 2*x*e^(-x + 1) - 2*x + 1)/(5*x^2 + x - 1) - 1)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (5 \, x^{3} + 6 \, x^{2} - x + 1\right )} e^{\left (-x + \frac {2 \, x e^{\left (-x + 1\right )}}{5 \, x^{2} + x - 1} + 1\right )}}{25 \, x^{4} + 10 \, x^{3} - 9 \, x^{2} - 2 \, x + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-12*x^2+2*x-2)*exp(-x+1)*exp(2*x*exp(-x+1)/(5*x^2+x-1))/(25*x^4+10*x^3-9*x^2-2*x+1),x, algor
ithm="giac")

[Out]

integrate(-2*(5*x^3 + 6*x^2 - x + 1)*e^(-x + 2*x*e^(-x + 1)/(5*x^2 + x - 1) + 1)/(25*x^4 + 10*x^3 - 9*x^2 - 2*
x + 1), x)

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maple [A]  time = 0.29, size = 21, normalized size = 0.91

method	result	size
risch	\({\mathrm e}^{\frac {2 x \,{\mathrm e}^{1-x}}{5 x^{2}+x -1}}\)	\(21\)
norman	\(\frac {x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{1-x}}{5 x^{2}+x -1}}+5 x^{2} {\mathrm e}^{\frac {2 x \,{\mathrm e}^{1-x}}{5 x^{2}+x -1}}-{\mathrm e}^{\frac {2 x \,{\mathrm e}^{1-x}}{5 x^{2}+x -1}}}{5 x^{2}+x -1}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-10*x^3-12*x^2+2*x-2)*exp(1-x)*exp(2*x*exp(1-x)/(5*x^2+x-1))/(25*x^4+10*x^3-9*x^2-2*x+1),x,method=_RETURN
VERBOSE)

[Out]

exp(2*x*exp(1-x)/(5*x^2+x-1))

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maxima [A]  time = 0.52, size = 20, normalized size = 0.87 \begin {gather*} e^{\left (\frac {2 \, x e^{\left (-x + 1\right )}}{5 \, x^{2} + x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x^3-12*x^2+2*x-2)*exp(-x+1)*exp(2*x*exp(-x+1)/(5*x^2+x-1))/(25*x^4+10*x^3-9*x^2-2*x+1),x, algor
ithm="maxima")

[Out]

e^(2*x*e^(-x + 1)/(5*x^2 + x - 1))

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mupad [B]  time = 5.53, size = 20, normalized size = 0.87 \begin {gather*} {\mathrm {e}}^{\frac {2\,x\,{\mathrm {e}}^{-x}\,\mathrm {e}}{5\,x^2+x-1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((2*x*exp(1 - x))/(x + 5*x^2 - 1))*exp(1 - x)*(12*x^2 - 2*x + 10*x^3 + 2))/(10*x^3 - 9*x^2 - 2*x + 25
*x^4 + 1),x)

[Out]

exp((2*x*exp(-x)*exp(1))/(x + 5*x^2 - 1))

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sympy [A]  time = 0.30, size = 17, normalized size = 0.74 \begin {gather*} e^{\frac {2 x e^{1 - x}}{5 x^{2} + x - 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*x**3-12*x**2+2*x-2)*exp(-x+1)*exp(2*x*exp(-x+1)/(5*x**2+x-1))/(25*x**4+10*x**3-9*x**2-2*x+1),x)

[Out]

exp(2*x*exp(1 - x)/(5*x**2 + x - 1))

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