3.64.25 \(\int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 (-45 x^2-60 x^3-150 x^4-90 x^5))}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 (-150 x+30 x^2+30 x^3+60 x^4+30 x^5)} \, dx\)

Optimal. Leaf size=30 \[ e^{\frac {x}{-1+\frac {5-\frac {15 e^3}{x}}{x+\left (x+x^2\right )^2}}} \]

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Rubi [F]  time = 12.00, 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 {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}\right ) \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-x^3 - x^4 - 2*x^5 - x^6)/(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5))*(10*x^3 + 14*x^4 + 38*x^5 + 20*x^
6 - 6*x^7 - 6*x^8 - 4*x^9 - x^10 + E^3*(-45*x^2 - 60*x^3 - 150*x^4 - 90*x^5)))/(225*E^6 + 25*x^2 - 10*x^3 - 9*
x^4 - 18*x^5 - 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 + x^10 + E^3*(-150*x + 30*x^2 + 30*x^3 + 60*x^4 + 30*x^5)),x]

[Out]

-Defer[Int][E^(-((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4)))), x] + 75*(2 + 15*E^3)
*Defer[Int][E^(3 - (x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4)))/(15*E^3 - 5*x + x^2
+ x^3 + 2*x^4 + x^5)^2, x] - 25*(2 + 27*E^3)*Defer[Int][x/(E^((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x
+ x^2 + 2*x^3 + x^4)))*(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5)^2), x] + 5*(22 + 9*E^3)*Defer[Int][x^2/(E^((x^
3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4)))*(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5)^2
), x] - 5*(1 - 6*E^3)*Defer[Int][x^3/(E^((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4))
)*(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5)^2), x] + 10*(1 + 3*E^3)*Defer[Int][x^4/(E^((x^3*(1 + x + 2*x^2 + x^
3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4)))*(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5)^2), x] - 10*(1 + 6*E^3
)*Defer[Int][1/(E^((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4)))*(15*E^3 - 5*x + x^2
+ x^3 + 2*x^4 + x^5)), x] + 15*Defer[Int][x/(E^((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3
+ x^4)))*(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x^2 \left (-15 e^3 \left (3+4 x+10 x^2+6 x^3\right )-x \left (-10-14 x-38 x^2-20 x^3+6 x^4+6 x^5+4 x^6+x^7\right )\right )}{\left (15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )\right )^2} \, dx\\ &=\int \left (-\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right )+\frac {5 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (15 e^3 \left (2+15 e^3\right )-5 \left (2+27 e^3\right ) x+\left (22+9 e^3\right ) x^2-\left (1-6 e^3\right ) x^3+2 \left (1+3 e^3\right ) x^4\right )}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}-\frac {5 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (2+12 e^3-3 x\right )}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}\right ) \, dx\\ &=5 \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (15 e^3 \left (2+15 e^3\right )-5 \left (2+27 e^3\right ) x+\left (22+9 e^3\right ) x^2-\left (1-6 e^3\right ) x^3+2 \left (1+3 e^3\right ) x^4\right )}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx-5 \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (2+12 e^3-3 x\right )}{15 e^3-5 x+x^2+x^3+2 x^4+x^5} \, dx-\int \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \, dx\\ &=5 \int \left (\frac {15 \exp \left (3-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (2+15 e^3\right )}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}-\frac {5 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (2+27 e^3\right ) x}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}+\frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (22+9 e^3\right ) x^2}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}+\frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (-1+6 e^3\right ) x^3}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}+\frac {2 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (1+3 e^3\right ) x^4}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2}\right ) \, dx-5 \int \left (\frac {2 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \left (1+6 e^3\right )}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}-\frac {3 \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}\right ) \, dx-\int \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \, dx\\ &=15 \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x}{15 e^3-5 x+x^2+x^3+2 x^4+x^5} \, dx-\left (5 \left (1-6 e^3\right )\right ) \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x^3}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx+\left (10 \left (1+3 e^3\right )\right ) \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x^4}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx-\left (10 \left (1+6 e^3\right )\right ) \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right )}{15 e^3-5 x+x^2+x^3+2 x^4+x^5} \, dx+\left (5 \left (22+9 e^3\right )\right ) \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x^2}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx+\left (75 \left (2+15 e^3\right )\right ) \int \frac {\exp \left (3-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right )}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx-\left (25 \left (2+27 e^3\right )\right ) \int \frac {\exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) x}{\left (15 e^3-5 x+x^2+x^3+2 x^4+x^5\right )^2} \, dx-\int \exp \left (-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 42, normalized size = 1.40 \begin {gather*} e^{-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-x^3 - x^4 - 2*x^5 - x^6)/(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5))*(10*x^3 + 14*x^4 + 38*x^5 +
20*x^6 - 6*x^7 - 6*x^8 - 4*x^9 - x^10 + E^3*(-45*x^2 - 60*x^3 - 150*x^4 - 90*x^5)))/(225*E^6 + 25*x^2 - 10*x^
3 - 9*x^4 - 18*x^5 - 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 + x^10 + E^3*(-150*x + 30*x^2 + 30*x^3 + 60*x^4 + 30*x^5)),
x]

[Out]

E^(-((x^3*(1 + x + 2*x^2 + x^3))/(15*E^3 + x*(-5 + x + x^2 + 2*x^3 + x^4))))

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fricas [A]  time = 0.57, size = 42, normalized size = 1.40 \begin {gather*} e^{\left (-\frac {x^{6} + 2 \, x^{5} + x^{4} + x^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^
6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x
^10+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="fricas")

[Out]

e^(-(x^6 + 2*x^5 + x^4 + x^3)/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3))

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giac [A]  time = 1.67, size = 42, normalized size = 1.40 \begin {gather*} e^{\left (-\frac {x^{6} + 2 \, x^{5} + x^{4} + x^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^
6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x
^10+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="giac")

[Out]

e^(-(x^6 + 2*x^5 + x^4 + x^3)/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3))

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maple [A]  time = 0.43, size = 42, normalized size = 1.40

method	result	size
gosper	\({\mathrm e}^{-\frac {x^{3} \left (x^{3}+2 x^{2}+x +1\right )}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}\)	\(42\)
risch	\({\mathrm e}^{-\frac {x^{3} \left (x^{3}+2 x^{2}+x +1\right )}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}\)	\(42\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+x^{3} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+x^{5} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}-5 x \,{\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+2 x^{4} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+15 \,{\mathrm e}^{3} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}\)	\(333\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^
5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^10+4*
x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(-x^3*(x^3+2*x^2+x+1)/(15*exp(3)+x^5+2*x^4+x^3+x^2-5*x))

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maxima [B]  time = 0.67, size = 63, normalized size = 2.10 \begin {gather*} e^{\left (-x - \frac {5 \, x^{2}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}} + \frac {15 \, x e^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^
6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x
^10+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="maxima")

[Out]

e^(-x - 5*x^2/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3) + 15*x*e^3/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3))

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mupad [B]  time = 5.19, size = 121, normalized size = 4.03 \begin {gather*} {\mathrm {e}}^{-\frac {x^3}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {x^4}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {2\,x^5}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {x^6}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-(x^3 + x^4 + 2*x^5 + x^6)/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))*(6*x^7 - 14*x^4 - 38*x^5 - 2
0*x^6 - 10*x^3 + 6*x^8 + 4*x^9 + x^10 + exp(3)*(45*x^2 + 60*x^3 + 150*x^4 + 90*x^5)))/(225*exp(6) + exp(3)*(30
*x^2 - 150*x + 30*x^3 + 60*x^4 + 30*x^5) + 25*x^2 - 10*x^3 - 9*x^4 - 18*x^5 - 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 +
x^10),x)

[Out]

exp(-x^3/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))*exp(-x^4/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))*ex
p(-(2*x^5)/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))*exp(-x^6/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))

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sympy [B]  time = 7.95, size = 41, normalized size = 1.37 \begin {gather*} e^{\frac {- x^{6} - 2 x^{5} - x^{4} - x^{3}}{x^{5} + 2 x^{4} + x^{3} + x^{2} - 5 x + 15 e^{3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x**5-150*x**4-60*x**3-45*x**2)*exp(3)-x**10-4*x**9-6*x**8-6*x**7+20*x**6+38*x**5+14*x**4+10*x*
*3)*exp((-x**6-2*x**5-x**4-x**3)/(15*exp(3)+x**5+2*x**4+x**3+x**2-5*x))/(225*exp(3)**2+(30*x**5+60*x**4+30*x**
3+30*x**2-150*x)*exp(3)+x**10+4*x**9+6*x**8+6*x**7-5*x**6-18*x**5-9*x**4-10*x**3+25*x**2),x)

[Out]

exp((-x**6 - 2*x**5 - x**4 - x**3)/(x**5 + 2*x**4 + x**3 + x**2 - 5*x + 15*exp(3)))

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