3.31.99 \(\int \frac {e^{-\frac {x^2}{e^x+x^2}} (20 x^5-45 x^6+e^{2 x} (20 x-45 x^2)+e^x (20 x^3-50 x^4-15 x^5))}{e^{2 x}+2 e^x x^2+x^4} \, dx\)

Optimal. Leaf size=29 \[ e^{-\frac {x^2}{e^x+x^2}} (1+3 (3-5 x)) x^2 \]

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Rubi [F]  time = 5.25, 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^{-\frac {x^2}{e^x+x^2}} \left (20 x^5-45 x^6+e^{2 x} \left (20 x-45 x^2\right )+e^x \left (20 x^3-50 x^4-15 x^5\right )\right )}{e^{2 x}+2 e^x x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*x^5 - 45*x^6 + E^(2*x)*(20*x - 45*x^2) + E^x*(20*x^3 - 50*x^4 - 15*x^5))/(E^(x^2/(E^x + x^2))*(E^(2*x)
+ 2*E^x*x^2 + x^4)),x]

[Out]

20*Defer[Int][x/E^(x^2/(E^x + x^2)), x] - 45*Defer[Int][x^2/E^(x^2/(E^x + x^2)), x] + 20*Defer[Int][x^5/(E^(x^
2/(E^x + x^2))*(E^x + x^2)^2), x] - 40*Defer[Int][x^6/(E^(x^2/(E^x + x^2))*(E^x + x^2)^2), x] + 15*Defer[Int][
x^7/(E^(x^2/(E^x + x^2))*(E^x + x^2)^2), x] - 20*Defer[Int][x^3/(E^(x^2/(E^x + x^2))*(E^x + x^2)), x] + 40*Def
er[Int][x^4/(E^(x^2/(E^x + x^2))*(E^x + x^2)), x] - 15*Defer[Int][x^5/(E^(x^2/(E^x + x^2))*(E^x + x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-\frac {x^2}{e^x+x^2}} \left (20 x^5-45 x^6+e^{2 x} \left (20 x-45 x^2\right )+e^x \left (20 x^3-50 x^4-15 x^5\right )\right )}{\left (e^x+x^2\right )^2} \, dx\\ &=\int \left (-5 e^{-\frac {x^2}{e^x+x^2}} x (-4+9 x)+\frac {5 e^{-\frac {x^2}{e^x+x^2}} x^5 \left (4-8 x+3 x^2\right )}{\left (e^x+x^2\right )^2}-\frac {5 e^{-\frac {x^2}{e^x+x^2}} x^3 \left (4-8 x+3 x^2\right )}{e^x+x^2}\right ) \, dx\\ &=-\left (5 \int e^{-\frac {x^2}{e^x+x^2}} x (-4+9 x) \, dx\right )+5 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^5 \left (4-8 x+3 x^2\right )}{\left (e^x+x^2\right )^2} \, dx-5 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^3 \left (4-8 x+3 x^2\right )}{e^x+x^2} \, dx\\ &=-\left (5 \int \left (-4 e^{-\frac {x^2}{e^x+x^2}} x+9 e^{-\frac {x^2}{e^x+x^2}} x^2\right ) \, dx\right )+5 \int \left (\frac {4 e^{-\frac {x^2}{e^x+x^2}} x^5}{\left (e^x+x^2\right )^2}-\frac {8 e^{-\frac {x^2}{e^x+x^2}} x^6}{\left (e^x+x^2\right )^2}+\frac {3 e^{-\frac {x^2}{e^x+x^2}} x^7}{\left (e^x+x^2\right )^2}\right ) \, dx-5 \int \left (\frac {4 e^{-\frac {x^2}{e^x+x^2}} x^3}{e^x+x^2}-\frac {8 e^{-\frac {x^2}{e^x+x^2}} x^4}{e^x+x^2}+\frac {3 e^{-\frac {x^2}{e^x+x^2}} x^5}{e^x+x^2}\right ) \, dx\\ &=15 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^7}{\left (e^x+x^2\right )^2} \, dx-15 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^5}{e^x+x^2} \, dx+20 \int e^{-\frac {x^2}{e^x+x^2}} x \, dx+20 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^5}{\left (e^x+x^2\right )^2} \, dx-20 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^3}{e^x+x^2} \, dx-40 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^6}{\left (e^x+x^2\right )^2} \, dx+40 \int \frac {e^{-\frac {x^2}{e^x+x^2}} x^4}{e^x+x^2} \, dx-45 \int e^{-\frac {x^2}{e^x+x^2}} x^2 \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.74, size = 26, normalized size = 0.90 \begin {gather*} -5 e^{-\frac {x^2}{e^x+x^2}} x^2 (-2+3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*x^5 - 45*x^6 + E^(2*x)*(20*x - 45*x^2) + E^x*(20*x^3 - 50*x^4 - 15*x^5))/(E^(x^2/(E^x + x^2))*(E
^(2*x) + 2*E^x*x^2 + x^4)),x]

[Out]

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

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fricas [A]  time = 0.88, size = 27, normalized size = 0.93 \begin {gather*} -5 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e^{\left (-\frac {x^{2}}{x^{2} + e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-45*x^2+20*x)*exp(x)^2+(-15*x^5-50*x^4+20*x^3)*exp(x)-45*x^6+20*x^5)/(exp(x)^2+2*exp(x)*x^2+x^4)/e
xp(x^2/(x^2+exp(x))),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.29, size = 27, normalized size = 0.93 \begin {gather*} -5 \, {\left (3 \, x^{3} - 2 \, x^{2}\right )} e^{\left (-\frac {x^{2}}{x^{2} + e^{x}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-45*x^2+20*x)*exp(x)^2+(-15*x^5-50*x^4+20*x^3)*exp(x)-45*x^6+20*x^5)/(exp(x)^2+2*exp(x)*x^2+x^4)/e
xp(x^2/(x^2+exp(x))),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.05, size = 27, normalized size = 0.93

method	result	size
risch	\(\left (-15 x^{3}+10 x^{2}\right ) {\mathrm e}^{-\frac {x^{2}}{x^{2}+{\mathrm e}^{x}}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-45*x^2+20*x)*exp(x)^2+(-15*x^5-50*x^4+20*x^3)*exp(x)-45*x^6+20*x^5)/(exp(x)^2+2*exp(x)*x^2+x^4)/exp(x^2
/(x^2+exp(x))),x,method=_RETURNVERBOSE)

[Out]

(-15*x^3+10*x^2)*exp(-x^2/(x^2+exp(x)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-45*x^2+20*x)*exp(x)^2+(-15*x^5-50*x^4+20*x^3)*exp(x)-45*x^6+20*x^5)/(exp(x)^2+2*exp(x)*x^2+x^4)/e
xp(x^2/(x^2+exp(x))),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {x^2}{{\mathrm {e}}^x+x^2}}\,\left ({\mathrm {e}}^{2\,x}\,\left (20\,x-45\,x^2\right )-{\mathrm {e}}^x\,\left (15\,x^5+50\,x^4-20\,x^3\right )+20\,x^5-45\,x^6\right )}{{\mathrm {e}}^{2\,x}+2\,x^2\,{\mathrm {e}}^x+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-x^2/(exp(x) + x^2))*(exp(2*x)*(20*x - 45*x^2) - exp(x)*(50*x^4 - 20*x^3 + 15*x^5) + 20*x^5 - 45*x^6)
)/(exp(2*x) + 2*x^2*exp(x) + x^4),x)

[Out]

int((exp(-x^2/(exp(x) + x^2))*(exp(2*x)*(20*x - 45*x^2) - exp(x)*(50*x^4 - 20*x^3 + 15*x^5) + 20*x^5 - 45*x^6)
)/(exp(2*x) + 2*x^2*exp(x) + x^4), x)

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sympy [A]  time = 1.97, size = 20, normalized size = 0.69 \begin {gather*} \left (- 15 x^{3} + 10 x^{2}\right ) e^{- \frac {x^{2}}{x^{2} + e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-45*x**2+20*x)*exp(x)**2+(-15*x**5-50*x**4+20*x**3)*exp(x)-45*x**6+20*x**5)/(exp(x)**2+2*exp(x)*x*
*2+x**4)/exp(x**2/(x**2+exp(x))),x)

[Out]

(-15*x**3 + 10*x**2)*exp(-x**2/(x**2 + exp(x)))

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