3.7.12 \(\int \frac {-2 e^{8+\frac {2 (2+3 x-e^x x)}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} (-10 e^{4+x} x^2+e^4 (-20+10 x))}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 (2+3 x-e^x x)}{x}} x^5} \, dx\)

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

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Rubi [F]  time = 3.56, 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 {-2 e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^3+e^{\frac {2+3 x-e^x x}{x}} \left (-10 e^{4+x} x^2+e^4 (-20+10 x)\right )}{25 x+10 e^{4+\frac {2+3 x-e^x x}{x}} x^3+e^{8+\frac {2 \left (2+3 x-e^x x\right )}{x}} x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^3 + E^((2 + 3*x - E^x*x)/x)*(-10*E^(4 + x)*x^2 + E^4*(-20 + 10*x)))/
(25*x + 10*E^(4 + (2 + 3*x - E^x*x)/x)*x^3 + E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^5),x]

[Out]

20*Defer[Int][E^(7 + E^x + 2/x)/(5*E^E^x + E^(7 + 2/x)*x^2)^2, x] - 20*Defer[Int][E^(7 + E^x + 2/x)/(x*(5*E^E^
x + E^(7 + 2/x)*x^2)^2), x] - 10*Defer[Int][(E^(7 + E^x + 2/x + x)*x)/(5*E^E^x + E^(7 + 2/x)*x^2)^2, x] - 2*De
fer[Int][E^(7 + 2/x)/(5*E^E^x + E^(7 + 2/x)*x^2), x]

Rubi steps

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

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Mathematica [A]  time = 0.87, size = 35, normalized size = 1.09 \begin {gather*} \frac {2 e^{7+\frac {2}{x}} x}{5 e^{e^x}+e^{7+\frac {2}{x}} x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^3 + E^((2 + 3*x - E^x*x)/x)*(-10*E^(4 + x)*x^2 + E^4*(-20 + 10
*x)))/(25*x + 10*E^(4 + (2 + 3*x - E^x*x)/x)*x^3 + E^(8 + (2*(2 + 3*x - E^x*x))/x)*x^5),x]

[Out]

(2*E^(7 + 2/x)*x)/(5*E^E^x + E^(7 + 2/x)*x^2)

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fricas [A]  time = 0.65, size = 57, normalized size = 1.78 \begin {gather*} \frac {2 \, x e^{\left (\frac {{\left ({\left (7 \, x + 2\right )} e^{4} - x e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{x}\right )}}{x^{2} e^{\left (\frac {{\left ({\left (7 \, x + 2\right )} e^{4} - x e^{\left (x + 4\right )}\right )} e^{\left (-4\right )}}{x}\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10*x-20)*exp(4))*exp((-exp(x)*x+
3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm=
"fricas")

[Out]

2*x*e^(((7*x + 2)*e^4 - x*e^(x + 4))*e^(-4)/x)/(x^2*e^(((7*x + 2)*e^4 - x*e^(x + 4))*e^(-4)/x) + 5)

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giac [A]  time = 0.55, size = 35, normalized size = 1.09 \begin {gather*} \frac {2 \, x e^{\left (\frac {7 \, x + 2}{x}\right )}}{x^{2} e^{\left (\frac {7 \, x + 2}{x}\right )} + 5 \, e^{\left (e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10*x-20)*exp(4))*exp((-exp(x)*x+
3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm=
"giac")

[Out]

2*x*e^((7*x + 2)/x)/(x^2*e^((7*x + 2)/x) + 5*e^(e^x))

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maple [A]  time = 0.34, size = 35, normalized size = 1.09

method	result	size
risch	\(\frac {2}{x}-\frac {10}{x \left ({\mathrm e}^{-\frac {{\mathrm e}^{x} x -7 x -2}{x}} x^{2}+5\right )}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10*x-20)*exp(4))*exp((-exp(x)*x+3*x+2)
/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x,method=_RETURNVER
BOSE)

[Out]

2/x-10/x/(exp(-(exp(x)*x-7*x-2)/x)*x^2+5)

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maxima [A]  time = 0.68, size = 31, normalized size = 0.97 \begin {gather*} \frac {2 \, x e^{\left (\frac {2}{x} + 7\right )}}{x^{2} e^{\left (\frac {2}{x} + 7\right )} + 5 \, e^{\left (e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^3*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+(-10*x^2*exp(4)*exp(x)+(10*x-20)*exp(4))*exp((-exp(x)*x+
3*x+2)/x))/(x^5*exp(4)^2*exp((-exp(x)*x+3*x+2)/x)^2+10*x^3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x, algorithm=
"maxima")

[Out]

2*x*e^(2/x + 7)/(x^2*e^(2/x + 7) + 5*e^(e^x))

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mupad [B]  time = 0.71, size = 69, normalized size = 2.16 \begin {gather*} \frac {2}{x}-\frac {10\,\left (x^2\,{\mathrm {e}}^x-2\,x+2\right )}{x\,\left ({\mathrm {e}}^{\frac {2}{x}-{\mathrm {e}}^x+3}+\frac {5\,{\mathrm {e}}^{-4}}{x^2}\right )\,\left (x^4\,{\mathrm {e}}^{x+4}+2\,x^2\,{\mathrm {e}}^4-2\,x^3\,{\mathrm {e}}^4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((3*x - x*exp(x) + 2)/x)*(exp(4)*(10*x - 20) - 10*x^2*exp(4)*exp(x)) - 2*x^3*exp(8)*exp((2*(3*x - x*ex
p(x) + 2))/x))/(25*x + 10*x^3*exp(4)*exp((3*x - x*exp(x) + 2)/x) + x^5*exp(8)*exp((2*(3*x - x*exp(x) + 2))/x))
,x)

[Out]

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

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sympy [A]  time = 0.23, size = 27, normalized size = 0.84 \begin {gather*} - \frac {10}{x^{3} e^{4} e^{\frac {- x e^{x} + 3 x + 2}{x}} + 5 x} + \frac {2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**3*exp(4)**2*exp((-exp(x)*x+3*x+2)/x)**2+(-10*x**2*exp(4)*exp(x)+(10*x-20)*exp(4))*exp((-exp(x
)*x+3*x+2)/x))/(x**5*exp(4)**2*exp((-exp(x)*x+3*x+2)/x)**2+10*x**3*exp(4)*exp((-exp(x)*x+3*x+2)/x)+25*x),x)

[Out]

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

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