3.46.32 \(\int \frac {-x^2+e^{\frac {-60+40 e^x+30 x+5 x^2}{x}} (60+5 x^2+e^x (-40+40 x))}{x^2} \, dx\) [4532]

Optimal. Leaf size=27 \[ 1+e^{5 \left (2-\frac {4 \left (3-2 e^x-x\right )}{x}+x\right )}-x \]

[Out]

1-x+exp(10-20*(3-2*exp(x)-x)/x+5*x)

________________________________________________________________________________________

Rubi [F]
time = 1.00, 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 {-x^2+e^{\frac {-60+40 e^x+30 x+5 x^2}{x}} \left (60+5 x^2+e^x (-40+40 x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x^2 + E^((-60 + 40*E^x + 30*x + 5*x^2)/x)*(60 + 5*x^2 + E^x*(-40 + 40*x)))/x^2,x]

[Out]

-x + 5*Defer[Int][E^((5*(-12 + 8*E^x + 6*x + x^2))/x), x] + 60*Defer[Int][E^((5*(-12 + 8*E^x + 6*x + x^2))/x)/
x^2, x] - 40*Defer[Int][E^((2*(-30 + 20*E^x + 15*x + 3*x^2))/x)/x^2, x] + 40*Defer[Int][E^((2*(-30 + 20*E^x +
15*x + 3*x^2))/x)/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-1+\frac {40 e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}} (-1+x)}{x^2}+\frac {5 e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}} \left (12+x^2\right )}{x^2}\right ) \, dx\\ &=-x+5 \int \frac {e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}} \left (12+x^2\right )}{x^2} \, dx+40 \int \frac {e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}} (-1+x)}{x^2} \, dx\\ &=-x+5 \int \left (e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}}+\frac {12 e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}}}{x^2}\right ) \, dx+40 \int \left (-\frac {e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}}}{x^2}+\frac {e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}}}{x}\right ) \, dx\\ &=-x+5 \int e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}} \, dx-40 \int \frac {e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}}}{x^2} \, dx+40 \int \frac {e^{\frac {2 \left (-30+20 e^x+15 x+3 x^2\right )}{x}}}{x} \, dx+60 \int \frac {e^{\frac {5 \left (-12+8 e^x+6 x+x^2\right )}{x}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.12, size = 24, normalized size = 0.89 \begin {gather*} e^{30-\frac {60}{x}+\frac {40 e^x}{x}+5 x}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-x^2 + E^((-60 + 40*E^x + 30*x + 5*x^2)/x)*(60 + 5*x^2 + E^x*(-40 + 40*x)))/x^2,x]

[Out]

E^(30 - 60/x + (40*E^x)/x + 5*x) - x

________________________________________________________________________________________

Maple [A]
time = 1.48, size = 23, normalized size = 0.85

method	result	size
risch	\(-x +{\mathrm e}^{\frac {40 \,{\mathrm e}^{x}+5 x^{2}+30 x -60}{x}}\)	\(23\)
norman	\(\frac {x \,{\mathrm e}^{\frac {40 \,{\mathrm e}^{x}+5 x^{2}+30 x -60}{x}}-x^{2}}{x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((40*x-40)*exp(x)+5*x^2+60)*exp((40*exp(x)+5*x^2+30*x-60)/x)-x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x+exp(5*(x^2+8*exp(x)+6*x-12)/x)

________________________________________________________________________________________

Maxima [A]
time = 0.34, size = 22, normalized size = 0.81 \begin {gather*} -x + e^{\left (5 \, x + \frac {40 \, e^{x}}{x} - \frac {60}{x} + 30\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((40*x-40)*exp(x)+5*x^2+60)*exp((40*exp(x)+5*x^2+30*x-60)/x)-x^2)/x^2,x, algorithm="maxima")

[Out]

-x + e^(5*x + 40*e^x/x - 60/x + 30)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 22, normalized size = 0.81 \begin {gather*} -x + e^{\left (\frac {5 \, {\left (x^{2} + 6 \, x + 8 \, e^{x} - 12\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((40*x-40)*exp(x)+5*x^2+60)*exp((40*exp(x)+5*x^2+30*x-60)/x)-x^2)/x^2,x, algorithm="fricas")

[Out]

-x + e^(5*(x^2 + 6*x + 8*e^x - 12)/x)

________________________________________________________________________________________

Sympy [A]
time = 0.12, size = 19, normalized size = 0.70 \begin {gather*} - x + e^{\frac {5 x^{2} + 30 x + 40 e^{x} - 60}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((40*x-40)*exp(x)+5*x**2+60)*exp((40*exp(x)+5*x**2+30*x-60)/x)-x**2)/x**2,x)

[Out]

-x + exp((5*x**2 + 30*x + 40*exp(x) - 60)/x)

________________________________________________________________________________________

Giac [A]
time = 0.42, size = 22, normalized size = 0.81 \begin {gather*} -x + e^{\left (\frac {5 \, {\left (x^{2} + 6 \, x + 8 \, e^{x} - 12\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((40*x-40)*exp(x)+5*x^2+60)*exp((40*exp(x)+5*x^2+30*x-60)/x)-x^2)/x^2,x, algorithm="giac")

[Out]

-x + e^(5*(x^2 + 6*x + 8*e^x - 12)/x)

________________________________________________________________________________________

Mupad [B]
time = 3.26, size = 25, normalized size = 0.93 \begin {gather*} {\mathrm {e}}^{5\,x}\,{\mathrm {e}}^{30}\,{\mathrm {e}}^{\frac {40\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{-\frac {60}{x}}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((30*x + 40*exp(x) + 5*x^2 - 60)/x)*(exp(x)*(40*x - 40) + 5*x^2 + 60) - x^2)/x^2,x)

[Out]

exp(5*x)*exp(30)*exp((40*exp(x))/x)*exp(-60/x) - x

________________________________________________________________________________________