3.8.88 \(\int \frac {x^2+e^{e^x+x} x^2+e^{3+e^x} (-5+5 e^x x)}{x^2} \, dx\)

Optimal. Leaf size=20 \[ -11+e^{e^x}+\frac {5 e^{3+e^x}}{x}+x \]

________________________________________________________________________________________

Rubi [F]  time = 0.22, 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^{e^x+x} x^2+e^{3+e^x} \left (-5+5 e^x x\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x^2 + E^(E^x + x)*x^2 + E^(3 + E^x)*(-5 + 5*E^x*x))/x^2,x]

[Out]

E^E^x + x - 5*Defer[Int][E^(3 + E^x)/x^2, x] + 5*Defer[Int][E^(3 + E^x + x)/x, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 19, normalized size = 0.95 \begin {gather*} e^{e^x}+\frac {5 e^{3+e^x}}{x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^(E^x + x)*x^2 + E^(3 + E^x)*(-5 + 5*E^x*x))/x^2,x]

[Out]

E^E^x + (5*E^(3 + E^x))/x + x

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 27, normalized size = 1.35 \begin {gather*} \frac {{\left (x^{2} e^{x} + {\left (x + 5 \, e^{3}\right )} e^{\left (x + e^{x}\right )}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.86, size = 30, normalized size = 1.50 \begin {gather*} \frac {{\left (x^{2} e^{x} + x e^{\left (x + e^{x}\right )} + 5 \, e^{\left (x + e^{x} + 3\right )}\right )} e^{\left (-x\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 16, normalized size = 0.80

method	result	size
risch	\(x +\frac {\left (5 \,{\mathrm e}^{3}+x \right ) {\mathrm e}^{{\mathrm e}^{x}}}{x}\)	\(16\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{{\mathrm e}^{x}}+5 \,{\mathrm e}^{3} {\mathrm e}^{{\mathrm e}^{x}}}{x}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*exp(x)*x-5)*exp(3+exp(x))+x^2*exp(x)*exp(exp(x))+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x+1/x*(5*exp(3)+x)*exp(exp(x))

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 15, normalized size = 0.75 \begin {gather*} x + \frac {{\left (x + 5 \, e^{3}\right )} e^{\left (e^{x}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + (x + 5*e^3)*e^(e^x)/x

________________________________________________________________________________________

mupad [B]  time = 0.65, size = 15, normalized size = 0.75 \begin {gather*} x+{\mathrm {e}}^{{\mathrm {e}}^x}+\frac {5\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^3}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x) + 3)*(5*x*exp(x) - 5) + x^2 + x^2*exp(exp(x))*exp(x))/x^2,x)

[Out]

x + exp(exp(x)) + (5*exp(exp(x))*exp(3))/x

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 14, normalized size = 0.70 \begin {gather*} x + \frac {\left (x + 5 e^{3}\right ) e^{e^{x}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(x)*x-5)*exp(3+exp(x))+x**2*exp(x)*exp(exp(x))+x**2)/x**2,x)

[Out]

x + (x + 5*exp(3))*exp(exp(x))/x

________________________________________________________________________________________