3.3.65 \(\int \frac {x^2+e^{e^2} (e^{-2+x} (-1+x)+5 x^2)}{x^2} \, dx\)

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

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Rubi [A]  time = 0.07, antiderivative size = 24, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {14, 2197} \begin {gather*} \left (1+5 e^{e^2}\right ) x+\frac {e^{x+e^2-2}}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^2 + E^E^2*(E^(-2 + x)*(-1 + x) + 5*x^2))/x^2,x]

[Out]

E^(-2 + E^2 + x)/x + (1 + 5*E^E^2)*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2197

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> With[{b = Coefficient[v, x, 1], d = Coefficient[u, x, 0],
e = Coefficient[u, x, 1], f = Coefficient[w, x, 0], g = Coefficient[w, x, 1]}, Simp[(g*u^(m + 1)*F^(c*v))/(b*c
*e*Log[F]), x] /; EqQ[e*g*(m + 1) - b*c*(e*f - d*g)*Log[F], 0]] /; FreeQ[{F, c, m}, x] && LinearQ[{u, v, w}, x
]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{-2+e^2+x}}{x}+x+5 e^{e^2} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^2 + E^E^2*(E^(-2 + x)*(-1 + x) + 5*x^2))/x^2,x]

[Out]

E^(-2 + E^2 + x)/x + x + 5*E^E^2*x

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fricas [A]  time = 0.50, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{2} + {\left (5 \, x^{2} + e^{\left (x - 2\right )}\right )} e^{\left (e^{2}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(x-2)+5*x^2)*exp(exp(2))+x^2)/x^2,x, algorithm="fricas")

[Out]

(x^2 + (5*x^2 + e^(x - 2))*e^(e^2))/x

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giac [A]  time = 0.73, size = 22, normalized size = 0.81 \begin {gather*} \frac {5 \, x^{2} e^{\left (e^{2}\right )} + x^{2} + e^{\left (x + e^{2} - 2\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(x-2)+5*x^2)*exp(exp(2))+x^2)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.36, size = 19, normalized size = 0.70

method	result	size
risch	\(5 x \,{\mathrm e}^{{\mathrm e}^{2}}+x +\frac {{\mathrm e}^{{\mathrm e}^{2}+x -2}}{x}\)	\(19\)
norman	\(\frac {\left (5 \,{\mathrm e}^{{\mathrm e}^{2}}+1\right ) x^{2}+{\mathrm e}^{x -2} {\mathrm e}^{{\mathrm e}^{2}}}{x}\)	\(25\)
derivativedivides	\(x -2+{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x -2}}{x}-{\mathrm e}^{-2} \expIntegralEi \left (1, -x \right )\right )+{\mathrm e}^{{\mathrm e}^{2}} \left ({\mathrm e}^{-2} \expIntegralEi \left (1, -x \right )+\frac {2 \,{\mathrm e}^{x -2}}{x}\right )+5 \,{\mathrm e}^{{\mathrm e}^{2}} \left (x -2\right )\)	\(57\)
default	\(x -2+{\mathrm e}^{{\mathrm e}^{2}} \left (-\frac {{\mathrm e}^{x -2}}{x}-{\mathrm e}^{-2} \expIntegralEi \left (1, -x \right )\right )+{\mathrm e}^{{\mathrm e}^{2}} \left ({\mathrm e}^{-2} \expIntegralEi \left (1, -x \right )+\frac {2 \,{\mathrm e}^{x -2}}{x}\right )+5 \,{\mathrm e}^{{\mathrm e}^{2}} \left (x -2\right )\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x-1)*exp(x-2)+5*x^2)*exp(exp(2))+x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

5*x*exp(exp(2))+x+1/x*exp(exp(2)+x-2)

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maxima [C]  time = 0.47, size = 28, normalized size = 1.04 \begin {gather*} {\rm Ei}\relax (x) e^{\left (e^{2} - 2\right )} + 5 \, x e^{\left (e^{2}\right )} - e^{\left (e^{2} - 2\right )} \Gamma \left (-1, -x\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(x-2)+5*x^2)*exp(exp(2))+x^2)/x^2,x, algorithm="maxima")

[Out]

Ei(x)*e^(e^2 - 2) + 5*x*e^(e^2) - e^(e^2 - 2)*gamma(-1, -x) + x

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mupad [B]  time = 0.33, size = 19, normalized size = 0.70 \begin {gather*} x+5\,x\,{\mathrm {e}}^{{\mathrm {e}}^2}+\frac {{\mathrm {e}}^{-2}\,{\mathrm {e}}^{{\mathrm {e}}^2}\,{\mathrm {e}}^x}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(2))*(exp(x - 2)*(x - 1) + 5*x^2) + x^2)/x^2,x)

[Out]

x + 5*x*exp(exp(2)) + (exp(-2)*exp(exp(2))*exp(x))/x

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sympy [A]  time = 0.12, size = 20, normalized size = 0.74 \begin {gather*} x \left (1 + 5 e^{e^{2}}\right ) + \frac {e^{x - 2} e^{e^{2}}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-1)*exp(x-2)+5*x**2)*exp(exp(2))+x**2)/x**2,x)

[Out]

x*(1 + 5*exp(exp(2))) + exp(x - 2)*exp(exp(2))/x

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