3.47.47 \(\int \frac {2 x^2+e^{\frac {3-6 x-2 x^2}{2 x}} (-3-2 x^2)}{2 x^2} \, dx\)

Optimal. Leaf size=17 \[ -1+e^{-3+\frac {3}{2 x}-x}+x \]

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Rubi [A]  time = 0.15, antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {12, 14, 6706} \begin {gather*} x+e^{-x+\frac {3}{2 x}-3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*x^2 + E^((3 - 6*x - 2*x^2)/(2*x))*(-3 - 2*x^2))/(2*x^2),x]

[Out]

E^(-3 + 3/(2*x) - x) + x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {2 x^2+e^{\frac {3-6 x-2 x^2}{2 x}} \left (-3-2 x^2\right )}{x^2} \, dx\\ &=\frac {1}{2} \int \left (2-\frac {e^{-3+\frac {3}{2 x}-x} \left (3+2 x^2\right )}{x^2}\right ) \, dx\\ &=x-\frac {1}{2} \int \frac {e^{-3+\frac {3}{2 x}-x} \left (3+2 x^2\right )}{x^2} \, dx\\ &=e^{-3+\frac {3}{2 x}-x}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 16, normalized size = 0.94 \begin {gather*} e^{-3+\frac {3}{2 x}-x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x^2 + E^((3 - 6*x - 2*x^2)/(2*x))*(-3 - 2*x^2))/(2*x^2),x]

[Out]

E^(-3 + 3/(2*x) - x) + x

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fricas [A]  time = 0.59, size = 18, normalized size = 1.06 \begin {gather*} x + e^{\left (-\frac {2 \, x^{2} + 6 \, x - 3}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2-3)*exp(1/2*(-2*x^2-6*x+3)/x)+2*x^2)/x^2,x, algorithm="fricas")

[Out]

x + e^(-1/2*(2*x^2 + 6*x - 3)/x)

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giac [A]  time = 0.27, size = 18, normalized size = 1.06 \begin {gather*} x + e^{\left (-\frac {2 \, x^{2} + 6 \, x - 3}{2 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2-3)*exp(1/2*(-2*x^2-6*x+3)/x)+2*x^2)/x^2,x, algorithm="giac")

[Out]

x + e^(-1/2*(2*x^2 + 6*x - 3)/x)

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maple [A]  time = 0.05, size = 19, normalized size = 1.12

method	result	size
risch	\(x +{\mathrm e}^{-\frac {2 x^{2}+6 x -3}{2 x}}\)	\(19\)
norman	\(\frac {x^{2}+x \,{\mathrm e}^{\frac {-2 x^{2}-6 x +3}{2 x}}}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((-2*x^2-3)*exp(1/2*(-2*x^2-6*x+3)/x)+2*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

x+exp(-1/2*(2*x^2+6*x-3)/x)

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maxima [A]  time = 0.44, size = 13, normalized size = 0.76 \begin {gather*} x + e^{\left (-x + \frac {3}{2 \, x} - 3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x^2-3)*exp(1/2*(-2*x^2-6*x+3)/x)+2*x^2)/x^2,x, algorithm="maxima")

[Out]

x + e^(-x + 3/2/x - 3)

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mupad [B]  time = 3.12, size = 13, normalized size = 0.76 \begin {gather*} x+{\mathrm {e}}^{\frac {3}{2\,x}-x-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x + exp(3/(2*x) - x - 3)

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sympy [A]  time = 0.13, size = 14, normalized size = 0.82 \begin {gather*} x + e^{\frac {- x^{2} - 3 x + \frac {3}{2}}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((-2*x**2-3)*exp(1/2*(-2*x**2-6*x+3)/x)+2*x**2)/x**2,x)

[Out]

x + exp((-x**2 - 3*x + 3/2)/x)

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