3.31.66 \(\int \frac {e^{\frac {30-12 x+2 x^2}{x}} (-30+2 x^2)}{x^2} \, dx\)

Optimal. Leaf size=21 \[ e^{\frac {2 \left (5 (3-x)-x+x^2\right )}{x}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.14, antiderivative size = 15, normalized size of antiderivative = 0.71, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {6706} \begin {gather*} e^{\frac {2 \left (x^2-6 x+15\right )}{x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((30 - 12*x + 2*x^2)/x)*(-30 + 2*x^2))/x^2,x]

[Out]

E^((2*(15 - 6*x + x^2))/x)

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} &=e^{\frac {2 \left (15-6 x+x^2\right )}{x}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 12, normalized size = 0.57 \begin {gather*} e^{-12+\frac {30}{x}+2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((30 - 12*x + 2*x^2)/x)*(-30 + 2*x^2))/x^2,x]

[Out]

E^(-12 + 30/x + 2*x)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 14, normalized size = 0.67 \begin {gather*} e^{\left (\frac {2 \, {\left (x^{2} - 6 \, x + 15\right )}}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-30)*exp((2*x^2-12*x+30)/x)/x^2,x, algorithm="fricas")

[Out]

e^(2*(x^2 - 6*x + 15)/x)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 11, normalized size = 0.52 \begin {gather*} e^{\left (2 \, x + \frac {30}{x} - 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-30)*exp((2*x^2-12*x+30)/x)/x^2,x, algorithm="giac")

[Out]

e^(2*x + 30/x - 12)

________________________________________________________________________________________

maple [A]  time = 0.13, size = 15, normalized size = 0.71

method	result	size
gosper	\({\mathrm e}^{\frac {2 x^{2}-12 x +30}{x}}\)	\(15\)
risch	\({\mathrm e}^{\frac {2 x^{2}-12 x +30}{x}}\)	\(15\)
norman	\({\mathrm e}^{\frac {2 x^{2}-12 x +30}{x}}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2-30)*exp((2*x^2-12*x+30)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

exp(2/x*(x^2-6*x+15))

________________________________________________________________________________________

maxima [A]  time = 0.83, size = 11, normalized size = 0.52 \begin {gather*} e^{\left (2 \, x + \frac {30}{x} - 12\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^2-30)*exp((2*x^2-12*x+30)/x)/x^2,x, algorithm="maxima")

[Out]

e^(2*x + 30/x - 12)

________________________________________________________________________________________

mupad [B]  time = 1.80, size = 13, normalized size = 0.62 \begin {gather*} {\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{-12}\,{\mathrm {e}}^{30/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x^2 - 12*x + 30)/x)*(2*x^2 - 30))/x^2,x)

[Out]

exp(2*x)*exp(-12)*exp(30/x)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 12, normalized size = 0.57 \begin {gather*} e^{\frac {2 x^{2} - 12 x + 30}{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**2-30)*exp((2*x**2-12*x+30)/x)/x**2,x)

[Out]

exp((2*x**2 - 12*x + 30)/x)

________________________________________________________________________________________