3.69.97 \(\int \frac {e^{\frac {67837+6 x-3 x^2}{768 x}} (-67837-3 x^2)}{3072 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac {1}{4} e^{\frac {\frac {265}{3}-\frac {1}{256} (1-x)^2}{x}} \]

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Rubi [A]  time = 0.25, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 6706} \begin {gather*} \frac {1}{4} e^{\frac {-3 x^2+6 x+67837}{768 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((67837 + 6*x - 3*x^2)/(768*x))*(-67837 - 3*x^2))/(3072*x^2),x]

[Out]

E^((67837 + 6*x - 3*x^2)/(768*x))/4

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {\int \frac {e^{\frac {67837+6 x-3 x^2}{768 x}} \left (-67837-3 x^2\right )}{x^2} \, dx}{3072}\\ &=\frac {1}{4} e^{\frac {67837+6 x-3 x^2}{768 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 22, normalized size = 0.88 \begin {gather*} \frac {1}{4} e^{\frac {1}{128}+\frac {67837}{768 x}-\frac {x}{256}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((67837 + 6*x - 3*x^2)/(768*x))*(-67837 - 3*x^2))/(3072*x^2),x]

[Out]

E^(1/128 + 67837/(768*x) - x/256)/4

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fricas [A]  time = 1.12, size = 18, normalized size = 0.72 \begin {gather*} \frac {1}{4} \, e^{\left (-\frac {3 \, x^{2} - 6 \, x - 67837}{768 \, x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3072*(-3*x^2-67837)*exp(1/768*(-3*x^2+6*x+67837)/x)/x^2,x, algorithm="fricas")

[Out]

1/4*e^(-1/768*(3*x^2 - 6*x - 67837)/x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3072*(-3*x^2-67837)*exp(1/768*(-3*x^2+6*x+67837)/x)/x^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Polynomial exponent overflow. Error: Bad Argument Value

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maple [A]  time = 0.04, size = 19, normalized size = 0.76

method	result	size
gosper	\(\frac {{\mathrm e}^{-\frac {3 x^{2}-6 x -67837}{768 x}}}{4}\)	\(19\)
norman	\(\frac {{\mathrm e}^{\frac {-3 x^{2}+6 x +67837}{768 x}}}{4}\)	\(19\)
risch	\(\frac {{\mathrm e}^{-\frac {3 x^{2}-6 x -67837}{768 x}}}{4}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/3072*(-3*x^2-67837)*exp(1/768*(-3*x^2+6*x+67837)/x)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/4*exp(-1/768*(3*x^2-6*x-67837)/x)

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maxima [A]  time = 2.19, size = 13, normalized size = 0.52 \begin {gather*} \frac {1}{4} \, e^{\left (-\frac {1}{256} \, x + \frac {67837}{768 \, x} + \frac {1}{128}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3072*(-3*x^2-67837)*exp(1/768*(-3*x^2+6*x+67837)/x)/x^2,x, algorithm="maxima")

[Out]

1/4*e^(-1/256*x + 67837/768/x + 1/128)

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mupad [B]  time = 4.14, size = 13, normalized size = 0.52 \begin {gather*} \frac {{\mathrm {e}}^{\frac {67837}{768\,x}-\frac {x}{256}+\frac {1}{128}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((x/128 - x^2/256 + 67837/768)/x)*(3*x^2 + 67837))/(3072*x^2),x)

[Out]

exp(67837/(768*x) - x/256 + 1/128)/4

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sympy [A]  time = 0.13, size = 15, normalized size = 0.60 \begin {gather*} \frac {e^{\frac {- \frac {x^{2}}{256} + \frac {x}{128} + \frac {67837}{768}}{x}}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/3072*(-3*x**2-67837)*exp(1/768*(-3*x**2+6*x+67837)/x)/x**2,x)

[Out]

exp((-x**2/256 + x/128 + 67837/768)/x)/4

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