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

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} $$\begin{array}{c}\frac{1}{4}{e}^{\frac{-3x^2+6x+67837}{768x}}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{\int \frac{e^{\frac{67837+6x-3x^2}{768x}}(-67837-3x^2)}{x^2}dx}{3072}\\ & =\frac{1}{4}{e}^{\frac{67837+6x-3x^2}{768x}}\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{1}{4}{e}^{(-\frac{3x^2-6x-67837}{768x})}\end{array}$$

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 $$\begin{array}{c}\text{Exception raised: TypeError}\end{array}$$

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	$\frac{{\mathrm{e}}^{-\frac{3x^2-6x-67837}{768x}}}{4}$	$19$
norman	$\frac{{\mathrm{e}}^{\frac{-3x^2+6x+67837}{768x}}}{4}$	$19$
risch	$\frac{{\mathrm{e}}^{-\frac{3x^2-6x-67837}{768x}}}{4}$	$19$

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 $$\begin{array}{c}\frac{1}{4}{e}^{(-\frac{1}{256}x+\frac{67837}{768x}+\frac{1}{128})}\end{array}$$

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 $$\begin{array}{c}\frac{{\mathrm{e}}^{\frac{67837}{768x}-\frac{x}{256}+\frac{1}{128}}}{4}\end{array}$$

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 $$\begin{array}{c}\frac{e^{\frac{-\frac{x^2}{256}+\frac{x}{128}+\frac{67837}{768}}{x}}}{4}\end{array}$$

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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