3.22.50 \(\int \frac {e^{-\frac {2-x-98 x^2-76 x^3-16 x^4}{x^2}} (16-4 x+304 x^3+128 x^4)}{x^3} \, dx\)

Optimal. Leaf size=30 \[ 3+4 e^{-2-\frac {2-x}{x^2}-4 x+4 (5+2 x)^2} \]

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Rubi [A]  time = 0.27, antiderivative size = 29, normalized size of antiderivative = 0.97, number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6706} \begin {gather*} 4 e^{-\frac {-16 x^4-76 x^3-98 x^2-x+2}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16 - 4*x + 304*x^3 + 128*x^4)/(E^((2 - x - 98*x^2 - 76*x^3 - 16*x^4)/x^2)*x^3),x]

[Out]

4/E^((2 - x - 98*x^2 - 76*x^3 - 16*x^4)/x^2)

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} &=4 e^{-\frac {2-x-98 x^2-76 x^3-16 x^4}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 22, normalized size = 0.73 \begin {gather*} 4 e^{98-\frac {2}{x^2}+\frac {1}{x}+76 x+16 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 - 4*x + 304*x^3 + 128*x^4)/(E^((2 - x - 98*x^2 - 76*x^3 - 16*x^4)/x^2)*x^3),x]

[Out]

4*E^(98 - 2/x^2 + x^(-1) + 76*x + 16*x^2)

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fricas [A]  time = 0.86, size = 25, normalized size = 0.83 \begin {gather*} 4 \, e^{\left (\frac {16 \, x^{4} + 76 \, x^{3} + 98 \, x^{2} + x - 2}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4+304*x^3-4*x+16)/x^3/exp((-16*x^4-76*x^3-98*x^2-x+2)/x^2),x, algorithm="fricas")

[Out]

4*e^((16*x^4 + 76*x^3 + 98*x^2 + x - 2)/x^2)

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giac [A]  time = 0.22, size = 21, normalized size = 0.70 \begin {gather*} 4 \, e^{\left (16 \, x^{2} + 76 \, x + \frac {1}{x} - \frac {2}{x^{2}} + 98\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4+304*x^3-4*x+16)/x^3/exp((-16*x^4-76*x^3-98*x^2-x+2)/x^2),x, algorithm="giac")

[Out]

4*e^(16*x^2 + 76*x + 1/x - 2/x^2 + 98)

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maple [A]  time = 0.16, size = 26, normalized size = 0.87

method	result	size
risch	\(4 \,{\mathrm e}^{\frac {16 x^{4}+76 x^{3}+98 x^{2}+x -2}{x^{2}}}\)	\(26\)
gosper	\(4 \,{\mathrm e}^{\frac {16 x^{4}+76 x^{3}+98 x^{2}+x -2}{x^{2}}}\)	\(29\)
norman	\(4 \,{\mathrm e}^{-\frac {-16 x^{4}-76 x^{3}-98 x^{2}-x +2}{x^{2}}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x^4+304*x^3-4*x+16)/x^3/exp((-16*x^4-76*x^3-98*x^2-x+2)/x^2),x,method=_RETURNVERBOSE)

[Out]

4*exp((16*x^4+76*x^3+98*x^2+x-2)/x^2)

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maxima [A]  time = 0.53, size = 21, normalized size = 0.70 \begin {gather*} 4 \, e^{\left (16 \, x^{2} + 76 \, x + \frac {1}{x} - \frac {2}{x^{2}} + 98\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4+304*x^3-4*x+16)/x^3/exp((-16*x^4-76*x^3-98*x^2-x+2)/x^2),x, algorithm="maxima")

[Out]

4*e^(16*x^2 + 76*x + 1/x - 2/x^2 + 98)

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mupad [B]  time = 1.22, size = 24, normalized size = 0.80 \begin {gather*} 4\,{\mathrm {e}}^{76\,x}\,{\mathrm {e}}^{1/x}\,{\mathrm {e}}^{98}\,{\mathrm {e}}^{-\frac {2}{x^2}}\,{\mathrm {e}}^{16\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x + 98*x^2 + 76*x^3 + 16*x^4 - 2)/x^2)*(304*x^3 - 4*x + 128*x^4 + 16))/x^3,x)

[Out]

4*exp(76*x)*exp(1/x)*exp(98)*exp(-2/x^2)*exp(16*x^2)

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sympy [A]  time = 0.15, size = 24, normalized size = 0.80 \begin {gather*} 4 e^{- \frac {- 16 x^{4} - 76 x^{3} - 98 x^{2} - x + 2}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x**4+304*x**3-4*x+16)/x**3/exp((-16*x**4-76*x**3-98*x**2-x+2)/x**2),x)

[Out]

4*exp(-(-16*x**4 - 76*x**3 - 98*x**2 - x + 2)/x**2)

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