3.82.36 \(\int \frac {e^{\frac {256+768 x+320 x^2-384 x^3+64 x^4}{x^2}} (-512-768 x-384 x^3+128 x^4)}{x^3} \, dx\)

Optimal. Leaf size=22 \[ -\frac {3}{5}+e^{16 \left (5-2 x+\frac {4+x}{x}\right )^2} \]

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Rubi [A]  time = 0.29, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 1, number of rules used = 1, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {6706} \begin {gather*} e^{\frac {64 \left (x^4-6 x^3+5 x^2+12 x+4\right )}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((256 + 768*x + 320*x^2 - 384*x^3 + 64*x^4)/x^2)*(-512 - 768*x - 384*x^3 + 128*x^4))/x^3,x]

[Out]

E^((64*(4 + 12*x + 5*x^2 - 6*x^3 + 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} &=e^{\frac {64 \left (4+12 x+5 x^2-6 x^3+x^4\right )}{x^2}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 17, normalized size = 0.77 \begin {gather*} e^{\frac {64 \left (-2-3 x+x^2\right )^2}{x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((256 + 768*x + 320*x^2 - 384*x^3 + 64*x^4)/x^2)*(-512 - 768*x - 384*x^3 + 128*x^4))/x^3,x]

[Out]

E^((64*(-2 - 3*x + x^2)^2)/x^2)

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fricas [A]  time = 0.97, size = 24, normalized size = 1.09 \begin {gather*} e^{\left (\frac {64 \, {\left (x^{4} - 6 \, x^{3} + 5 \, x^{2} + 12 \, x + 4\right )}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4-384*x^3-768*x-512)*exp((64*x^4-384*x^3+320*x^2+768*x+256)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(64*(x^4 - 6*x^3 + 5*x^2 + 12*x + 4)/x^2)

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giac [A]  time = 0.13, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (64 \, x^{2} - 384 \, x + \frac {768}{x} + \frac {256}{x^{2}} + 320\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4-384*x^3-768*x-512)*exp((64*x^4-384*x^3+320*x^2+768*x+256)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(64*x^2 - 384*x + 768/x + 256/x^2 + 320)

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maple [A]  time = 0.21, size = 17, normalized size = 0.77

method	result	size
risch	\({\mathrm e}^{\frac {64 \left (x^{2}-3 x -2\right )^{2}}{x^{2}}}\)	\(17\)
gosper	\({\mathrm e}^{\frac {64 x^{4}-384 x^{3}+320 x^{2}+768 x +256}{x^{2}}}\)	\(25\)
norman	\({\mathrm e}^{\frac {64 x^{4}-384 x^{3}+320 x^{2}+768 x +256}{x^{2}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x^4-384*x^3-768*x-512)*exp((64*x^4-384*x^3+320*x^2+768*x+256)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

exp(64*(x^2-3*x-2)^2/x^2)

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maxima [A]  time = 0.51, size = 21, normalized size = 0.95 \begin {gather*} e^{\left (64 \, x^{2} - 384 \, x + \frac {768}{x} + \frac {256}{x^{2}} + 320\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x^4-384*x^3-768*x-512)*exp((64*x^4-384*x^3+320*x^2+768*x+256)/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(64*x^2 - 384*x + 768/x + 256/x^2 + 320)

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mupad [B]  time = 5.45, size = 25, normalized size = 1.14 \begin {gather*} {\mathrm {e}}^{-384\,x}\,{\mathrm {e}}^{320}\,{\mathrm {e}}^{64\,x^2}\,{\mathrm {e}}^{\frac {256}{x^2}}\,{\mathrm {e}}^{768/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((768*x + 320*x^2 - 384*x^3 + 64*x^4 + 256)/x^2)*(768*x + 384*x^3 - 128*x^4 + 512))/x^3,x)

[Out]

exp(-384*x)*exp(320)*exp(64*x^2)*exp(256/x^2)*exp(768/x)

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sympy [A]  time = 0.16, size = 24, normalized size = 1.09 \begin {gather*} e^{\frac {64 x^{4} - 384 x^{3} + 320 x^{2} + 768 x + 256}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((128*x**4-384*x**3-768*x-512)*exp((64*x**4-384*x**3+320*x**2+768*x+256)/x**2)/x**3,x)

[Out]

exp((64*x**4 - 384*x**3 + 320*x**2 + 768*x + 256)/x**2)

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