3.49.88 \(\int \frac {e^{\frac {324-72 x+4 x^2+4 e^6 x^2+e^3 (72 x-8 x^2)}{x^4}} (-2592+1728 x-232 x^2+8 x^3-x^5+e^3 (-432 x+248 x^2-16 x^3)+e^6 (-16 x^2+8 x^3))}{x^5} \, dx\)

Optimal. Leaf size=29 \[ -2+e^{\frac {4 \left (1-e^3-\frac {9}{x}\right )^2}{x^2}} (2-x) \]

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Rubi [B]  time = 0.65, antiderivative size = 150, normalized size of antiderivative = 5.17, number of steps used = 1, number of rules used = 1, integrand size = 94, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {2288} \begin {gather*} \frac {\left (-x^3+29 x^2+e^6 \left (2 x^2-x^3\right )+e^3 \left (2 x^3-31 x^2+54 x\right )-216 x+324\right ) \exp \left (\frac {4 \left (e^6 x^2+x^2+2 e^3 \left (9 x-x^2\right )-18 x+81\right )}{x^4}\right )}{x^5 \left (\frac {-e^3 (9-2 x)-e^6 x-x+9}{x^4}+\frac {2 \left (e^6 x^2+x^2+2 e^3 \left (9 x-x^2\right )-18 x+81\right )}{x^5}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((324 - 72*x + 4*x^2 + 4*E^6*x^2 + E^3*(72*x - 8*x^2))/x^4)*(-2592 + 1728*x - 232*x^2 + 8*x^3 - x^5 + E
^3*(-432*x + 248*x^2 - 16*x^3) + E^6*(-16*x^2 + 8*x^3)))/x^5,x]

[Out]

(E^((4*(81 - 18*x + x^2 + E^6*x^2 + 2*E^3*(9*x - x^2)))/x^4)*(324 - 216*x + 29*x^2 - x^3 + E^6*(2*x^2 - x^3) +
E^3*(54*x - 31*x^2 + 2*x^3)))/(x^5*((9 - E^3*(9 - 2*x) - x - E^6*x)/x^4 + (2*(81 - 18*x + x^2 + E^6*x^2 + 2*E
^3*(9*x - x^2)))/x^5))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\exp \left (\frac {4 \left (81-18 x+x^2+e^6 x^2+2 e^3 \left (9 x-x^2\right )\right )}{x^4}\right ) \left (324-216 x+29 x^2-x^3+e^6 \left (2 x^2-x^3\right )+e^3 \left (54 x-31 x^2+2 x^3\right )\right )}{x^5 \left (\frac {9-e^3 (9-2 x)-x-e^6 x}{x^4}+\frac {2 \left (81-18 x+x^2+e^6 x^2+2 e^3 \left (9 x-x^2\right )\right )}{x^5}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 23, normalized size = 0.79 \begin {gather*} -e^{\frac {4 \left (9+\left (-1+e^3\right ) x\right )^2}{x^4}} (-2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((324 - 72*x + 4*x^2 + 4*E^6*x^2 + E^3*(72*x - 8*x^2))/x^4)*(-2592 + 1728*x - 232*x^2 + 8*x^3 - x
^5 + E^3*(-432*x + 248*x^2 - 16*x^3) + E^6*(-16*x^2 + 8*x^3)))/x^5,x]

[Out]

-(E^((4*(9 + (-1 + E^3)*x)^2)/x^4)*(-2 + x))

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fricas [A]  time = 0.65, size = 36, normalized size = 1.24 \begin {gather*} -{\left (x - 2\right )} e^{\left (\frac {4 \, {\left (x^{2} e^{6} + x^{2} - 2 \, {\left (x^{2} - 9 \, x\right )} e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232*x^2+1728*x-2592)*exp((4*x^2*ex
p(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/x^4)/x^5,x, algorithm="fricas")

[Out]

-(x - 2)*e^(4*(x^2*e^6 + x^2 - 2*(x^2 - 9*x)*e^3 - 18*x + 81)/x^4)

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giac [B]  time = 0.24, size = 87, normalized size = 3.00 \begin {gather*} -{\left (x e^{\left (\frac {3 \, x^{4} + 4 \, x^{2} e^{6} - 8 \, x^{2} e^{3} + 4 \, x^{2} + 72 \, x e^{3} - 72 \, x + 324}{x^{4}}\right )} - 2 \, e^{\left (\frac {3 \, x^{4} + 4 \, x^{2} e^{6} - 8 \, x^{2} e^{3} + 4 \, x^{2} + 72 \, x e^{3} - 72 \, x + 324}{x^{4}}\right )}\right )} e^{\left (-3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232*x^2+1728*x-2592)*exp((4*x^2*ex
p(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/x^4)/x^5,x, algorithm="giac")

[Out]

-(x*e^((3*x^4 + 4*x^2*e^6 - 8*x^2*e^3 + 4*x^2 + 72*x*e^3 - 72*x + 324)/x^4) - 2*e^((3*x^4 + 4*x^2*e^6 - 8*x^2*
e^3 + 4*x^2 + 72*x*e^3 - 72*x + 324)/x^4))*e^(-3)

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maple [A]  time = 0.29, size = 40, normalized size = 1.38

method	result	size
gosper	\(-{\mathrm e}^{\frac {-8 x^{2} {\mathrm e}^{3}+4 x^{2} {\mathrm e}^{6}+72 x \,{\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}} \left (x -2\right )\)	\(40\)
risch	\(\left (2-x \right ) {\mathrm e}^{-\frac {4 \left (2 x^{2} {\mathrm e}^{3}-x^{2} {\mathrm e}^{6}-18 x \,{\mathrm e}^{3}-x^{2}+18 x -81\right )}{x^{4}}}\)	\(42\)
norman	\(\frac {2 x^{4} {\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}-x^{5} {\mathrm e}^{\frac {4 x^{2} {\mathrm e}^{6}+\left (-8 x^{2}+72 x \right ) {\mathrm e}^{3}+4 x^{2}-72 x +324}{x^{4}}}}{x^{4}}\)	\(88\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232*x^2+1728*x-2592)*exp((4*x^2*exp(3)^2
+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/x^4)/x^5,x,method=_RETURNVERBOSE)

[Out]

-exp(4*(x^2*exp(3)^2-2*x^2*exp(3)+18*x*exp(3)+x^2-18*x+81)/x^4)*(x-2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{5} - 8 \, x^{3} + 232 \, x^{2} - 8 \, {\left (x^{3} - 2 \, x^{2}\right )} e^{6} + 8 \, {\left (2 \, x^{3} - 31 \, x^{2} + 54 \, x\right )} e^{3} - 1728 \, x + 2592\right )} e^{\left (\frac {4 \, {\left (x^{2} e^{6} + x^{2} - 2 \, {\left (x^{2} - 9 \, x\right )} e^{3} - 18 \, x + 81\right )}}{x^{4}}\right )}}{x^{5}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^3-16*x^2)*exp(3)^2+(-16*x^3+248*x^2-432*x)*exp(3)-x^5+8*x^3-232*x^2+1728*x-2592)*exp((4*x^2*ex
p(3)^2+(-8*x^2+72*x)*exp(3)+4*x^2-72*x+324)/x^4)/x^5,x, algorithm="maxima")

[Out]

-integrate((x^5 - 8*x^3 + 232*x^2 - 8*(x^3 - 2*x^2)*e^6 + 8*(2*x^3 - 31*x^2 + 54*x)*e^3 - 1728*x + 2592)*e^(4*
(x^2*e^6 + x^2 - 2*(x^2 - 9*x)*e^3 - 18*x + 81)/x^4)/x^5, x)

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mupad [B]  time = 3.86, size = 47, normalized size = 1.62 \begin {gather*} -{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^6}{x^2}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^3}{x^2}}\,{\mathrm {e}}^{\frac {72\,{\mathrm {e}}^3}{x^3}}\,{\mathrm {e}}^{\frac {4}{x^2}}\,{\mathrm {e}}^{-\frac {72}{x^3}}\,{\mathrm {e}}^{\frac {324}{x^4}}\,\left (x-2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(3)*(72*x - 8*x^2) - 72*x + 4*x^2*exp(6) + 4*x^2 + 324)/x^4)*(exp(3)*(432*x - 248*x^2 + 16*x^3)
- 1728*x + exp(6)*(16*x^2 - 8*x^3) + 232*x^2 - 8*x^3 + x^5 + 2592))/x^5,x)

[Out]

-exp((4*exp(6))/x^2)*exp(-(8*exp(3))/x^2)*exp((72*exp(3))/x^3)*exp(4/x^2)*exp(-72/x^3)*exp(324/x^4)*(x - 2)

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sympy [A]  time = 0.89, size = 37, normalized size = 1.28 \begin {gather*} \left (2 - x\right ) e^{\frac {4 x^{2} + 4 x^{2} e^{6} - 72 x + \left (- 8 x^{2} + 72 x\right ) e^{3} + 324}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**3-16*x**2)*exp(3)**2+(-16*x**3+248*x**2-432*x)*exp(3)-x**5+8*x**3-232*x**2+1728*x-2592)*exp((
4*x**2*exp(3)**2+(-8*x**2+72*x)*exp(3)+4*x**2-72*x+324)/x**4)/x**5,x)

[Out]

(2 - x)*exp((4*x**2 + 4*x**2*exp(6) - 72*x + (-8*x**2 + 72*x)*exp(3) + 324)/x**4)

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