3.39.30 \(\int \frac {e^{\frac {e^{2 x}+2 e^x x+x^2}{x^2}} (-46 x^3+6 x^4+e^{2 x} (-60+44 x+28 x^2-12 x^3)+e^x (-60 x+44 x^2+28 x^3-12 x^4))}{125 x^3+225 x^4+135 x^5+27 x^6} \, dx\)

Optimal. Leaf size=27 \[ \frac {2 e^{\frac {\left (e^x+x\right )^2}{x^2}} (3-x)}{(5+3 x)^2} \]

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Rubi [B]  time = 0.35, antiderivative size = 133, normalized size of antiderivative = 4.93, number of steps used = 1, number of rules used = 1, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2288} \begin {gather*} -\frac {2 e^{\frac {x^2+2 e^x x+e^{2 x}}{x^2}} \left (e^{2 x} \left (3 x^3-7 x^2-11 x+15\right )+e^x \left (3 x^4-7 x^3-11 x^2+15 x\right )\right )}{\left (27 x^6+135 x^5+225 x^4+125 x^3\right ) \left (\frac {e^x x+x+e^x+e^{2 x}}{x^2}-\frac {x^2+2 e^x x+e^{2 x}}{x^3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((E^(2*x) + 2*E^x*x + x^2)/x^2)*(-46*x^3 + 6*x^4 + E^(2*x)*(-60 + 44*x + 28*x^2 - 12*x^3) + E^x*(-60*x
+ 44*x^2 + 28*x^3 - 12*x^4)))/(125*x^3 + 225*x^4 + 135*x^5 + 27*x^6),x]

[Out]

(-2*E^((E^(2*x) + 2*E^x*x + x^2)/x^2)*(E^(2*x)*(15 - 11*x - 7*x^2 + 3*x^3) + E^x*(15*x - 11*x^2 - 7*x^3 + 3*x^
4)))/((125*x^3 + 225*x^4 + 135*x^5 + 27*x^6)*((E^x + E^(2*x) + x + E^x*x)/x^2 - (E^(2*x) + 2*E^x*x + x^2)/x^3)
)

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 {2 e^{\frac {e^{2 x}+2 e^x x+x^2}{x^2}} \left (e^{2 x} \left (15-11 x-7 x^2+3 x^3\right )+e^x \left (15 x-11 x^2-7 x^3+3 x^4\right )\right )}{\left (125 x^3+225 x^4+135 x^5+27 x^6\right ) \left (\frac {e^x+e^{2 x}+x+e^x x}{x^2}-\frac {e^{2 x}+2 e^x x+x^2}{x^3}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 25, normalized size = 0.93 \begin {gather*} -\frac {2 e^{\frac {\left (e^x+x\right )^2}{x^2}} (-3+x)}{(5+3 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((E^(2*x) + 2*E^x*x + x^2)/x^2)*(-46*x^3 + 6*x^4 + E^(2*x)*(-60 + 44*x + 28*x^2 - 12*x^3) + E^x*(
-60*x + 44*x^2 + 28*x^3 - 12*x^4)))/(125*x^3 + 225*x^4 + 135*x^5 + 27*x^6),x]

[Out]

(-2*E^((E^x + x)^2/x^2)*(-3 + x))/(5 + 3*x)^2

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fricas [A]  time = 0.71, size = 35, normalized size = 1.30 \begin {gather*} -\frac {2 \, {\left (x - 3\right )} e^{\left (\frac {x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}}{x^{2}}\right )}}{9 \, x^{2} + 30 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+28*x^2+44*x-60)*exp(x)^2+(-12*x^4+28*x^3+44*x^2-60*x)*exp(x)+6*x^4-46*x^3)*exp((exp(x)^2+2
*exp(x)*x+x^2)/x^2)/(27*x^6+135*x^5+225*x^4+125*x^3),x, algorithm="fricas")

[Out]

-2*(x - 3)*e^((x^2 + 2*x*e^x + e^(2*x))/x^2)/(9*x^2 + 30*x + 25)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, {\left (3 \, x^{4} - 23 \, x^{3} - 2 \, {\left (3 \, x^{3} - 7 \, x^{2} - 11 \, x + 15\right )} e^{\left (2 \, x\right )} - 2 \, {\left (3 \, x^{4} - 7 \, x^{3} - 11 \, x^{2} + 15 \, x\right )} e^{x}\right )} e^{\left (\frac {x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}}{x^{2}}\right )}}{27 \, x^{6} + 135 \, x^{5} + 225 \, x^{4} + 125 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+28*x^2+44*x-60)*exp(x)^2+(-12*x^4+28*x^3+44*x^2-60*x)*exp(x)+6*x^4-46*x^3)*exp((exp(x)^2+2
*exp(x)*x+x^2)/x^2)/(27*x^6+135*x^5+225*x^4+125*x^3),x, algorithm="giac")

[Out]

integrate(2*(3*x^4 - 23*x^3 - 2*(3*x^3 - 7*x^2 - 11*x + 15)*e^(2*x) - 2*(3*x^4 - 7*x^3 - 11*x^2 + 15*x)*e^x)*e
^((x^2 + 2*x*e^x + e^(2*x))/x^2)/(27*x^6 + 135*x^5 + 225*x^4 + 125*x^3), x)

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maple [A]  time = 0.05, size = 36, normalized size = 1.33

method	result	size
risch	\(-\frac {2 \left (x -3\right ) {\mathrm e}^{\frac {{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}{x^{2}}}}{9 x^{2}+30 x +25}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-12*x^3+28*x^2+44*x-60)*exp(x)^2+(-12*x^4+28*x^3+44*x^2-60*x)*exp(x)+6*x^4-46*x^3)*exp((exp(x)^2+2*exp(x
)*x+x^2)/x^2)/(27*x^6+135*x^5+225*x^4+125*x^3),x,method=_RETURNVERBOSE)

[Out]

-2*(x-3)/(9*x^2+30*x+25)*exp((exp(2*x)+2*exp(x)*x+x^2)/x^2)

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maxima [A]  time = 0.58, size = 40, normalized size = 1.48 \begin {gather*} -\frac {2 \, {\left (x e - 3 \, e\right )} e^{\left (\frac {2 \, e^{x}}{x} + \frac {e^{\left (2 \, x\right )}}{x^{2}}\right )}}{9 \, x^{2} + 30 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x^3+28*x^2+44*x-60)*exp(x)^2+(-12*x^4+28*x^3+44*x^2-60*x)*exp(x)+6*x^4-46*x^3)*exp((exp(x)^2+2
*exp(x)*x+x^2)/x^2)/(27*x^6+135*x^5+225*x^4+125*x^3),x, algorithm="maxima")

[Out]

-2*(x*e - 3*e)*e^(2*e^x/x + e^(2*x)/x^2)/(9*x^2 + 30*x + 25)

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mupad [B]  time = 2.49, size = 31, normalized size = 1.15 \begin {gather*} -\frac {2\,\mathrm {e}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^x}{x}}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x}}{x^2}}\,\left (x-3\right )}{{\left (3\,x+5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((exp(2*x) + 2*x*exp(x) + x^2)/x^2)*(exp(x)*(60*x - 44*x^2 - 28*x^3 + 12*x^4) - exp(2*x)*(44*x + 28*x
^2 - 12*x^3 - 60) + 46*x^3 - 6*x^4))/(125*x^3 + 225*x^4 + 135*x^5 + 27*x^6),x)

[Out]

-(2*exp(1)*exp((2*exp(x))/x)*exp(exp(2*x)/x^2)*(x - 3))/(3*x + 5)^2

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sympy [A]  time = 0.30, size = 34, normalized size = 1.26 \begin {gather*} \frac {\left (6 - 2 x\right ) e^{\frac {x^{2} + 2 x e^{x} + e^{2 x}}{x^{2}}}}{9 x^{2} + 30 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-12*x**3+28*x**2+44*x-60)*exp(x)**2+(-12*x**4+28*x**3+44*x**2-60*x)*exp(x)+6*x**4-46*x**3)*exp((ex
p(x)**2+2*exp(x)*x+x**2)/x**2)/(27*x**6+135*x**5+225*x**4+125*x**3),x)

[Out]

(6 - 2*x)*exp((x**2 + 2*x*exp(x) + exp(2*x))/x**2)/(9*x**2 + 30*x + 25)

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