3.16.39 \(\int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 (-8+8 e^{\frac {2}{x^2}}-2 x^3)}{(-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3)^2}+\frac {x^2 (-12 x+10 x^3+e^{\frac {2}{x^2}} (8 x+2 x^3))}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx\)

Optimal. Leaf size=23 \[ 2+\left (x+\frac {1}{5+e^{\frac {2}{x^2}}-\frac {2}{x^2}+x}\right )^2 \]

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Rubi [F]  time = 5.46, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-4 x^2+10 x^4+2 e^{\frac {2}{x^2}} x^4+2 x^5+\frac {x^4 \left (-8+8 e^{\frac {2}{x^2}}-2 x^3\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^2 \left (-12 x+10 x^3+e^{\frac {2}{x^2}} \left (8 x+2 x^3\right )\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}}{-2 x+5 x^3+e^{\frac {2}{x^2}} x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4*x^2 + 10*x^4 + 2*E^(2/x^2)*x^4 + 2*x^5 + (x^4*(-8 + 8*E^(2/x^2) - 2*x^3))/(-2 + 5*x^2 + E^(2/x^2)*x^2
+ x^3)^2 + (x^2*(-12*x + 10*x^3 + E^(2/x^2)*(8*x + 2*x^3)))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3))/(-2*x + 5*x^3
+ E^(2/x^2)*x^3 + x^4),x]

[Out]

x^2 + 16*Defer[Int][x/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^3, x] - 48*Defer[Int][x^3/(-2 + 5*x^2 + E^(2/x^2)*x^2
+ x^3)^3, x] - 8*Defer[Int][x^4/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^3, x] - 2*Defer[Int][x^6/(-2 + 5*x^2 + E^(
2/x^2)*x^2 + x^3)^3, x] + 16*Defer[Int][(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^(-2), x] + 8*Defer[Int][x/(-2 + 5*x
^2 + E^(2/x^2)*x^2 + x^3)^2, x] - 48*Defer[Int][x^2/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^2, x] - 8*Defer[Int][x^
3/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^2, x] - 2*Defer[Int][x^5/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^2, x] + 8*Def
er[Int][(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3)^(-1), x] + 2*Defer[Int][x^2/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2 x+\frac {2 \left (4+x^2\right )}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}-\frac {2 x \left (-8+24 x^2+4 x^3+x^5\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3}-\frac {2 \left (-8-4 x+24 x^2+4 x^3+x^5\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}\right ) \, dx\\ &=x^2+2 \int \frac {4+x^2}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3} \, dx-2 \int \frac {x \left (-8+24 x^2+4 x^3+x^5\right )}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3} \, dx-2 \int \frac {-8-4 x+24 x^2+4 x^3+x^5}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx\\ &=x^2-2 \int \left (-\frac {8 x}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3}+\frac {24 x^3}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3}+\frac {4 x^4}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3}+\frac {x^6}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3}\right ) \, dx-2 \int \left (-\frac {8}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}-\frac {4 x}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {24 x^2}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {4 x^3}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}+\frac {x^5}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2}\right ) \, dx+2 \int \left (\frac {4}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}+\frac {x^2}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3}\right ) \, dx\\ &=x^2-2 \int \frac {x^6}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3} \, dx-2 \int \frac {x^5}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx+2 \int \frac {x^2}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3} \, dx-8 \int \frac {x^4}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3} \, dx+8 \int \frac {x}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx-8 \int \frac {x^3}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx+8 \int \frac {1}{-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3} \, dx+16 \int \frac {x}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3} \, dx+16 \int \frac {1}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx-48 \int \frac {x^3}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^3} \, dx-48 \int \frac {x^2}{\left (-2+5 x^2+e^{\frac {2}{x^2}} x^2+x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 45, normalized size = 1.96 \begin {gather*} \frac {x^2 \left (-2+x+\left (5+e^{\frac {2}{x^2}}\right ) x^2+x^3\right )^2}{\left (-2+\left (5+e^{\frac {2}{x^2}}\right ) x^2+x^3\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + 10*x^4 + 2*E^(2/x^2)*x^4 + 2*x^5 + (x^4*(-8 + 8*E^(2/x^2) - 2*x^3))/(-2 + 5*x^2 + E^(2/x^2
)*x^2 + x^3)^2 + (x^2*(-12*x + 10*x^3 + E^(2/x^2)*(8*x + 2*x^3)))/(-2 + 5*x^2 + E^(2/x^2)*x^2 + x^3))/(-2*x +
5*x^3 + E^(2/x^2)*x^3 + x^4),x]

[Out]

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

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fricas [B]  time = 0.95, size = 129, normalized size = 5.61 \begin {gather*} \frac {x^{8} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 27 \, x^{6} + 6 \, x^{5} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{7} + 5 \, x^{6} + x^{5} - 2 \, x^{4}\right )} e^{\left (\frac {2}{x^{2}}\right )}}{x^{6} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 20 \, x^{2} + 2 \, {\left (x^{5} + 5 \, x^{4} - 2 \, x^{2}\right )} e^{\left (\frac {2}{x^{2}}\right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2
*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm=
"fricas")

[Out]

(x^8 + 10*x^7 + x^6*e^(4/x^2) + 27*x^6 + 6*x^5 - 19*x^4 - 4*x^3 + 4*x^2 + 2*(x^7 + 5*x^6 + x^5 - 2*x^4)*e^(2/x
^2))/(x^6 + 10*x^5 + x^4*e^(4/x^2) + 25*x^4 - 4*x^3 - 20*x^2 + 2*(x^5 + 5*x^4 - 2*x^2)*e^(2/x^2) + 4)

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giac [B]  time = 0.20, size = 159, normalized size = 6.91 \begin {gather*} \frac {x^{8} + 2 \, x^{7} e^{\left (\frac {2}{x^{2}}\right )} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 10 \, x^{6} e^{\left (\frac {2}{x^{2}}\right )} + 27 \, x^{6} + 2 \, x^{5} e^{\left (\frac {2}{x^{2}}\right )} + 6 \, x^{5} - 4 \, x^{4} e^{\left (\frac {2}{x^{2}}\right )} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2}}{x^{6} + 2 \, x^{5} e^{\left (\frac {2}{x^{2}}\right )} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 10 \, x^{4} e^{\left (\frac {2}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 4 \, x^{2} e^{\left (\frac {2}{x^{2}}\right )} - 20 \, x^{2} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2
*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm=
"giac")

[Out]

(x^8 + 2*x^7*e^(2/x^2) + 10*x^7 + x^6*e^(4/x^2) + 10*x^6*e^(2/x^2) + 27*x^6 + 2*x^5*e^(2/x^2) + 6*x^5 - 4*x^4*
e^(2/x^2) - 19*x^4 - 4*x^3 + 4*x^2)/(x^6 + 2*x^5*e^(2/x^2) + 10*x^5 + x^4*e^(4/x^2) + 10*x^4*e^(2/x^2) + 25*x^
4 - 4*x^3 - 4*x^2*e^(2/x^2) - 20*x^2 + 4)

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maple [B]  time = 0.14, size = 55, normalized size = 2.39

method	result	size
risch	\(x^{2}+\frac {\left (2 x^{3}+2 x^{2} {\mathrm e}^{\frac {2}{x^{2}}}+10 x^{2}+x -4\right ) x^{3}}{\left (x^{2} {\mathrm e}^{\frac {2}{x^{2}}}+x^{3}+5 x^{2}-2\right )^{2}}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2*exp(2
/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x,method=_RETURNVER
BOSE)

[Out]

x^2+(2*x^3+2*x^2*exp(2/x^2)+10*x^2+x-4)*x^3/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2

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maxima [B]  time = 0.55, size = 129, normalized size = 5.61 \begin {gather*} \frac {x^{8} + 10 \, x^{7} + x^{6} e^{\left (\frac {4}{x^{2}}\right )} + 27 \, x^{6} + 6 \, x^{5} - 19 \, x^{4} - 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{7} + 5 \, x^{6} + x^{5} - 2 \, x^{4}\right )} e^{\left (\frac {2}{x^{2}}\right )}}{x^{6} + 10 \, x^{5} + x^{4} e^{\left (\frac {4}{x^{2}}\right )} + 25 \, x^{4} - 4 \, x^{3} - 20 \, x^{2} + 2 \, {\left (x^{5} + 5 \, x^{4} - 2 \, x^{2}\right )} e^{\left (\frac {2}{x^{2}}\right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(2/x^2)-2*x^3-8)/(x^2*exp(2/x^2)+x^3+5*x^2-2)^2*x^4+((2*x^3+8*x)*exp(2/x^2)+10*x^3-12*x)/(x^2
*exp(2/x^2)+x^3+5*x^2-2)*x^2+2*x^4*exp(2/x^2)+2*x^5+10*x^4-4*x^2)/(x^3*exp(2/x^2)+x^4+5*x^3-2*x),x, algorithm=
"maxima")

[Out]

(x^8 + 10*x^7 + x^6*e^(4/x^2) + 27*x^6 + 6*x^5 - 19*x^4 - 4*x^3 + 4*x^2 + 2*(x^7 + 5*x^6 + x^5 - 2*x^4)*e^(2/x
^2))/(x^6 + 10*x^5 + x^4*e^(4/x^2) + 25*x^4 - 4*x^3 - 20*x^2 + 2*(x^5 + 5*x^4 - 2*x^2)*e^(2/x^2) + 4)

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mupad [B]  time = 1.32, size = 49, normalized size = 2.13 \begin {gather*} \frac {x^2\,{\left (x+x^2\,{\mathrm {e}}^{\frac {2}{x^2}}+5\,x^2+x^3-2\right )}^2}{{\left (x^2\,{\mathrm {e}}^{\frac {2}{x^2}}+5\,x^2+x^3-2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^4*exp(2/x^2) - 4*x^2 + 10*x^4 + 2*x^5 - (x^4*(2*x^3 - 8*exp(2/x^2) + 8))/(x^2*exp(2/x^2) + 5*x^2 + x^
3 - 2)^2 + (x^2*(exp(2/x^2)*(8*x + 2*x^3) - 12*x + 10*x^3))/(x^2*exp(2/x^2) + 5*x^2 + x^3 - 2))/(x^3*exp(2/x^2
) - 2*x + 5*x^3 + x^4),x)

[Out]

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

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sympy [B]  time = 0.39, size = 90, normalized size = 3.91 \begin {gather*} x^{2} + \frac {2 x^{6} + 2 x^{5} e^{\frac {2}{x^{2}}} + 10 x^{5} + x^{4} - 4 x^{3}}{x^{6} + 10 x^{5} + x^{4} e^{\frac {4}{x^{2}}} + 25 x^{4} - 4 x^{3} - 20 x^{2} + \left (2 x^{5} + 10 x^{4} - 4 x^{2}\right ) e^{\frac {2}{x^{2}}} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*exp(2/x**2)-2*x**3-8)/(x**2*exp(2/x**2)+x**3+5*x**2-2)**2*x**4+((2*x**3+8*x)*exp(2/x**2)+10*x**3
-12*x)/(x**2*exp(2/x**2)+x**3+5*x**2-2)*x**2+2*x**4*exp(2/x**2)+2*x**5+10*x**4-4*x**2)/(x**3*exp(2/x**2)+x**4+
5*x**3-2*x),x)

[Out]

x**2 + (2*x**6 + 2*x**5*exp(2/x**2) + 10*x**5 + x**4 - 4*x**3)/(x**6 + 10*x**5 + x**4*exp(4/x**2) + 25*x**4 -
4*x**3 - 20*x**2 + (2*x**5 + 10*x**4 - 4*x**2)*exp(2/x**2) + 4)

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