3.57.86 \(\int \frac {40 e^{2 x}+40 e^{10 x^2} x^2-60 x^3+40 x^4+e^x (20 x-60 x^2)+e^{5 x^2} (-80 e^x x-40 x^2+80 x^3-200 x^4)}{e^{2 x} x^3-2 e^x x^5+e^{10 x^2} x^5+x^7+e^{5 x^2} (-2 e^x x^4+2 x^6)} \, dx\)

Optimal. Leaf size=28 \[ \frac {4 \left (-5+\frac {5}{e^{5 x^2}-\frac {e^x}{x}+x}\right )}{x^2} \]

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Rubi [F]  time = 2.79, 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 {40 e^{2 x}+40 e^{10 x^2} x^2-60 x^3+40 x^4+e^x \left (20 x-60 x^2\right )+e^{5 x^2} \left (-80 e^x x-40 x^2+80 x^3-200 x^4\right )}{e^{2 x} x^3-2 e^x x^5+e^{10 x^2} x^5+x^7+e^{5 x^2} \left (-2 e^x x^4+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(40*E^(2*x) + 40*E^(10*x^2)*x^2 - 60*x^3 + 40*x^4 + E^x*(20*x - 60*x^2) + E^(5*x^2)*(-80*E^x*x - 40*x^2 +
80*x^3 - 200*x^4))/(E^(2*x)*x^3 - 2*E^x*x^5 + E^(10*x^2)*x^5 + x^7 + E^(5*x^2)*(-2*E^x*x^4 + 2*x^6)),x]

[Out]

-20/x^2 - 20*Defer[Int][(E^x - E^(5*x^2)*x - x^2)^(-2), x] - 200*Defer[Int][E^x/(E^x - E^(5*x^2)*x - x^2)^2, x
] - 20*Defer[Int][E^x/(x^2*(E^x - E^(5*x^2)*x - x^2)^2), x] + 20*Defer[Int][E^x/(x*(E^x - E^(5*x^2)*x - x^2)^2
), x] + 200*Defer[Int][(E^x - E^(5*x^2)*x - x^2)^(-1), x] + 200*Defer[Int][x^2/(-E^x + E^(5*x^2)*x + x^2)^2, x
] - 40*Defer[Int][1/(x^2*(-E^x + E^(5*x^2)*x + x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {40 e^{2 x}+40 e^{10 x^2} x^2-60 x^3+40 x^4+e^x \left (20 x-60 x^2\right )+e^{5 x^2} \left (-80 e^x x-40 x^2+80 x^3-200 x^4\right )}{x^3 \left (e^x-e^{5 x^2} x-x^2\right )^2} \, dx\\ &=\int \left (\frac {40}{x^3}-\frac {40 \left (1+5 x^2\right )}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )}+\frac {20 \left (-e^x+e^x x-x^2-10 e^x x^2+10 x^4\right )}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )^2}\right ) \, dx\\ &=-\frac {20}{x^2}+20 \int \frac {-e^x+e^x x-x^2-10 e^x x^2+10 x^4}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )^2} \, dx-40 \int \frac {1+5 x^2}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )} \, dx\\ &=-\frac {20}{x^2}+20 \int \left (-\frac {1}{\left (e^x-e^{5 x^2} x-x^2\right )^2}-\frac {10 e^x}{\left (e^x-e^{5 x^2} x-x^2\right )^2}-\frac {e^x}{x^2 \left (e^x-e^{5 x^2} x-x^2\right )^2}+\frac {e^x}{x \left (e^x-e^{5 x^2} x-x^2\right )^2}+\frac {10 x^2}{\left (-e^x+e^{5 x^2} x+x^2\right )^2}\right ) \, dx-40 \int \left (-\frac {5}{e^x-e^{5 x^2} x-x^2}+\frac {1}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )}\right ) \, dx\\ &=-\frac {20}{x^2}-20 \int \frac {1}{\left (e^x-e^{5 x^2} x-x^2\right )^2} \, dx-20 \int \frac {e^x}{x^2 \left (e^x-e^{5 x^2} x-x^2\right )^2} \, dx+20 \int \frac {e^x}{x \left (e^x-e^{5 x^2} x-x^2\right )^2} \, dx-40 \int \frac {1}{x^2 \left (-e^x+e^{5 x^2} x+x^2\right )} \, dx-200 \int \frac {e^x}{\left (e^x-e^{5 x^2} x-x^2\right )^2} \, dx+200 \int \frac {1}{e^x-e^{5 x^2} x-x^2} \, dx+200 \int \frac {x^2}{\left (-e^x+e^{5 x^2} x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 32, normalized size = 1.14 \begin {gather*} 20 \left (-\frac {1}{x^2}+\frac {1}{x \left (-e^x+e^{5 x^2} x+x^2\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40*E^(2*x) + 40*E^(10*x^2)*x^2 - 60*x^3 + 40*x^4 + E^x*(20*x - 60*x^2) + E^(5*x^2)*(-80*E^x*x - 40*
x^2 + 80*x^3 - 200*x^4))/(E^(2*x)*x^3 - 2*E^x*x^5 + E^(10*x^2)*x^5 + x^7 + E^(5*x^2)*(-2*E^x*x^4 + 2*x^6)),x]

[Out]

20*(-x^(-2) + 1/(x*(-E^x + E^(5*x^2)*x + x^2)))

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fricas [A]  time = 0.57, size = 44, normalized size = 1.57 \begin {gather*} -\frac {20 \, {\left (x^{2} + x e^{\left (5 \, x^{2}\right )} - x - e^{x}\right )}}{x^{4} + x^{3} e^{\left (5 \, x^{2}\right )} - x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^2*exp(5*x^2)^2+(-80*exp(x)*x-200*x^4+80*x^3-40*x^2)*exp(5*x^2)+40*exp(x)^2+(-60*x^2+20*x)*exp(
x)+40*x^4-60*x^3)/(x^5*exp(5*x^2)^2+(-2*exp(x)*x^4+2*x^6)*exp(5*x^2)+exp(x)^2*x^3-2*x^5*exp(x)+x^7),x, algorit
hm="fricas")

[Out]

-20*(x^2 + x*e^(5*x^2) - x - e^x)/(x^4 + x^3*e^(5*x^2) - x^2*e^x)

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giac [A]  time = 0.37, size = 44, normalized size = 1.57 \begin {gather*} -\frac {20 \, {\left (x^{2} + x e^{\left (5 \, x^{2}\right )} - x - e^{x}\right )}}{x^{4} + x^{3} e^{\left (5 \, x^{2}\right )} - x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^2*exp(5*x^2)^2+(-80*exp(x)*x-200*x^4+80*x^3-40*x^2)*exp(5*x^2)+40*exp(x)^2+(-60*x^2+20*x)*exp(
x)+40*x^4-60*x^3)/(x^5*exp(5*x^2)^2+(-2*exp(x)*x^4+2*x^6)*exp(5*x^2)+exp(x)^2*x^3-2*x^5*exp(x)+x^7),x, algorit
hm="giac")

[Out]

-20*(x^2 + x*e^(5*x^2) - x - e^x)/(x^4 + x^3*e^(5*x^2) - x^2*e^x)

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maple [A]  time = 0.06, size = 30, normalized size = 1.07

method	result	size
risch	\(-\frac {20}{x^{2}}+\frac {20}{x \left (x^{2}+x \,{\mathrm e}^{5 x^{2}}-{\mathrm e}^{x}\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x^2*exp(5*x^2)^2+(-80*exp(x)*x-200*x^4+80*x^3-40*x^2)*exp(5*x^2)+40*exp(x)^2+(-60*x^2+20*x)*exp(x)+40*
x^4-60*x^3)/(x^5*exp(5*x^2)^2+(-2*exp(x)*x^4+2*x^6)*exp(5*x^2)+exp(x)^2*x^3-2*x^5*exp(x)+x^7),x,method=_RETURN
VERBOSE)

[Out]

-20/x^2+20/x/(x^2+x*exp(5*x^2)-exp(x))

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maxima [A]  time = 0.40, size = 44, normalized size = 1.57 \begin {gather*} -\frac {20 \, {\left (x^{2} + x e^{\left (5 \, x^{2}\right )} - x - e^{x}\right )}}{x^{4} + x^{3} e^{\left (5 \, x^{2}\right )} - x^{2} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x^2*exp(5*x^2)^2+(-80*exp(x)*x-200*x^4+80*x^3-40*x^2)*exp(5*x^2)+40*exp(x)^2+(-60*x^2+20*x)*exp(
x)+40*x^4-60*x^3)/(x^5*exp(5*x^2)^2+(-2*exp(x)*x^4+2*x^6)*exp(5*x^2)+exp(x)^2*x^3-2*x^5*exp(x)+x^7),x, algorit
hm="maxima")

[Out]

-20*(x^2 + x*e^(5*x^2) - x - e^x)/(x^4 + x^3*e^(5*x^2) - x^2*e^x)

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mupad [B]  time = 3.60, size = 29, normalized size = 1.04 \begin {gather*} \frac {20}{x\,\left (x\,{\mathrm {e}}^{5\,x^2}-{\mathrm {e}}^x+x^2\right )}-\frac {20}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*exp(2*x) + exp(x)*(20*x - 60*x^2) - exp(5*x^2)*(80*x*exp(x) + 40*x^2 - 80*x^3 + 200*x^4) + 40*x^2*exp(
10*x^2) - 60*x^3 + 40*x^4)/(x^3*exp(2*x) - 2*x^5*exp(x) - exp(5*x^2)*(2*x^4*exp(x) - 2*x^6) + x^5*exp(10*x^2)
+ x^7),x)

[Out]

20/(x*(x*exp(5*x^2) - exp(x) + x^2)) - 20/x^2

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sympy [A]  time = 0.19, size = 24, normalized size = 0.86 \begin {gather*} \frac {20}{x^{3} + x^{2} e^{5 x^{2}} - x e^{x}} - \frac {20}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((40*x**2*exp(5*x**2)**2+(-80*exp(x)*x-200*x**4+80*x**3-40*x**2)*exp(5*x**2)+40*exp(x)**2+(-60*x**2+2
0*x)*exp(x)+40*x**4-60*x**3)/(x**5*exp(5*x**2)**2+(-2*exp(x)*x**4+2*x**6)*exp(5*x**2)+exp(x)**2*x**3-2*x**5*ex
p(x)+x**7),x)

[Out]

20/(x**3 + x**2*exp(5*x**2) - x*exp(x)) - 20/x**2

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