3.10.10 \(\int \frac {-2 e^2 x-2 e^{2 x} x-6 x^2-50 x^3+40 x^4-8 x^5+e^x (3-3 x+4 e x+20 x^2-8 x^3)+e (-3-20 x^2+8 x^3)}{e^2+e^{2 x}+25 x^2-20 x^3+4 x^4+e (10 x-4 x^2)+e^x (-2 e-10 x+4 x^2)} \, dx\)

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

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Rubi [F]  time = 1.08, 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 {-2 e^2 x-2 e^{2 x} x-6 x^2-50 x^3+40 x^4-8 x^5+e^x \left (3-3 x+4 e x+20 x^2-8 x^3\right )+e \left (-3-20 x^2+8 x^3\right )}{e^2+e^{2 x}+25 x^2-20 x^3+4 x^4+e \left (10 x-4 x^2\right )+e^x \left (-2 e-10 x+4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*E^2*x - 2*E^(2*x)*x - 6*x^2 - 50*x^3 + 40*x^4 - 8*x^5 + E^x*(3 - 3*x + 4*E*x + 20*x^2 - 8*x^3) + E*(-3
- 20*x^2 + 8*x^3))/(E^2 + E^(2*x) + 25*x^2 - 20*x^3 + 4*x^4 + E*(10*x - 4*x^2) + E^x*(-2*E - 10*x + 4*x^2)),x
]

[Out]

-x^2 + 3*(5 - E)*Defer[Int][x/(E - E^x + 5*x - 2*x^2)^2, x] - 27*Defer[Int][x^2/(-E + E^x - 5*x + 2*x^2)^2, x]
+ 6*Defer[Int][x^3/(-E + E^x - 5*x + 2*x^2)^2, x] + 3*Defer[Int][(-E + E^x - 5*x + 2*x^2)^(-1), x] - 3*Defer[
Int][x/(-E + E^x - 5*x + 2*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 e^2 x-2 e^{2 x} x+4 e^{1+x} x+e^x \left (3-3 x+20 x^2-8 x^3\right )-2 x^2 \left (3+25 x-20 x^2+4 x^3\right )+e \left (-3-20 x^2+8 x^3\right )}{\left (e-e^x-x (-5+2 x)\right )^2} \, dx\\ &=\int \left (-2 x+\frac {3 x \left (5-e-9 x+2 x^2\right )}{\left (-e+e^x-5 x+2 x^2\right )^2}-\frac {3 (-1+x)}{-e+e^x-5 x+2 x^2}\right ) \, dx\\ &=-x^2+3 \int \frac {x \left (5-e-9 x+2 x^2\right )}{\left (-e+e^x-5 x+2 x^2\right )^2} \, dx-3 \int \frac {-1+x}{-e+e^x-5 x+2 x^2} \, dx\\ &=-x^2+3 \int \left (-\frac {(-5+e) x}{\left (e-e^x+5 x-2 x^2\right )^2}-\frac {9 x^2}{\left (-e+e^x-5 x+2 x^2\right )^2}+\frac {2 x^3}{\left (-e+e^x-5 x+2 x^2\right )^2}\right ) \, dx-3 \int \left (-\frac {1}{-e+e^x-5 x+2 x^2}+\frac {x}{-e+e^x-5 x+2 x^2}\right ) \, dx\\ &=-x^2+3 \int \frac {1}{-e+e^x-5 x+2 x^2} \, dx-3 \int \frac {x}{-e+e^x-5 x+2 x^2} \, dx+6 \int \frac {x^3}{\left (-e+e^x-5 x+2 x^2\right )^2} \, dx-27 \int \frac {x^2}{\left (-e+e^x-5 x+2 x^2\right )^2} \, dx+(3 (5-e)) \int \frac {x}{\left (e-e^x+5 x-2 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 26, normalized size = 0.90 \begin {gather*} -x^2+\frac {3 x}{-e+e^x-5 x+2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*E^2*x - 2*E^(2*x)*x - 6*x^2 - 50*x^3 + 40*x^4 - 8*x^5 + E^x*(3 - 3*x + 4*E*x + 20*x^2 - 8*x^3) +
E*(-3 - 20*x^2 + 8*x^3))/(E^2 + E^(2*x) + 25*x^2 - 20*x^3 + 4*x^4 + E*(10*x - 4*x^2) + E^x*(-2*E - 10*x + 4*x
^2)),x]

[Out]

-x^2 + (3*x)/(-E + E^x - 5*x + 2*x^2)

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fricas [A]  time = 0.48, size = 46, normalized size = 1.59 \begin {gather*} -\frac {2 \, x^{4} - 5 \, x^{3} - x^{2} e + x^{2} e^{x} - 3 \, x}{2 \, x^{2} - 5 \, x - e + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)^2+(4*x*exp(1)-8*x^3+20*x^2-3*x+3)*exp(x)-2*x*exp(1)^2+(8*x^3-20*x^2-3)*exp(1)-8*x^5+40*
x^4-50*x^3-6*x^2)/(exp(x)^2+(-2*exp(1)+4*x^2-10*x)*exp(x)+exp(1)^2+(-4*x^2+10*x)*exp(1)+4*x^4-20*x^3+25*x^2),x
, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.82, size = 46, normalized size = 1.59 \begin {gather*} -\frac {2 \, x^{4} - 5 \, x^{3} - x^{2} e + x^{2} e^{x} - 3 \, x}{2 \, x^{2} - 5 \, x - e + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)^2+(4*x*exp(1)-8*x^3+20*x^2-3*x+3)*exp(x)-2*x*exp(1)^2+(8*x^3-20*x^2-3)*exp(1)-8*x^5+40*
x^4-50*x^3-6*x^2)/(exp(x)^2+(-2*exp(1)+4*x^2-10*x)*exp(x)+exp(1)^2+(-4*x^2+10*x)*exp(1)+4*x^4-20*x^3+25*x^2),x
, algorithm="giac")

[Out]

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

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maple [A]  time = 0.18, size = 27, normalized size = 0.93

method	result	size
risch	\(-x^{2}-\frac {3 x}{-2 x^{2}+{\mathrm e}-{\mathrm e}^{x}+5 x}\)	\(27\)
norman	\(\frac {\frac {{\mathrm e} \,{\mathrm e}^{x}}{2}+\left (-3-\frac {5 \,{\mathrm e}}{2}\right ) x +{\mathrm e}^{x} x^{2}-5 x^{3}+2 x^{4}-\frac {{\mathrm e}^{2}}{2}}{-2 x^{2}+{\mathrm e}-{\mathrm e}^{x}+5 x}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x*exp(x)^2+(4*x*exp(1)-8*x^3+20*x^2-3*x+3)*exp(x)-2*x*exp(1)^2+(8*x^3-20*x^2-3)*exp(1)-8*x^5+40*x^4-50
*x^3-6*x^2)/(exp(x)^2+(-2*exp(1)+4*x^2-10*x)*exp(x)+exp(1)^2+(-4*x^2+10*x)*exp(1)+4*x^4-20*x^3+25*x^2),x,metho
d=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.65, size = 46, normalized size = 1.59 \begin {gather*} -\frac {2 \, x^{4} - 5 \, x^{3} - x^{2} e + x^{2} e^{x} - 3 \, x}{2 \, x^{2} - 5 \, x - e + e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)^2+(4*x*exp(1)-8*x^3+20*x^2-3*x+3)*exp(x)-2*x*exp(1)^2+(8*x^3-20*x^2-3)*exp(1)-8*x^5+40*
x^4-50*x^3-6*x^2)/(exp(x)^2+(-2*exp(1)+4*x^2-10*x)*exp(x)+exp(1)^2+(-4*x^2+10*x)*exp(1)+4*x^4-20*x^3+25*x^2),x
, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.94, size = 26, normalized size = 0.90 \begin {gather*} -\frac {3\,x}{5\,x+\mathrm {e}-{\mathrm {e}}^x-2\,x^2}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x*exp(2*x) - exp(x)*(4*x*exp(1) - 3*x + 20*x^2 - 8*x^3 + 3) + 2*x*exp(2) + exp(1)*(20*x^2 - 8*x^3 + 3)
+ 6*x^2 + 50*x^3 - 40*x^4 + 8*x^5)/(exp(2*x) + exp(2) + exp(1)*(10*x - 4*x^2) - exp(x)*(10*x + 2*exp(1) - 4*x
^2) + 25*x^2 - 20*x^3 + 4*x^4),x)

[Out]

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

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sympy [A]  time = 0.15, size = 20, normalized size = 0.69 \begin {gather*} - x^{2} + \frac {3 x}{2 x^{2} - 5 x + e^{x} - e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x*exp(x)**2+(4*x*exp(1)-8*x**3+20*x**2-3*x+3)*exp(x)-2*x*exp(1)**2+(8*x**3-20*x**2-3)*exp(1)-8*x
**5+40*x**4-50*x**3-6*x**2)/(exp(x)**2+(-2*exp(1)+4*x**2-10*x)*exp(x)+exp(1)**2+(-4*x**2+10*x)*exp(1)+4*x**4-2
0*x**3+25*x**2),x)

[Out]

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

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