3.87.52 \(\int \frac {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} (2-2 x-30 x^2+10 x^3)}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} (-6 x^2+2 x^3)} \, dx\)

Optimal. Leaf size=33 \[ \frac {2}{\left (3-e^{4-x}-x\right ) x}+x+(4-x) x+x^2 \]

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Rubi [F]  time = 2.33, 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 {-6+4 x+45 x^2+5 e^{8-2 x} x^2-30 x^3+5 x^4+e^{4-x} \left (2-2 x-30 x^2+10 x^3\right )}{9 x^2+e^{8-2 x} x^2-6 x^3+x^4+e^{4-x} \left (-6 x^2+2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 4*x + 45*x^2 + 5*E^(8 - 2*x)*x^2 - 30*x^3 + 5*x^4 + E^(4 - x)*(2 - 2*x - 30*x^2 + 10*x^3))/(9*x^2 +
E^(8 - 2*x)*x^2 - 6*x^3 + x^4 + E^(4 - x)*(-6*x^2 + 2*x^3)),x]

[Out]

2/(3*(3 - x)) + 2/(3*x) + 5*x + (2*E^8*Defer[Int][1/((-3 + x)^2*(E^4 - 3*E^x + E^x*x)^2), x])/3 + (4*E^8*Defer
[Int][1/((-3 + x)*(E^4 - 3*E^x + E^x*x)^2), x])/9 - (4*E^8*Defer[Int][1/(x*(E^4 - 3*E^x + E^x*x)^2), x])/9 - (
4*E^4*Defer[Int][1/((-3 + x)^2*(E^4 - 3*E^x + E^x*x)), x])/3 - (4*E^4*Defer[Int][1/((-3 + x)*(E^4 - 3*E^x + E^
x*x)), x])/9 + (2*E^4*Defer[Int][1/(x^2*(E^4 - 3*E^x + E^x*x)), x])/3 + (4*E^4*Defer[Int][1/(x*(E^4 - 3*E^x +
E^x*x)), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^8 x^2+2 e^{4+x} \left (1-x-15 x^2+5 x^3\right )+e^{2 x} \left (-6+4 x+45 x^2-30 x^3+5 x^4\right )}{\left (e^4+e^x (-3+x)\right )^2 x^2} \, dx\\ &=\int \left (\frac {2 e^8 (-2+x)}{(-3+x)^2 x \left (e^4-3 e^x+e^x x\right )^2}-\frac {2 e^4 \left (-3+x^2\right )}{(-3+x)^2 x^2 \left (e^4-3 e^x+e^x x\right )}+\frac {-6+4 x+45 x^2-30 x^3+5 x^4}{(-3+x)^2 x^2}\right ) \, dx\\ &=-\left (\left (2 e^4\right ) \int \frac {-3+x^2}{(-3+x)^2 x^2 \left (e^4-3 e^x+e^x x\right )} \, dx\right )+\left (2 e^8\right ) \int \frac {-2+x}{(-3+x)^2 x \left (e^4-3 e^x+e^x x\right )^2} \, dx+\int \frac {-6+4 x+45 x^2-30 x^3+5 x^4}{(-3+x)^2 x^2} \, dx\\ &=-\left (\left (2 e^4\right ) \int \left (\frac {2}{3 (-3+x)^2 \left (e^4-3 e^x+e^x x\right )}+\frac {2}{9 (-3+x) \left (e^4-3 e^x+e^x x\right )}-\frac {1}{3 x^2 \left (e^4-3 e^x+e^x x\right )}-\frac {2}{9 x \left (e^4-3 e^x+e^x x\right )}\right ) \, dx\right )+\left (2 e^8\right ) \int \left (\frac {1}{3 (-3+x)^2 \left (e^4-3 e^x+e^x x\right )^2}+\frac {2}{9 (-3+x) \left (e^4-3 e^x+e^x x\right )^2}-\frac {2}{9 x \left (e^4-3 e^x+e^x x\right )^2}\right ) \, dx+\int \left (5+\frac {2}{3 (-3+x)^2}-\frac {2}{3 x^2}\right ) \, dx\\ &=\frac {2}{3 (3-x)}+\frac {2}{3 x}+5 x-\frac {1}{9} \left (4 e^4\right ) \int \frac {1}{(-3+x) \left (e^4-3 e^x+e^x x\right )} \, dx+\frac {1}{9} \left (4 e^4\right ) \int \frac {1}{x \left (e^4-3 e^x+e^x x\right )} \, dx+\frac {1}{3} \left (2 e^4\right ) \int \frac {1}{x^2 \left (e^4-3 e^x+e^x x\right )} \, dx-\frac {1}{3} \left (4 e^4\right ) \int \frac {1}{(-3+x)^2 \left (e^4-3 e^x+e^x x\right )} \, dx+\frac {1}{9} \left (4 e^8\right ) \int \frac {1}{(-3+x) \left (e^4-3 e^x+e^x x\right )^2} \, dx-\frac {1}{9} \left (4 e^8\right ) \int \frac {1}{x \left (e^4-3 e^x+e^x x\right )^2} \, dx+\frac {1}{3} \left (2 e^8\right ) \int \frac {1}{(-3+x)^2 \left (e^4-3 e^x+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 42, normalized size = 1.27 \begin {gather*} \frac {5 e^4 x^2+e^x \left (-2-15 x^2+5 x^3\right )}{\left (e^4+e^x (-3+x)\right ) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 4*x + 45*x^2 + 5*E^(8 - 2*x)*x^2 - 30*x^3 + 5*x^4 + E^(4 - x)*(2 - 2*x - 30*x^2 + 10*x^3))/(9*
x^2 + E^(8 - 2*x)*x^2 - 6*x^3 + x^4 + E^(4 - x)*(-6*x^2 + 2*x^3)),x]

[Out]

(5*E^4*x^2 + E^x*(-2 - 15*x^2 + 5*x^3))/((E^4 + E^x*(-3 + x))*x)

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fricas [A]  time = 0.46, size = 41, normalized size = 1.24 \begin {gather*} \frac {5 \, x^{3} + 5 \, x^{2} e^{\left (-x + 4\right )} - 15 \, x^{2} - 2}{x^{2} + x e^{\left (-x + 4\right )} - 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^
3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="fricas")

[Out]

(5*x^3 + 5*x^2*e^(-x + 4) - 15*x^2 - 2)/(x^2 + x*e^(-x + 4) - 3*x)

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giac [B]  time = 0.15, size = 180, normalized size = 5.45 \begin {gather*} \frac {5 \, x^{6} e^{x} + 10 \, x^{5} e^{4} - 45 \, x^{5} e^{x} - 60 \, x^{4} e^{4} + 145 \, x^{4} e^{x} + 5 \, x^{4} e^{\left (-x + 8\right )} + 110 \, x^{3} e^{4} - 197 \, x^{3} e^{x} - 15 \, x^{3} e^{\left (-x + 8\right )} - 62 \, x^{2} e^{4} + 100 \, x^{2} e^{x} + 10 \, x^{2} e^{\left (-x + 8\right )} + 4 \, x e^{4} - 12 \, x e^{x}}{x^{5} e^{x} + 2 \, x^{4} e^{4} - 8 \, x^{4} e^{x} - 10 \, x^{3} e^{4} + 21 \, x^{3} e^{x} + x^{3} e^{\left (-x + 8\right )} + 12 \, x^{2} e^{4} - 18 \, x^{2} e^{x} - 2 \, x^{2} e^{\left (-x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^
3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="giac")

[Out]

(5*x^6*e^x + 10*x^5*e^4 - 45*x^5*e^x - 60*x^4*e^4 + 145*x^4*e^x + 5*x^4*e^(-x + 8) + 110*x^3*e^4 - 197*x^3*e^x
- 15*x^3*e^(-x + 8) - 62*x^2*e^4 + 100*x^2*e^x + 10*x^2*e^(-x + 8) + 4*x*e^4 - 12*x*e^x)/(x^5*e^x + 2*x^4*e^4
- 8*x^4*e^x - 10*x^3*e^4 + 21*x^3*e^x + x^3*e^(-x + 8) + 12*x^2*e^4 - 18*x^2*e^x - 2*x^2*e^(-x + 8))

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maple [A]  time = 0.21, size = 21, normalized size = 0.64

method	result	size
risch	\(5 x -\frac {2}{x \left (-3+{\mathrm e}^{-x +4}+x \right )}\)	\(21\)
norman	\(\frac {-2-15 x^{2}+5 x^{3}+5 x^{2} {\mathrm e}^{-x +4}}{x \left (-3+{\mathrm e}^{-x +4}+x \right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^3-6*x^
2)*exp(-x+4)+x^4-6*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 41, normalized size = 1.24 \begin {gather*} \frac {5 \, x^{2} e^{4} + {\left (5 \, x^{3} - 15 \, x^{2} - 2\right )} e^{x}}{x e^{4} + {\left (x^{2} - 3 \, x\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2*exp(-x+4)^2+(10*x^3-30*x^2-2*x+2)*exp(-x+4)+5*x^4-30*x^3+45*x^2+4*x-6)/(x^2*exp(-x+4)^2+(2*x^
3-6*x^2)*exp(-x+4)+x^4-6*x^3+9*x^2),x, algorithm="maxima")

[Out]

(5*x^2*e^4 + (5*x^3 - 15*x^2 - 2)*e^x)/(x*e^4 + (x^2 - 3*x)*e^x)

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mupad [B]  time = 5.32, size = 23, normalized size = 0.70 \begin {gather*} 5\,x-\frac {2}{x\,{\mathrm {e}}^{4-x}-3\,x+x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x - exp(4 - x)*(2*x + 30*x^2 - 10*x^3 - 2) + 5*x^2*exp(8 - 2*x) + 45*x^2 - 30*x^3 + 5*x^4 - 6)/(x^2*exp
(8 - 2*x) - exp(4 - x)*(6*x^2 - 2*x^3) + 9*x^2 - 6*x^3 + x^4),x)

[Out]

5*x - 2/(x*exp(4 - x) - 3*x + x^2)

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sympy [A]  time = 0.14, size = 17, normalized size = 0.52 \begin {gather*} 5 x - \frac {2}{x^{2} + x e^{4 - x} - 3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2*exp(-x+4)**2+(10*x**3-30*x**2-2*x+2)*exp(-x+4)+5*x**4-30*x**3+45*x**2+4*x-6)/(x**2*exp(-x+4)
**2+(2*x**3-6*x**2)*exp(-x+4)+x**4-6*x**3+9*x**2),x)

[Out]

5*x - 2/(x**2 + x*exp(4 - x) - 3*x)

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