3.53.70 \(\int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} (9-4 x+2 x^2+e^{-4+x} (8-2 x+2 x^2))}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} (50+50 x^2)} \, dx\)

Optimal. Leaf size=31 \[ \frac {-4+x-x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \]

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Rubi [F]  time = 5.60, 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 {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(25 + 150*x - 25*x^2 + E^(2*E^(-4 + x) + 2*x)*(9 - 4*x + 2*x^2 + E^(-4 + x)*(8 - 2*x + 2*x^2)))/(625 + E^(
4*E^(-4 + x) + 4*x) + 1250*x^2 + 625*x^4 + E^(2*E^(-4 + x) + 2*x)*(50 + 50*x^2)),x]

[Out]

-200*Defer[Int][(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^(-2), x] - (200*Defer[Int][E^x/(25 + E^(2*E^(-4 + x) +
2*x) + 25*x^2)^2, x])/E^4 + 250*Defer[Int][x/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x] + (50*Defer[Int][(E^
x*x)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x])/E^4 - 300*Defer[Int][x^2/(25 + E^(2*E^(-4 + x) + 2*x) + 25*
x^2)^2, x] - (250*Defer[Int][(E^x*x^2)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x])/E^4 + 100*Defer[Int][x^3/
(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x] + (50*Defer[Int][(E^x*x^3)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)
^2, x])/E^4 - 50*Defer[Int][x^4/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x] - (50*Defer[Int][(E^x*x^4)/(25 +
E^(2*E^(-4 + x) + 2*x) + 25*x^2)^2, x])/E^4 + 9*Defer[Int][(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2)^(-1), x] + (
8*Defer[Int][E^x/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2), x])/E^4 - 4*Defer[Int][x/(25 + E^(2*E^(-4 + x) + 2*x)
+ 25*x^2), x] - (2*Defer[Int][(E^x*x)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2), x])/E^4 + 2*Defer[Int][x^2/(25
+ E^(2*E^(-4 + x) + 2*x) + 25*x^2), x] + (2*Defer[Int][(E^x*x^2)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2), x])/E
^4

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx\\ &=\int \left (-\frac {50 \left (4-x+x^2\right ) \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{e^4 \left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {9 e^4+8 e^x-4 e^4 x-2 e^x x+2 e^4 x^2+2 e^x x^2}{e^4 \left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {9 e^4+8 e^x-4 e^4 x-2 e^x x+2 e^4 x^2+2 e^x x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}-\frac {50 \int \frac {\left (4-x+x^2\right ) \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (\frac {9 e^4}{25+e^{2 e^{-4+x}+2 x}+25 x^2}+\frac {8 e^x}{25+e^{2 e^{-4+x}+2 x}+25 x^2}-\frac {4 e^4 x}{25+e^{2 e^{-4+x}+2 x}+25 x^2}-\frac {2 e^x x}{25+e^{2 e^{-4+x}+2 x}+25 x^2}+\frac {2 e^4 x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2}+\frac {2 e^x x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2}\right ) \, dx}{e^4}-\frac {50 \int \left (\frac {4 \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}-\frac {x \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {x^2 \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}\right ) \, dx}{e^4}\\ &=2 \int \frac {x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx-4 \int \frac {x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx+9 \int \frac {1}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx-\frac {2 \int \frac {e^x x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {2 \int \frac {e^x x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {8 \int \frac {e^x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {50 \int \frac {x \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {50 \int \frac {x^2 \left (e^4+e^x-e^4 x+e^4 x^2+e^x x^2\right )}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {200 \int \frac {e^4+e^x-e^4 x+e^4 x^2+e^x x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}\\ &=2 \int \frac {x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx-4 \int \frac {x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx+9 \int \frac {1}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx-\frac {2 \int \frac {e^x x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {2 \int \frac {e^x x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {8 \int \frac {e^x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {50 \int \left (\frac {e^4 x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}-\frac {e^4 x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^4 x^3}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x x^3}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}\right ) \, dx}{e^4}-\frac {50 \int \left (\frac {e^4 x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}-\frac {e^4 x^3}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^4 x^4}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x x^4}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}\right ) \, dx}{e^4}-\frac {200 \int \left (\frac {e^4}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}-\frac {e^4 x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^4 x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}+\frac {e^x x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2}\right ) \, dx}{e^4}\\ &=2 \int \frac {x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx-4 \int \frac {x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx+9 \int \frac {1}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx+50 \int \frac {x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx-2 \left (50 \int \frac {x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx\right )+2 \left (50 \int \frac {x^3}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx\right )-50 \int \frac {x^4}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx-200 \int \frac {1}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx+200 \int \frac {x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx-200 \int \frac {x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx-\frac {2 \int \frac {e^x x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {2 \int \frac {e^x x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {8 \int \frac {e^x}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \, dx}{e^4}+\frac {50 \int \frac {e^x x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {50 \int \frac {e^x x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}+\frac {50 \int \frac {e^x x^3}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {50 \int \frac {e^x x^4}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {200 \int \frac {e^x}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}-\frac {200 \int \frac {e^x x^2}{\left (25+e^{2 e^{-4+x}+2 x}+25 x^2\right )^2} \, dx}{e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 32, normalized size = 1.03 \begin {gather*} -\frac {4-x+x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(25 + 150*x - 25*x^2 + E^(2*E^(-4 + x) + 2*x)*(9 - 4*x + 2*x^2 + E^(-4 + x)*(8 - 2*x + 2*x^2)))/(625
+ E^(4*E^(-4 + x) + 4*x) + 1250*x^2 + 625*x^4 + E^(2*E^(-4 + x) + 2*x)*(50 + 50*x^2)),x]

[Out]

-((4 - x + x^2)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*x^2))

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fricas [A]  time = 0.47, size = 30, normalized size = 0.97 \begin {gather*} -\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x+8)*exp(x-4)+2*x^2-4*x+9)*exp(x+exp(x-4))^2-25*x^2+150*x+25)/(exp(x+exp(x-4))^4+(50*x^2+
50)*exp(x+exp(x-4))^2+625*x^4+1250*x^2+625),x, algorithm="fricas")

[Out]

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)

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giac [A]  time = 0.26, size = 30, normalized size = 0.97 \begin {gather*} -\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x+8)*exp(x-4)+2*x^2-4*x+9)*exp(x+exp(x-4))^2-25*x^2+150*x+25)/(exp(x+exp(x-4))^4+(50*x^2+
50)*exp(x+exp(x-4))^2+625*x^4+1250*x^2+625),x, algorithm="giac")

[Out]

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)

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maple [A]  time = 0.05, size = 31, normalized size = 1.00

method	result	size
risch	\(-\frac {x^{2}-x +4}{25 x^{2}+25+{\mathrm e}^{2 x +2 \,{\mathrm e}^{x -4}}}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^2-2*x+8)*exp(x-4)+2*x^2-4*x+9)*exp(x+exp(x-4))^2-25*x^2+150*x+25)/(exp(x+exp(x-4))^4+(50*x^2+50)*ex
p(x+exp(x-4))^2+625*x^4+1250*x^2+625),x,method=_RETURNVERBOSE)

[Out]

-(x^2-x+4)/(25*x^2+25+exp(2*x+2*exp(x-4)))

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maxima [A]  time = 0.41, size = 30, normalized size = 0.97 \begin {gather*} -\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^2-2*x+8)*exp(x-4)+2*x^2-4*x+9)*exp(x+exp(x-4))^2-25*x^2+150*x+25)/(exp(x+exp(x-4))^4+(50*x^2+
50)*exp(x+exp(x-4))^2+625*x^4+1250*x^2+625),x, algorithm="maxima")

[Out]

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {150\,x+{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^{x-4}}\,\left ({\mathrm {e}}^{x-4}\,\left (2\,x^2-2\,x+8\right )-4\,x+2\,x^2+9\right )-25\,x^2+25}{{\mathrm {e}}^{4\,x+4\,{\mathrm {e}}^{x-4}}+{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^{x-4}}\,\left (50\,x^2+50\right )+1250\,x^2+625\,x^4+625} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((150*x + exp(2*x + 2*exp(x - 4))*(exp(x - 4)*(2*x^2 - 2*x + 8) - 4*x + 2*x^2 + 9) - 25*x^2 + 25)/(exp(4*x
+ 4*exp(x - 4)) + exp(2*x + 2*exp(x - 4))*(50*x^2 + 50) + 1250*x^2 + 625*x^4 + 625),x)

[Out]

int((150*x + exp(2*x + 2*exp(x - 4))*(exp(x - 4)*(2*x^2 - 2*x + 8) - 4*x + 2*x^2 + 9) - 25*x^2 + 25)/(exp(4*x
+ 4*exp(x - 4)) + exp(2*x + 2*exp(x - 4))*(50*x^2 + 50) + 1250*x^2 + 625*x^4 + 625), x)

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sympy [A]  time = 0.20, size = 24, normalized size = 0.77 \begin {gather*} \frac {- x^{2} + x - 4}{25 x^{2} + e^{2 x + 2 e^{x - 4}} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**2-2*x+8)*exp(x-4)+2*x**2-4*x+9)*exp(x+exp(x-4))**2-25*x**2+150*x+25)/(exp(x+exp(x-4))**4+(50
*x**2+50)*exp(x+exp(x-4))**2+625*x**4+1250*x**2+625),x)

[Out]

(-x**2 + x - 4)/(25*x**2 + exp(2*x + 2*exp(x - 4)) + 25)

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