3.41.3 $\int \frac{e^{-4+e^{\frac{x^2}{5+x}}-x}(-500-200x-20x^2+e^4(-50x-20x^2-2x^3)+e^{\frac{x^2}{5+x}}(200x+20x^2+e^4(20x+22x^2+2x^3)))}{25+10x+x^2}dx$

Optimal. Leaf size=27 $$2e^{e^{\frac{x^2}{5+x}}-x}(1+\frac{10}{e^4}+x)$$

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Rubi [F]  time = 3.64, 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 = {} $$\begin{array}{c}\int \frac{e^{-4+e^{\frac{x^2}{5+x}}-x}(-500-200x-20x^2+e^4(-50x-20x^2-2x^3)+e^{\frac{x^2}{5+x}}(200x+20x^2+e^4(20x+22x^2+2x^3)))}{25+10x+x^2}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[(E^(-4 + E^(x^2/(5 + x)) - x)*(-500 - 200*x - 20*x^2 + E^4*(-50*x - 20*x^2 - 2*x^3) + E^(x^2/(5 + x))*(200
*x + 20*x^2 + E^4*(20*x + 22*x^2 + 2*x^3))))/(25 + 10*x + x^2),x]

[Out]

-20*Defer[Int][E^(-4 + E^(x^2/(5 + x)) - x), x] + 2*(10 + E^4)*Defer[Int][E^(-4 + E^(x^2/(5 + x)) - x + x^2/(5
+ x)), x] - 2*Defer[Int][E^(E^(x^2/(5 + x)) - x)*x, x] + 2*Defer[Int][E^(E^(x^2/(5 + x)) - x + x^2/(5 + x))*x
, x] - 100*(5 - 2*E^4)*Defer[Int][E^(-4 + E^(x^2/(5 + x)) - x + x^2/(5 + x))/(5 + x)^2, x] - 50*Defer[Int][E^(
E^(x^2/(5 + x)) - x + x^2/(5 + x))/(5 + x), x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{-4+e^{\frac{x^2}{5+x}}-x}(-500-200x-20x^2+e^4(-50x-20x^2-2x^3)+e^{\frac{x^2}{5+x}}(200x+20x^2+e^4(20x+22x^2+2x^3)))}{(5+x{)}^2}dx\\ & =\int (-2e^{-4+e^{\frac{x^2}{5+x}}-x}(10+e^{4}x)+\frac{2e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}x(10+x)(10+e^4+e^{4}x)}{(5+x{)}^2})dx\\ & =-(2\int e^{-4+e^{\frac{x^2}{5+x}}-x}(10+e^{4}x)dx)+2\int \frac{e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}x(10+x)(10+e^4+e^{4}x)}{(5+x{)}^2}dx\\ & =-(2\int (10e^{-4+e^{\frac{x^2}{5+x}}-x}+e^{e^{\frac{x^2}{5+x}}-x}x)dx)+2\int (10e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}(1+\frac{e^4}{10})+e^{e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}x+\frac{50e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}(-5+2e^4)}{(5+x{)}^2}-\frac{25e^{e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}}{5+x})dx\\ & =-(2\int e^{e^{\frac{x^2}{5+x}}-x}xdx)+2\int e^{e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}xdx-20\int e^{-4+e^{\frac{x^2}{5+x}}-x}dx-50\int \frac{e^{e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}}{5+x}dx-\left(100(5-2e^4)\right)\int \frac{e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}}{(5+x{)}^2}dx+\left(2(10+e^4)\right)\int e^{-4+e^{\frac{x^2}{5+x}}-x+\frac{x^2}{5+x}}dx\end{array}\end{array}$$

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Mathematica [A]  time = 1.53, size = 29, normalized size = 1.07 $$\begin{array}{c}{e}^{-4+e^{\frac{x^2}{5+x}}-x}(20+2e^4(1+x))\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(E^(-4 + E^(x^2/(5 + x)) - x)*(-500 - 200*x - 20*x^2 + E^4*(-50*x - 20*x^2 - 2*x^3) + E^(x^2/(5 + x)
)*(200*x + 20*x^2 + E^4*(20*x + 22*x^2 + 2*x^3))))/(25 + 10*x + x^2),x]

[Out]

E^(-4 + E^(x^2/(5 + x)) - x)*(20 + 2*E^4*(1 + x))

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fricas [A]  time = 0.70, size = 26, normalized size = 0.96 $$\begin{array}{c}2((x+1)e^4+10)e^{(-x+e^{\left(\frac{x^2}{x+5}\right)}-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+22*x^2+20*x)*exp(4)+20*x^2+200*x)*exp(x^2/(5+x))+(-2*x^3-20*x^2-50*x)*exp(4)-20*x^2-200*x-5
00)/(x^2+10*x+25)/exp(4)/exp(-exp(x^2/(5+x))+x),x, algorithm="fricas")

[Out]

2*((x + 1)*e^4 + 10)*e^(-x + e^(x^2/(x + 5)) - 4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 $$\begin{array}{c}\int -\frac{2(10x^2+(x^3+10x^2+25x)e^4-(10x^2+(x^3+11x^2+10x)e^4+100x)e^{\left(\frac{x^2}{x+5}\right)}+100x+250)e^{(-x+e^{\left(\frac{x^2}{x+5}\right)}-4)}}{x^2+10x+25}dx\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+22*x^2+20*x)*exp(4)+20*x^2+200*x)*exp(x^2/(5+x))+(-2*x^3-20*x^2-50*x)*exp(4)-20*x^2-200*x-5
00)/(x^2+10*x+25)/exp(4)/exp(-exp(x^2/(5+x))+x),x, algorithm="giac")

[Out]

integrate(-2*(10*x^2 + (x^3 + 10*x^2 + 25*x)*e^4 - (10*x^2 + (x^3 + 11*x^2 + 10*x)*e^4 + 100*x)*e^(x^2/(x + 5)
) + 100*x + 250)*e^(-x + e^(x^2/(x + 5)) - 4)/(x^2 + 10*x + 25), x)

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maple [A]  time = 0.13, size = 29, normalized size = 1.07

method	result	size
risch	$(2x{\mathrm{e}}^4+2{\mathrm{e}}^4+20){\mathrm{e}}^{-4+{\mathrm{e}}^{\frac{x^2}{5+x}}-x}$	$29$
norman	$\frac{(2x^2+10(10+{\mathrm{e}}^4){\mathrm{e}}^{-4}+4(5+3{\mathrm{e}}^4){\mathrm{e}}^{-4}x){\mathrm{e}}^{{\mathrm{e}}^{\frac{x^2}{5+x}}-x}}{5+x}$	$53$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x^3+22*x^2+20*x)*exp(4)+20*x^2+200*x)*exp(x^2/(5+x))+(-2*x^3-20*x^2-50*x)*exp(4)-20*x^2-200*x-500)/(x
^2+10*x+25)/exp(4)/exp(-exp(x^2/(5+x))+x),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.45, size = 27, normalized size = 1.00 $$\begin{array}{c}2(x{e}^4+e^4+10)e^{(-x+e^{(x+\frac{25}{x+5}-5)}-4)}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x^3+22*x^2+20*x)*exp(4)+20*x^2+200*x)*exp(x^2/(5+x))+(-2*x^3-20*x^2-50*x)*exp(4)-20*x^2-200*x-5
00)/(x^2+10*x+25)/exp(4)/exp(-exp(x^2/(5+x))+x),x, algorithm="maxima")

[Out]

2*(x*e^4 + e^4 + 10)*e^(-x + e^(x + 25/(x + 5) - 5) - 4)

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mupad [B]  time = 2.89, size = 29, normalized size = 1.07 $$\begin{array}{c}{\mathrm{e}}^{{\mathrm{e}}^{\frac{x^2}{x+5}}-x}(2x+{\mathrm{e}}^{-4}(2{\mathrm{e}}^4+20))\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-4)*exp(exp(x^2/(x + 5)) - x)*(200*x - exp(x^2/(x + 5))*(200*x + exp(4)*(20*x + 22*x^2 + 2*x^3) + 20
*x^2) + exp(4)*(50*x + 20*x^2 + 2*x^3) + 20*x^2 + 500))/(10*x + x^2 + 25),x)

[Out]

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

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sympy [A]  time = 17.47, size = 27, normalized size = 1.00 $$\begin{array}{c}\frac{(2x{e}^4+20+2e^4)e^{-x+e^{\frac{x^2}{x+5}}}}{e^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x**3+22*x**2+20*x)*exp(4)+20*x**2+200*x)*exp(x**2/(5+x))+(-2*x**3-20*x**2-50*x)*exp(4)-20*x**2-
200*x-500)/(x**2+10*x+25)/exp(4)/exp(-exp(x**2/(5+x))+x),x)

[Out]

(2*x*exp(4) + 20 + 2*exp(4))*exp(-4)*exp(-x + exp(x**2/(x + 5)))

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