3.1.7 \(\int \frac {e^{\frac {2 x^2}{10+7 x+x^2+e^{5+x} (10+2 x)}} (40 x+14 x^2+e^{5+x} (40 x-16 x^2-4 x^3))}{100+140 x+69 x^2+14 x^3+x^4+e^{10+2 x} (100+40 x+4 x^2)+e^{5+x} (200+180 x+48 x^2+4 x^3)} \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} \]

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Rubi [F]  time = 8.57, 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 {e^{\frac {2 x^2}{10+7 x+x^2+e^{5+x} (10+2 x)}} \left (40 x+14 x^2+e^{5+x} \left (40 x-16 x^2-4 x^3\right )\right )}{100+140 x+69 x^2+14 x^3+x^4+e^{10+2 x} \left (100+40 x+4 x^2\right )+e^{5+x} \left (200+180 x+48 x^2+4 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*x^2)/(10 + 7*x + x^2 + E^(5 + x)*(10 + 2*x)))*(40*x + 14*x^2 + E^(5 + x)*(40*x - 16*x^2 - 4*x^3)))/
(100 + 140*x + 69*x^2 + 14*x^3 + x^4 + E^(10 + 2*x)*(100 + 40*x + 4*x^2) + E^(5 + x)*(200 + 180*x + 48*x^2 + 4
*x^3)),x]

[Out]

40*Defer[Int][E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))/(2 + 2*E^(5 + x) + x)^2, x] - 8*Defer[Int][(E^((2*x^
2)/((5 + x)*(2 + 2*E^(5 + x) + x)))*x)/(2 + 2*E^(5 + x) + x)^2, x] + 2*Defer[Int][(E^((2*x^2)/((5 + x)*(2 + 2*
E^(5 + x) + x)))*x^2)/(2 + 2*E^(5 + x) + x)^2, x] - 200*Defer[Int][E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))
/((5 + x)*(2 + 2*E^(5 + x) + x)^2), x] + 12*Defer[Int][E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))/(2 + 2*E^(5
+ x) + x), x] - 2*Defer[Int][(E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))*x)/(2 + 2*E^(5 + x) + x), x] - 50*D
efer[Int][E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))/((5 + x)^2*(2 + 2*E^(5 + x) + x)), x] - 50*Defer[Int][E^
((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))/((5 + x)*(2 + 2*E^(5 + x) + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x \left (20+7 x-2 e^{5+x} \left (-10+4 x+x^2\right )\right )}{(5+x)^2 \left (2+2 e^{5+x}+x\right )^2} \, dx\\ &=2 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x \left (20+7 x-2 e^{5+x} \left (-10+4 x+x^2\right )\right )}{(5+x)^2 \left (2+2 e^{5+x}+x\right )^2} \, dx\\ &=2 \int \left (\frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x^2 (1+x)}{(5+x) \left (2+2 e^{5+x}+x\right )^2}-\frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x \left (-10+4 x+x^2\right )}{(5+x)^2 \left (2+2 e^{5+x}+x\right )}\right ) \, dx\\ &=2 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x^2 (1+x)}{(5+x) \left (2+2 e^{5+x}+x\right )^2} \, dx-2 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x \left (-10+4 x+x^2\right )}{(5+x)^2 \left (2+2 e^{5+x}+x\right )} \, dx\\ &=2 \int \left (\frac {20 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{\left (2+2 e^{5+x}+x\right )^2}-\frac {4 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x}{\left (2+2 e^{5+x}+x\right )^2}+\frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x^2}{\left (2+2 e^{5+x}+x\right )^2}-\frac {100 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x) \left (2+2 e^{5+x}+x\right )^2}\right ) \, dx-2 \int \left (-\frac {6 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{2+2 e^{5+x}+x}+\frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x}{2+2 e^{5+x}+x}+\frac {25 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x)^2 \left (2+2 e^{5+x}+x\right )}+\frac {25 e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x) \left (2+2 e^{5+x}+x\right )}\right ) \, dx\\ &=2 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x^2}{\left (2+2 e^{5+x}+x\right )^2} \, dx-2 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x}{2+2 e^{5+x}+x} \, dx-8 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} x}{\left (2+2 e^{5+x}+x\right )^2} \, dx+12 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{2+2 e^{5+x}+x} \, dx+40 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{\left (2+2 e^{5+x}+x\right )^2} \, dx-50 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x)^2 \left (2+2 e^{5+x}+x\right )} \, dx-50 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x) \left (2+2 e^{5+x}+x\right )} \, dx-200 \int \frac {e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}}}{(5+x) \left (2+2 e^{5+x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.96, size = 24, normalized size = 1.00 \begin {gather*} e^{\frac {2 x^2}{(5+x) \left (2+2 e^{5+x}+x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*x^2)/(10 + 7*x + x^2 + E^(5 + x)*(10 + 2*x)))*(40*x + 14*x^2 + E^(5 + x)*(40*x - 16*x^2 - 4*x
^3)))/(100 + 140*x + 69*x^2 + 14*x^3 + x^4 + E^(10 + 2*x)*(100 + 40*x + 4*x^2) + E^(5 + x)*(200 + 180*x + 48*x
^2 + 4*x^3)),x]

[Out]

E^((2*x^2)/((5 + x)*(2 + 2*E^(5 + x) + x)))

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fricas [A]  time = 0.52, size = 25, normalized size = 1.04 \begin {gather*} e^{\left (\frac {2 \, x^{2}}{x^{2} + 2 \, {\left (x + 5\right )} e^{\left (x + 5\right )} + 7 \, x + 10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-16*x^2+40*x)*exp(5)*exp(x)+14*x^2+40*x)*exp(2*x^2/((2*x+10)*exp(5)*exp(x)+x^2+7*x+10))/((4*
x^2+40*x+100)*exp(5)^2*exp(x)^2+(4*x^3+48*x^2+180*x+200)*exp(5)*exp(x)+x^4+14*x^3+69*x^2+140*x+100),x, algorit
hm="fricas")

[Out]

e^(2*x^2/(x^2 + 2*(x + 5)*e^(x + 5) + 7*x + 10))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-16*x^2+40*x)*exp(5)*exp(x)+14*x^2+40*x)*exp(2*x^2/((2*x+10)*exp(5)*exp(x)+x^2+7*x+10))/((4*
x^2+40*x+100)*exp(5)^2*exp(x)^2+(4*x^3+48*x^2+180*x+200)*exp(5)*exp(x)+x^4+14*x^3+69*x^2+140*x+100),x, algorit
hm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{229376,[0,8,31,8]%%%}+%%%{18087936,[0,8,30,8]%%%}+%%%{67
1416320,[0,

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maple [A]  time = 0.63, size = 23, normalized size = 0.96

method	result	size
risch	\({\mathrm e}^{\frac {2 x^{2}}{\left (5+x \right ) \left (x +2 \,{\mathrm e}^{5+x}+2\right )}}\)	\(23\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {2 x^{2}}{\left (2 x +10\right ) {\mathrm e}^{5} {\mathrm e}^{x}+x^{2}+7 x +10}}+7 x \,{\mathrm e}^{\frac {2 x^{2}}{\left (2 x +10\right ) {\mathrm e}^{5} {\mathrm e}^{x}+x^{2}+7 x +10}}+10 \,{\mathrm e}^{5} {\mathrm e}^{x} {\mathrm e}^{\frac {2 x^{2}}{\left (2 x +10\right ) {\mathrm e}^{5} {\mathrm e}^{x}+x^{2}+7 x +10}}+2 x \,{\mathrm e}^{5} {\mathrm e}^{x} {\mathrm e}^{\frac {2 x^{2}}{\left (2 x +10\right ) {\mathrm e}^{5} {\mathrm e}^{x}+x^{2}+7 x +10}}+10 \,{\mathrm e}^{\frac {2 x^{2}}{\left (2 x +10\right ) {\mathrm e}^{5} {\mathrm e}^{x}+x^{2}+7 x +10}}}{\left (5+x \right ) \left (2 \,{\mathrm e}^{5} {\mathrm e}^{x}+2+x \right )}\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^3-16*x^2+40*x)*exp(5)*exp(x)+14*x^2+40*x)*exp(2*x^2/((2*x+10)*exp(5)*exp(x)+x^2+7*x+10))/((4*x^2+40
*x+100)*exp(5)^2*exp(x)^2+(4*x^3+48*x^2+180*x+200)*exp(5)*exp(x)+x^4+14*x^3+69*x^2+140*x+100),x,method=_RETURN
VERBOSE)

[Out]

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

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maxima [B]  time = 1.03, size = 125, normalized size = 5.21 \begin {gather*} e^{\left (-\frac {8 \, e^{\left (2 \, x + 10\right )}}{2 \, {\left (x e^{5} - e^{5}\right )} e^{x} - 3 \, x + 4 \, e^{\left (2 \, x + 10\right )} - 6} - \frac {16 \, e^{\left (x + 5\right )}}{2 \, {\left (x e^{5} - e^{5}\right )} e^{x} - 3 \, x + 4 \, e^{\left (2 \, x + 10\right )} - 6} + \frac {50}{2 \, {\left (x e^{5} + 5 \, e^{5}\right )} e^{x} - 3 \, x - 15} - \frac {8}{2 \, {\left (x e^{5} - e^{5}\right )} e^{x} - 3 \, x + 4 \, e^{\left (2 \, x + 10\right )} - 6} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^3-16*x^2+40*x)*exp(5)*exp(x)+14*x^2+40*x)*exp(2*x^2/((2*x+10)*exp(5)*exp(x)+x^2+7*x+10))/((4*
x^2+40*x+100)*exp(5)^2*exp(x)^2+(4*x^3+48*x^2+180*x+200)*exp(5)*exp(x)+x^4+14*x^3+69*x^2+140*x+100),x, algorit
hm="maxima")

[Out]

e^(-8*e^(2*x + 10)/(2*(x*e^5 - e^5)*e^x - 3*x + 4*e^(2*x + 10) - 6) - 16*e^(x + 5)/(2*(x*e^5 - e^5)*e^x - 3*x
+ 4*e^(2*x + 10) - 6) + 50/(2*(x*e^5 + 5*e^5)*e^x - 3*x - 15) - 8/(2*(x*e^5 - e^5)*e^x - 3*x + 4*e^(2*x + 10)
- 6) + 2)

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mupad [B]  time = 0.95, size = 29, normalized size = 1.21 \begin {gather*} {\mathrm {e}}^{\frac {2\,x^2}{7\,x+10\,{\mathrm {e}}^5\,{\mathrm {e}}^x+x^2+2\,x\,{\mathrm {e}}^5\,{\mathrm {e}}^x+10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x^2)/(7*x + x^2 + exp(5)*exp(x)*(2*x + 10) + 10))*(40*x + 14*x^2 - exp(5)*exp(x)*(16*x^2 - 40*x +
4*x^3)))/(140*x + 69*x^2 + 14*x^3 + x^4 + exp(5)*exp(x)*(180*x + 48*x^2 + 4*x^3 + 200) + exp(2*x)*exp(10)*(40*
x + 4*x^2 + 100) + 100),x)

[Out]

exp((2*x^2)/(7*x + 10*exp(5)*exp(x) + x^2 + 2*x*exp(5)*exp(x) + 10))

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sympy [A]  time = 0.57, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {2 x^{2}}{x^{2} + 7 x + \left (2 x + 10\right ) e^{5} e^{x} + 10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**3-16*x**2+40*x)*exp(5)*exp(x)+14*x**2+40*x)*exp(2*x**2/((2*x+10)*exp(5)*exp(x)+x**2+7*x+10))
/((4*x**2+40*x+100)*exp(5)**2*exp(x)**2+(4*x**3+48*x**2+180*x+200)*exp(5)*exp(x)+x**4+14*x**3+69*x**2+140*x+10
0),x)

[Out]

exp(2*x**2/(x**2 + 7*x + (2*x + 10)*exp(5)*exp(x) + 10))

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