3.103.20 \(\int \frac {e^{5+\frac {2 x}{e^5}} (12+12 x+3 x^2)+e^5 (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e (-16 x^2-6 x^3))+e^{\frac {x}{e^5}} (e (8 x+4 x^2)+e^5 (-24 x-24 x^2-6 x^3+e (-8+12 x+6 x^2)))}{e^{5+\frac {2 x}{e^5}} (12+12 x+3 x^2)+e^{5+\frac {x}{e^5}} (-24 x-24 x^2-6 x^3+e (12 x+6 x^2))+e^5 (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e (-12 x^2-6 x^3))} \, dx\)

Optimal. Leaf size=30 \[ x+\frac {4}{3+\frac {3 e x}{\left (e^{\frac {x}{e^5}}-x\right ) (2+x)}} \]

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Rubi [F]  time = 3.37, 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^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(5 + (2*x)/E^5)*(12 + 12*x + 3*x^2) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 + 3*x^4 + E*(-16*x^2 - 6*x^3)) +
E^(x/E^5)*(E*(8*x + 4*x^2) + E^5*(-24*x - 24*x^2 - 6*x^3 + E*(-8 + 12*x + 6*x^2))))/(E^(5 + (2*x)/E^5)*(12 +
12*x + 3*x^2) + E^(5 + x/E^5)*(-24*x - 24*x^2 - 6*x^3 + E*(12*x + 6*x^2)) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 +
3*x^4 + E*(-12*x^2 - 6*x^3))),x]

[Out]

x + (8*E^2*Defer[Int][(E*x + E^(x/E^5)*(2 + x) - x*(2 + x))^(-2), x])/3 + (16*E^2*Defer[Int][1/((-2 - x)*(E*x
+ E^(x/E^5)*(2 + x) - x*(2 + x))^2), x])/3 - (8*E*Defer[Int][x/(E*x + E^(x/E^5)*(2 + x) - x*(2 + x))^2, x])/3
+ (4*(2 - E - E^5)*Defer[Int][x^2/(E*x + E^(x/E^5)*(2 + x) - x*(2 + x))^2, x])/(3*E^4) + (4*Defer[Int][x^3/(E*
x + E^(x/E^5)*(2 + x) - x*(2 + x))^2, x])/(3*E^4) + (8*E*Defer[Int][1/((-2 - x)*(E*x + E^(x/E^5)*(2 + x) - x*(
2 + x))), x])/3 + (4*Defer[Int][x/(E*x + E^(x/E^5)*(2 + x) - x*(2 + x)), x])/(3*E^4)

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^6 x^2+4 e^{\frac {x}{e^5}} x (2+x)+3 e^{4+\frac {2 x}{e^5}} (2+x)^2-6 e^{4+\frac {x}{e^5}} x (2+x)^2+3 e^4 x^2 (2+x)^2-2 e^5 x^2 (8+3 x)+2 e^{5+\frac {x}{e^5}} \left (-4+6 x+3 x^2\right )}{3 e^4 \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx\\ &=\frac {\int \frac {3 e^6 x^2+4 e^{\frac {x}{e^5}} x (2+x)+3 e^{4+\frac {2 x}{e^5}} (2+x)^2-6 e^{4+\frac {x}{e^5}} x (2+x)^2+3 e^4 x^2 (2+x)^2-2 e^5 x^2 (8+3 x)+2 e^{5+\frac {x}{e^5}} \left (-4+6 x+3 x^2\right )}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}\\ &=\frac {\int \left (3 e^4+\frac {4 \left (-2 e^5+2 x+x^2\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )}+\frac {4 x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}\right ) \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {-2 e^5+2 x+x^2}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )} \, dx}{3 e^4}+\frac {4 \int \frac {x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {-2 e^5+2 x+x^2}{(2+x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )} \, dx}{3 e^4}\\ &=x+\frac {4 \int \left (\frac {2 e^6}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {4 e^6}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}-\frac {2 e^5 x}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {\left (2-e-e^5\right ) x^2}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {x^3}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}\right ) \, dx}{3 e^4}+\frac {4 \int \left (\frac {2 e^5}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )}+\frac {x}{2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2}\right ) \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x^3}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {x}{2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2} \, dx}{3 e^4}-\frac {1}{3} (8 e) \int \frac {x}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {1}{3} (8 e) \int \frac {1}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )} \, dx+\frac {1}{3} \left (8 e^2\right ) \int \frac {1}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {1}{3} \left (16 e^2\right ) \int \frac {1}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {\left (4 \left (2-e-e^5\right )\right ) \int \frac {x^2}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x^3}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {x}{e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)} \, dx}{3 e^4}-\frac {1}{3} (8 e) \int \frac {x}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {1}{3} (8 e) \int \frac {1}{(-2-x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )} \, dx+\frac {1}{3} \left (8 e^2\right ) \int \frac {1}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {1}{3} \left (16 e^2\right ) \int \frac {1}{(-2-x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {\left (4 \left (2-e-e^5\right )\right ) \int \frac {x^2}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 33, normalized size = 1.10 \begin {gather*} \frac {1}{3} x \left (3-\frac {4 e}{e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(5 + (2*x)/E^5)*(12 + 12*x + 3*x^2) + E^5*(12*x^2 + 3*E^2*x^2 + 12*x^3 + 3*x^4 + E*(-16*x^2 - 6*x
^3)) + E^(x/E^5)*(E*(8*x + 4*x^2) + E^5*(-24*x - 24*x^2 - 6*x^3 + E*(-8 + 12*x + 6*x^2))))/(E^(5 + (2*x)/E^5)*
(12 + 12*x + 3*x^2) + E^(5 + x/E^5)*(-24*x - 24*x^2 - 6*x^3 + E*(12*x + 6*x^2)) + E^5*(12*x^2 + 3*E^2*x^2 + 12
*x^3 + 3*x^4 + E*(-12*x^2 - 6*x^3))),x]

[Out]

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

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fricas [B]  time = 0.82, size = 79, normalized size = 2.63 \begin {gather*} \frac {{\left (3 \, x^{2} - 4 \, x\right )} e^{6} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{5} + 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}}{3 \, {\left (x e^{6} - {\left (x^{2} + 2 \, x\right )} e^{5} + {\left (x + 2\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x
)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*e
xp(5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*
x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x, algorithm="fricas")

[Out]

1/3*((3*x^2 - 4*x)*e^6 - 3*(x^3 + 2*x^2)*e^5 + 3*(x^2 + 2*x)*e^((x + 5*e^5)*e^(-5)))/(x*e^6 - (x^2 + 2*x)*e^5
+ (x + 2)*e^((x + 5*e^5)*e^(-5)))

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giac [B]  time = 0.51, size = 551, normalized size = 18.37 \begin {gather*} \frac {3 \, x^{6} e^{4} - 3 \, x^{5} e^{9} - 6 \, x^{5} e^{5} + 18 \, x^{5} e^{4} - 3 \, x^{5} e^{\left (x e^{\left (-5\right )} + 4\right )} + 3 \, x^{4} e^{10} - 18 \, x^{4} e^{9} + 3 \, x^{4} e^{6} - 20 \, x^{4} e^{5} + 36 \, x^{4} e^{4} + 3 \, x^{4} e^{\left (x e^{\left (-5\right )} + 9\right )} + 3 \, x^{4} e^{\left (x e^{\left (-5\right )} + 5\right )} - 18 \, x^{4} e^{\left (x e^{\left (-5\right )} + 4\right )} + 14 \, x^{3} e^{10} - 36 \, x^{3} e^{9} + 2 \, x^{3} e^{6} - 8 \, x^{3} e^{5} + 24 \, x^{3} e^{4} + 18 \, x^{3} e^{\left (x e^{\left (-5\right )} + 9\right )} + 12 \, x^{3} e^{\left (x e^{\left (-5\right )} + 5\right )} - 36 \, x^{3} e^{\left (x e^{\left (-5\right )} + 4\right )} - 6 \, x^{2} e^{11} + 8 \, x^{2} e^{10} - 24 \, x^{2} e^{9} - 8 \, x^{2} e^{6} + 16 \, x^{2} e^{5} - 6 \, x^{2} e^{\left (x e^{\left (-5\right )} + 10\right )} + 36 \, x^{2} e^{\left (x e^{\left (-5\right )} + 9\right )} + 12 \, x^{2} e^{\left (x e^{\left (-5\right )} + 5\right )} - 24 \, x^{2} e^{\left (x e^{\left (-5\right )} + 4\right )} + 8 \, x e^{11} - 16 \, x e^{10} - 12 \, x e^{\left (x e^{\left (-5\right )} + 10\right )} + 24 \, x e^{\left (x e^{\left (-5\right )} + 9\right )}}{3 \, {\left (x^{5} e^{4} - x^{4} e^{9} - 2 \, x^{4} e^{5} + 6 \, x^{4} e^{4} - x^{4} e^{\left (x e^{\left (-5\right )} + 4\right )} + x^{3} e^{10} - 6 \, x^{3} e^{9} + x^{3} e^{6} - 8 \, x^{3} e^{5} + 12 \, x^{3} e^{4} + x^{3} e^{\left (x e^{\left (-5\right )} + 9\right )} + x^{3} e^{\left (x e^{\left (-5\right )} + 5\right )} - 6 \, x^{3} e^{\left (x e^{\left (-5\right )} + 4\right )} + 6 \, x^{2} e^{10} - 12 \, x^{2} e^{9} + 2 \, x^{2} e^{6} - 8 \, x^{2} e^{5} + 8 \, x^{2} e^{4} + 6 \, x^{2} e^{\left (x e^{\left (-5\right )} + 9\right )} + 4 \, x^{2} e^{\left (x e^{\left (-5\right )} + 5\right )} - 12 \, x^{2} e^{\left (x e^{\left (-5\right )} + 4\right )} - 2 \, x e^{11} + 8 \, x e^{10} - 8 \, x e^{9} - 2 \, x e^{\left (x e^{\left (-5\right )} + 10\right )} + 12 \, x e^{\left (x e^{\left (-5\right )} + 9\right )} + 4 \, x e^{\left (x e^{\left (-5\right )} + 5\right )} - 8 \, x e^{\left (x e^{\left (-5\right )} + 4\right )} - 4 \, e^{\left (x e^{\left (-5\right )} + 10\right )} + 8 \, e^{\left (x e^{\left (-5\right )} + 9\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x
)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*e
xp(5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*
x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x, algorithm="giac")

[Out]

1/3*(3*x^6*e^4 - 3*x^5*e^9 - 6*x^5*e^5 + 18*x^5*e^4 - 3*x^5*e^(x*e^(-5) + 4) + 3*x^4*e^10 - 18*x^4*e^9 + 3*x^4
*e^6 - 20*x^4*e^5 + 36*x^4*e^4 + 3*x^4*e^(x*e^(-5) + 9) + 3*x^4*e^(x*e^(-5) + 5) - 18*x^4*e^(x*e^(-5) + 4) + 1
4*x^3*e^10 - 36*x^3*e^9 + 2*x^3*e^6 - 8*x^3*e^5 + 24*x^3*e^4 + 18*x^3*e^(x*e^(-5) + 9) + 12*x^3*e^(x*e^(-5) +
5) - 36*x^3*e^(x*e^(-5) + 4) - 6*x^2*e^11 + 8*x^2*e^10 - 24*x^2*e^9 - 8*x^2*e^6 + 16*x^2*e^5 - 6*x^2*e^(x*e^(-
5) + 10) + 36*x^2*e^(x*e^(-5) + 9) + 12*x^2*e^(x*e^(-5) + 5) - 24*x^2*e^(x*e^(-5) + 4) + 8*x*e^11 - 16*x*e^10
- 12*x*e^(x*e^(-5) + 10) + 24*x*e^(x*e^(-5) + 9))/(x^5*e^4 - x^4*e^9 - 2*x^4*e^5 + 6*x^4*e^4 - x^4*e^(x*e^(-5)
+ 4) + x^3*e^10 - 6*x^3*e^9 + x^3*e^6 - 8*x^3*e^5 + 12*x^3*e^4 + x^3*e^(x*e^(-5) + 9) + x^3*e^(x*e^(-5) + 5)
- 6*x^3*e^(x*e^(-5) + 4) + 6*x^2*e^10 - 12*x^2*e^9 + 2*x^2*e^6 - 8*x^2*e^5 + 8*x^2*e^4 + 6*x^2*e^(x*e^(-5) + 9
) + 4*x^2*e^(x*e^(-5) + 5) - 12*x^2*e^(x*e^(-5) + 4) - 2*x*e^11 + 8*x*e^10 - 8*x*e^9 - 2*x*e^(x*e^(-5) + 10) +
12*x*e^(x*e^(-5) + 9) + 4*x*e^(x*e^(-5) + 5) - 8*x*e^(x*e^(-5) + 4) - 4*e^(x*e^(-5) + 10) + 8*e^(x*e^(-5) + 9
))

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maple [A]  time = 0.86, size = 37, normalized size = 1.23

method	result	size
risch	\(x -\frac {4 \,{\mathrm e} x}{3 \left (x \,{\mathrm e}-x^{2}+x \,{\mathrm e}^{x \,{\mathrm e}^{-5}}-2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}\right )}\)	\(37\)
norman	\(\frac {\left (2 \,{\mathrm e}-4\right ) {\mathrm e}^{x \,{\mathrm e}^{-5}}+\left ({\mathrm e}^{2}-\frac {16 \,{\mathrm e}}{3}+4\right ) x +{\mathrm e}^{x \,{\mathrm e}^{-5}} x^{2}+{\mathrm e}^{x \,{\mathrm e}^{-5}} {\mathrm e} x -x^{3}}{x \,{\mathrm e}-x^{2}+x \,{\mathrm e}^{x \,{\mathrm e}^{-5}}-2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}}\)	\(89\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x)*exp(
1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*exp(5)*
exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*x^2)*e
xp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x,method=_RETURNVERBOSE)

[Out]

x-4/3*exp(1)*x/(x*exp(1)-x^2+x*exp(x*exp(-5))-2*x+2*exp(x*exp(-5)))

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maxima [B]  time = 0.44, size = 59, normalized size = 1.97 \begin {gather*} \frac {3 \, x^{3} - 3 \, x^{2} {\left (e - 2\right )} + 4 \, x e - 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x e^{\left (-5\right )}\right )}}{3 \, {\left (x^{2} - x {\left (e - 2\right )} - {\left (x + 2\right )} e^{\left (x e^{\left (-5\right )}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+12*x+12)*exp(5)*exp(x/exp(5))^2+(((6*x^2+12*x-8)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)+(4*x^2+8*x
)*exp(1))*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-16*x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5))/((3*x^2+12*x+12)*e
xp(5)*exp(x/exp(5))^2+((6*x^2+12*x)*exp(1)-6*x^3-24*x^2-24*x)*exp(5)*exp(x/exp(5))+(3*x^2*exp(1)^2+(-6*x^3-12*
x^2)*exp(1)+3*x^4+12*x^3+12*x^2)*exp(5)),x, algorithm="maxima")

[Out]

1/3*(3*x^3 - 3*x^2*(e - 2) + 4*x*e - 3*(x^2 + 2*x)*e^(x*e^(-5)))/(x^2 - x*(e - 2) - (x + 2)*e^(x*e^(-5)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}}\,\left (\mathrm {e}\,\left (4\,x^2+8\,x\right )-{\mathrm {e}}^5\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x-8\right )+24\,x^2+6\,x^3\right )\right )+{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+16\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}+5}\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x\right )+24\,x^2+6\,x^3\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+12\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*exp(-5))*(exp(1)*(8*x + 4*x^2) - exp(5)*(24*x - exp(1)*(12*x + 6*x^2 - 8) + 24*x^2 + 6*x^3)) + exp(
5)*(3*x^2*exp(2) - exp(1)*(16*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^4) + exp(5)*exp(2*x*exp(-5))*(12*x + 3*x^2
+ 12))/(exp(5)*(3*x^2*exp(2) - exp(1)*(12*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^4) - exp(5)*exp(x*exp(-5))*(24*
x - exp(1)*(12*x + 6*x^2) + 24*x^2 + 6*x^3) + exp(5)*exp(2*x*exp(-5))*(12*x + 3*x^2 + 12)),x)

[Out]

int((exp(x*exp(-5))*(exp(1)*(8*x + 4*x^2) - exp(5)*(24*x - exp(1)*(12*x + 6*x^2 - 8) + 24*x^2 + 6*x^3)) + exp(
2*x*exp(-5) + 5)*(12*x + 3*x^2 + 12) + exp(5)*(3*x^2*exp(2) - exp(1)*(16*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^
4))/(exp(2*x*exp(-5) + 5)*(12*x + 3*x^2 + 12) - exp(x*exp(-5) + 5)*(24*x - exp(1)*(12*x + 6*x^2) + 24*x^2 + 6*
x^3) + exp(5)*(3*x^2*exp(2) - exp(1)*(12*x^2 + 6*x^3) + 12*x^2 + 12*x^3 + 3*x^4)), x)

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sympy [A]  time = 0.40, size = 34, normalized size = 1.13 \begin {gather*} x - \frac {4 e x}{- 3 x^{2} - 6 x + 3 e x + \left (3 x + 6\right ) e^{\frac {x}{e^{5}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+12*x+12)*exp(5)*exp(x/exp(5))**2+(((6*x**2+12*x-8)*exp(1)-6*x**3-24*x**2-24*x)*exp(5)+(4*x*
*2+8*x)*exp(1))*exp(x/exp(5))+(3*x**2*exp(1)**2+(-6*x**3-16*x**2)*exp(1)+3*x**4+12*x**3+12*x**2)*exp(5))/((3*x
**2+12*x+12)*exp(5)*exp(x/exp(5))**2+((6*x**2+12*x)*exp(1)-6*x**3-24*x**2-24*x)*exp(5)*exp(x/exp(5))+(3*x**2*e
xp(1)**2+(-6*x**3-12*x**2)*exp(1)+3*x**4+12*x**3+12*x**2)*exp(5)),x)

[Out]

x - 4*E*x/(-3*x**2 - 6*x + 3*E*x + (3*x + 6)*exp(x*exp(-5)))

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