3.20 \(\int \frac{e^{2 x}}{\left (a+b e^x\right )^3} \, dx\)

Optimal. Leaf size=21 \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]

[Out]

E^(2*x)/(2*a*(a + b*E^x)^2)

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Rubi [A]  time = 0.0408097, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{e^{2 x}}{2 a \left (a+b e^x\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[E^(2*x)/(a + b*E^x)^3,x]

[Out]

E^(2*x)/(2*a*(a + b*E^x)^2)

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Rubi in Sympy [A]  time = 6.89898, size = 15, normalized size = 0.71 \[ \frac{e^{2 x}}{2 a \left (a + b e^{x}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(2*x)/(a+b*exp(x))**3,x)

[Out]

exp(2*x)/(2*a*(a + b*exp(x))**2)

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Mathematica [A]  time = 0.0156651, size = 24, normalized size = 1.14 \[ -\frac{a+2 b e^x}{2 b^2 \left (a+b e^x\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[E^(2*x)/(a + b*E^x)^3,x]

[Out]

-(a + 2*b*E^x)/(2*b^2*(a + b*E^x)^2)

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Maple [A]  time = 0.01, size = 29, normalized size = 1.4 \[{\frac{a}{2\,{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) ^{2}}}-{\frac{1}{{b}^{2} \left ( a+b{{\rm e}^{x}} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(2*x)/(a+b*exp(x))^3,x)

[Out]

1/2*a/b^2/(a+b*exp(x))^2-1/b^2/(a+b*exp(x))

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Maxima [A]  time = 0.765769, size = 82, normalized size = 3.9 \[ -\frac{b e^{x}}{b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}} - \frac{a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.265172, size = 47, normalized size = 2.24 \[ -\frac{2 \, b e^{x} + a}{2 \,{\left (b^{4} e^{\left (2 \, x\right )} + 2 \, a b^{3} e^{x} + a^{2} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.249224, size = 37, normalized size = 1.76 \[ \frac{- a - 2 b e^{x}}{2 a^{2} b^{2} + 4 a b^{3} e^{x} + 2 b^{4} e^{2 x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(2*x)/(a+b*exp(x))**3,x)

[Out]

(-a - 2*b*exp(x))/(2*a**2*b**2 + 4*a*b**3*exp(x) + 2*b**4*exp(2*x))

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GIAC/XCAS [A]  time = 0.323473, size = 27, normalized size = 1.29 \[ -\frac{2 \, b e^{x} + a}{2 \,{\left (b e^{x} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(2*x)/(b*e^x + a)^3,x, algorithm="giac")

[Out]

-1/2*(2*b*e^x + a)/((b*e^x + a)^2*b^2)