3.83.68 \(\int \frac {20 e^x-8 e^{2 x}}{25+e^x (30+20 x)+e^{2 x} (9+12 x+4 x^2)} \, dx\)

Optimal. Leaf size=18 \[ -5+\frac {4}{3+5 e^{-x}+2 x} \]

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Rubi [F]  time = 0.93, 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 {20 e^x-8 e^{2 x}}{25+e^x (30+20 x)+e^{2 x} \left (9+12 x+4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(20*E^x - 8*E^(2*x))/(25 + E^x*(30 + 20*x) + E^(2*x)*(9 + 12*x + 4*x^2)),x]

[Out]

20*Defer[Int][E^x/(5 + 3*E^x + 2*E^x*x)^2, x] + 40*Defer[Int][E^x/((3 + 2*x)*(5 + 3*E^x + 2*E^x*x)^2), x] - 8*
Defer[Int][E^x/((3 + 2*x)*(5 + 3*E^x + 2*E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 e^x \left (5-2 e^x\right )}{\left (5+e^x (3+2 x)\right )^2} \, dx\\ &=4 \int \frac {e^x \left (5-2 e^x\right )}{\left (5+e^x (3+2 x)\right )^2} \, dx\\ &=4 \int \left (\frac {5 e^x (5+2 x)}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2}-\frac {2 e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )}\right ) \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \frac {e^x (5+2 x)}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2} \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \left (\frac {e^x}{\left (5+3 e^x+2 e^x x\right )^2}+\frac {2 e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )} \, dx\right )+20 \int \frac {e^x}{\left (5+3 e^x+2 e^x x\right )^2} \, dx+40 \int \frac {e^x}{(3+2 x) \left (5+3 e^x+2 e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 e^x}{5+e^x (3+2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(20*E^x - 8*E^(2*x))/(25 + E^x*(30 + 20*x) + E^(2*x)*(9 + 12*x + 4*x^2)),x]

[Out]

(4*E^x)/(5 + E^x*(3 + 2*x))

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fricas [A]  time = 1.08, size = 16, normalized size = 0.89 \begin {gather*} \frac {4 \, e^{x}}{{\left (2 \, x + 3\right )} e^{x} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(x)^2+20*exp(x))/((4*x^2+12*x+9)*exp(x)^2+(20*x+30)*exp(x)+25),x, algorithm="fricas")

[Out]

4*e^x/((2*x + 3)*e^x + 5)

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giac [A]  time = 0.16, size = 17, normalized size = 0.94 \begin {gather*} \frac {4 \, e^{x}}{2 \, x e^{x} + 3 \, e^{x} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(x)^2+20*exp(x))/((4*x^2+12*x+9)*exp(x)^2+(20*x+30)*exp(x)+25),x, algorithm="giac")

[Out]

4*e^x/(2*x*e^x + 3*e^x + 5)

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maple [A]  time = 0.05, size = 18, normalized size = 1.00

method	result	size
norman	\(\frac {4 \,{\mathrm e}^{x}}{2 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}+5}\)	\(18\)
risch	\(\frac {2}{x +\frac {3}{2}}-\frac {20}{\left (2 x +3\right ) \left (2 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{x}+5\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*exp(x)^2+20*exp(x))/((4*x^2+12*x+9)*exp(x)^2+(20*x+30)*exp(x)+25),x,method=_RETURNVERBOSE)

[Out]

4*exp(x)/(2*exp(x)*x+3*exp(x)+5)

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maxima [A]  time = 0.41, size = 16, normalized size = 0.89 \begin {gather*} \frac {4 \, e^{x}}{{\left (2 \, x + 3\right )} e^{x} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(x)^2+20*exp(x))/((4*x^2+12*x+9)*exp(x)^2+(20*x+30)*exp(x)+25),x, algorithm="maxima")

[Out]

4*e^x/((2*x + 3)*e^x + 5)

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mupad [B]  time = 5.70, size = 31, normalized size = 1.72 \begin {gather*} \frac {4}{2\,x+3}-\frac {20}{\left (2\,x+3\right )\,\left ({\mathrm {e}}^x\,\left (2\,x+3\right )+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*exp(2*x) - 20*exp(x))/(exp(2*x)*(12*x + 4*x^2 + 9) + exp(x)*(20*x + 30) + 25),x)

[Out]

4/(2*x + 3) - 20/((2*x + 3)*(exp(x)*(2*x + 3) + 5))

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sympy [B]  time = 0.17, size = 26, normalized size = 1.44 \begin {gather*} - \frac {20}{10 x + \left (4 x^{2} + 12 x + 9\right ) e^{x} + 15} + \frac {8}{4 x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*exp(x)**2+20*exp(x))/((4*x**2+12*x+9)*exp(x)**2+(20*x+30)*exp(x)+25),x)

[Out]

-20/(10*x + (4*x**2 + 12*x + 9)*exp(x) + 15) + 8/(4*x + 6)

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