3.92.51 \(\int \frac {75-30 x-15 x^2+e^x (15+30 x+15 x^2)}{225+300 x+130 x^2+20 x^3+x^4+e^{2 x} (1+2 x+x^2)+e^x (-30-50 x-22 x^2-2 x^3)} \, dx\)

Optimal. Leaf size=19 \[ -\frac {15}{-9+e^x-x-\frac {6}{1+x}} \]

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Rubi [F]  time = 0.85, 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 {75-30 x-15 x^2+e^x \left (15+30 x+15 x^2\right )}{225+300 x+130 x^2+20 x^3+x^4+e^{2 x} \left (1+2 x+x^2\right )+e^x \left (-30-50 x-22 x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(75 - 30*x - 15*x^2 + E^x*(15 + 30*x + 15*x^2))/(225 + 300*x + 130*x^2 + 20*x^3 + x^4 + E^(2*x)*(1 + 2*x +
x^2) + E^x*(-30 - 50*x - 22*x^2 - 2*x^3)),x]

[Out]

300*Defer[Int][(-15 + E^x - 10*x + E^x*x - x^2)^(-2), x] + 15*Defer[Int][(-15 + E^x - 10*x + E^x*x - x^2)^(-1)
, x] + 345*Defer[Int][x/(15 - E^x + 10*x - E^x*x + x^2)^2, x] + 150*Defer[Int][x^2/(15 - E^x + 10*x - E^x*x +
x^2)^2, x] + 15*Defer[Int][x^3/(15 - E^x + 10*x - E^x*x + x^2)^2, x] - 15*Defer[Int][x/(15 - E^x + 10*x - E^x*
x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {15 \left (5-2 x-x^2+e^x (1+x)^2\right )}{\left (15+10 x+x^2-e^x (1+x)\right )^2} \, dx\\ &=15 \int \frac {5-2 x-x^2+e^x (1+x)^2}{\left (15+10 x+x^2-e^x (1+x)\right )^2} \, dx\\ &=15 \int \left (-\frac {1+x}{15-e^x+10 x-e^x x+x^2}+\frac {20+23 x+10 x^2+x^3}{\left (-15+e^x-10 x+e^x x-x^2\right )^2}\right ) \, dx\\ &=-\left (15 \int \frac {1+x}{15-e^x+10 x-e^x x+x^2} \, dx\right )+15 \int \frac {20+23 x+10 x^2+x^3}{\left (-15+e^x-10 x+e^x x-x^2\right )^2} \, dx\\ &=15 \int \left (\frac {20}{\left (-15+e^x-10 x+e^x x-x^2\right )^2}+\frac {23 x}{\left (15-e^x+10 x-e^x x+x^2\right )^2}+\frac {10 x^2}{\left (15-e^x+10 x-e^x x+x^2\right )^2}+\frac {x^3}{\left (15-e^x+10 x-e^x x+x^2\right )^2}\right ) \, dx-15 \int \left (-\frac {1}{-15+e^x-10 x+e^x x-x^2}+\frac {x}{15-e^x+10 x-e^x x+x^2}\right ) \, dx\\ &=15 \int \frac {1}{-15+e^x-10 x+e^x x-x^2} \, dx+15 \int \frac {x^3}{\left (15-e^x+10 x-e^x x+x^2\right )^2} \, dx-15 \int \frac {x}{15-e^x+10 x-e^x x+x^2} \, dx+150 \int \frac {x^2}{\left (15-e^x+10 x-e^x x+x^2\right )^2} \, dx+300 \int \frac {1}{\left (-15+e^x-10 x+e^x x-x^2\right )^2} \, dx+345 \int \frac {x}{\left (15-e^x+10 x-e^x x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 23, normalized size = 1.21 \begin {gather*} \frac {15 (1+x)}{15+10 x+x^2-e^x (1+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75 - 30*x - 15*x^2 + E^x*(15 + 30*x + 15*x^2))/(225 + 300*x + 130*x^2 + 20*x^3 + x^4 + E^(2*x)*(1 +
2*x + x^2) + E^x*(-30 - 50*x - 22*x^2 - 2*x^3)),x]

[Out]

(15*(1 + x))/(15 + 10*x + x^2 - E^x*(1 + x))

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fricas [A]  time = 0.60, size = 22, normalized size = 1.16 \begin {gather*} \frac {15 \, {\left (x + 1\right )}}{x^{2} - {\left (x + 1\right )} e^{x} + 10 \, x + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x^2+30*x+15)*exp(x)-15*x^2-30*x+75)/((x^2+2*x+1)*exp(x)^2+(-2*x^3-22*x^2-50*x-30)*exp(x)+x^4+20
*x^3+130*x^2+300*x+225),x, algorithm="fricas")

[Out]

15*(x + 1)/(x^2 - (x + 1)*e^x + 10*x + 15)

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giac [A]  time = 0.18, size = 24, normalized size = 1.26 \begin {gather*} \frac {15 \, {\left (x + 1\right )}}{x^{2} - x e^{x} + 10 \, x - e^{x} + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x^2+30*x+15)*exp(x)-15*x^2-30*x+75)/((x^2+2*x+1)*exp(x)^2+(-2*x^3-22*x^2-50*x-30)*exp(x)+x^4+20
*x^3+130*x^2+300*x+225),x, algorithm="giac")

[Out]

15*(x + 1)/(x^2 - x*e^x + 10*x - e^x + 15)

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maple [A]  time = 0.08, size = 25, normalized size = 1.32

method	result	size
risch	\(\frac {15 x +15}{x^{2}-{\mathrm e}^{x} x +10 x -{\mathrm e}^{x}+15}\)	\(25\)
norman	\(\frac {15 x +15}{x^{2}-{\mathrm e}^{x} x +10 x -{\mathrm e}^{x}+15}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((15*x^2+30*x+15)*exp(x)-15*x^2-30*x+75)/((x^2+2*x+1)*exp(x)^2+(-2*x^3-22*x^2-50*x-30)*exp(x)+x^4+20*x^3+1
30*x^2+300*x+225),x,method=_RETURNVERBOSE)

[Out]

15*(x+1)/(x^2-exp(x)*x+10*x-exp(x)+15)

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maxima [A]  time = 0.40, size = 22, normalized size = 1.16 \begin {gather*} \frac {15 \, {\left (x + 1\right )}}{x^{2} - {\left (x + 1\right )} e^{x} + 10 \, x + 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x^2+30*x+15)*exp(x)-15*x^2-30*x+75)/((x^2+2*x+1)*exp(x)^2+(-2*x^3-22*x^2-50*x-30)*exp(x)+x^4+20
*x^3+130*x^2+300*x+225),x, algorithm="maxima")

[Out]

15*(x + 1)/(x^2 - (x + 1)*e^x + 10*x + 15)

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mupad [B]  time = 7.12, size = 24, normalized size = 1.26 \begin {gather*} -\frac {15\,x+15}{{\mathrm {e}}^x+x\,\left ({\mathrm {e}}^x-10\right )-x^2-15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(30*x - exp(x)*(30*x + 15*x^2 + 15) + 15*x^2 - 75)/(300*x + exp(2*x)*(2*x + x^2 + 1) + 130*x^2 + 20*x^3 +
x^4 - exp(x)*(50*x + 22*x^2 + 2*x^3 + 30) + 225),x)

[Out]

-(15*x + 15)/(exp(x) + x*(exp(x) - 10) - x^2 - 15)

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sympy [A]  time = 0.23, size = 20, normalized size = 1.05 \begin {gather*} \frac {- 15 x - 15}{- x^{2} - 10 x + \left (x + 1\right ) e^{x} - 15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((15*x**2+30*x+15)*exp(x)-15*x**2-30*x+75)/((x**2+2*x+1)*exp(x)**2+(-2*x**3-22*x**2-50*x-30)*exp(x)+
x**4+20*x**3+130*x**2+300*x+225),x)

[Out]

(-15*x - 15)/(-x**2 - 10*x + (x + 1)*exp(x) - 15)

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