3.103.4 \(\int \frac {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} (24 x-588 x^2)+e^x (36 x^2-1764 x^3+21609 x^4)} \, dx\)

Optimal. Leaf size=27 \[ \frac {3 e^{-x}}{x-\frac {2 \left (\frac {e^x}{3}+x\right )}{49 x}} \]

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Rubi [F]  time = 1.48, 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 {-62181 x^2-64827 x^3+e^x (-882+1764 x)}{4 e^{3 x}+e^{2 x} \left (24 x-588 x^2\right )+e^x \left (36 x^2-1764 x^3+21609 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-62181*x^2 - 64827*x^3 + E^x*(-882 + 1764*x))/(4*E^(3*x) + E^(2*x)*(24*x - 588*x^2) + E^x*(36*x^2 - 1764*
x^3 + 21609*x^4)),x]

[Out]

-441*Defer[Int][1/(E^x*(2*E^x + 6*x - 147*x^2)), x] + 2646*Defer[Int][x/(E^x*(-2*E^x - 6*x + 147*x^2)^2), x] -
132300*Defer[Int][x^2/(E^x*(-2*E^x - 6*x + 147*x^2)^2), x] + 64827*Defer[Int][x^3/(E^x*(-2*E^x - 6*x + 147*x^
2)^2), x] - 882*Defer[Int][x/(E^x*(-2*E^x - 6*x + 147*x^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {441 e^{-x} \left (e^x (-2+4 x)-3 x^2 (47+49 x)\right )}{\left (2 e^x+3 (2-49 x) x\right )^2} \, dx\\ &=441 \int \frac {e^{-x} \left (e^x (-2+4 x)-3 x^2 (47+49 x)\right )}{\left (2 e^x+3 (2-49 x) x\right )^2} \, dx\\ &=441 \int \left (\frac {3 e^{-x} x \left (2-100 x+49 x^2\right )}{\left (-2 e^x-6 x+147 x^2\right )^2}-\frac {e^{-x} (-1+2 x)}{-2 e^x-6 x+147 x^2}\right ) \, dx\\ &=-\left (441 \int \frac {e^{-x} (-1+2 x)}{-2 e^x-6 x+147 x^2} \, dx\right )+1323 \int \frac {e^{-x} x \left (2-100 x+49 x^2\right )}{\left (-2 e^x-6 x+147 x^2\right )^2} \, dx\\ &=-\left (441 \int \left (\frac {e^{-x}}{2 e^x+6 x-147 x^2}+\frac {2 e^{-x} x}{-2 e^x-6 x+147 x^2}\right ) \, dx\right )+1323 \int \left (\frac {2 e^{-x} x}{\left (-2 e^x-6 x+147 x^2\right )^2}-\frac {100 e^{-x} x^2}{\left (-2 e^x-6 x+147 x^2\right )^2}+\frac {49 e^{-x} x^3}{\left (-2 e^x-6 x+147 x^2\right )^2}\right ) \, dx\\ &=-\left (441 \int \frac {e^{-x}}{2 e^x+6 x-147 x^2} \, dx\right )-882 \int \frac {e^{-x} x}{-2 e^x-6 x+147 x^2} \, dx+2646 \int \frac {e^{-x} x}{\left (-2 e^x-6 x+147 x^2\right )^2} \, dx+64827 \int \frac {e^{-x} x^3}{\left (-2 e^x-6 x+147 x^2\right )^2} \, dx-132300 \int \frac {e^{-x} x^2}{\left (-2 e^x-6 x+147 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 24, normalized size = 0.89 \begin {gather*} -\frac {441 e^{-x} x}{2 e^x+3 (2-49 x) x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-62181*x^2 - 64827*x^3 + E^x*(-882 + 1764*x))/(4*E^(3*x) + E^(2*x)*(24*x - 588*x^2) + E^x*(36*x^2 -
1764*x^3 + 21609*x^4)),x]

[Out]

(-441*x)/(E^x*(2*E^x + 3*(2 - 49*x)*x))

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fricas [A]  time = 0.48, size = 25, normalized size = 0.93 \begin {gather*} \frac {441 \, x}{3 \, {\left (49 \, x^{2} - 2 \, x\right )} e^{x} - 2 \, e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)*exp(x)^2+(21609*x^4-1764*x^3+3
6*x^2)*exp(x)),x, algorithm="fricas")

[Out]

441*x/(3*(49*x^2 - 2*x)*e^x - 2*e^(2*x))

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giac [A]  time = 0.22, size = 24, normalized size = 0.89 \begin {gather*} \frac {441 \, x}{147 \, x^{2} e^{x} - 6 \, x e^{x} - 2 \, e^{\left (2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)*exp(x)^2+(21609*x^4-1764*x^3+3
6*x^2)*exp(x)),x, algorithm="giac")

[Out]

441*x/(147*x^2*e^x - 6*x*e^x - 2*e^(2*x))

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maple [A]  time = 0.08, size = 23, normalized size = 0.85

method	result	size
norman	\(\frac {441 x \,{\mathrm e}^{-x}}{147 x^{2}-6 x -2 \,{\mathrm e}^{x}}\)	\(23\)
risch	\(\frac {147 \,{\mathrm e}^{-x}}{49 x -2}+\frac {294}{\left (49 x -2\right ) \left (147 x^{2}-6 x -2 \,{\mathrm e}^{x}\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)*exp(x)^2+(21609*x^4-1764*x^3+36*x^2)
*exp(x)),x,method=_RETURNVERBOSE)

[Out]

441*x/exp(x)/(147*x^2-6*x-2*exp(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 441 \, \int \frac {147 \, x^{3} + 141 \, x^{2} - 2 \, {\left (2 \, x - 1\right )} e^{x}}{12 \, {\left (49 \, x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )} - 9 \, {\left (2401 \, x^{4} - 196 \, x^{3} + 4 \, x^{2}\right )} e^{x} - 4 \, e^{\left (3 \, x\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1764*x-882)*exp(x)-64827*x^3-62181*x^2)/(4*exp(x)^3+(-588*x^2+24*x)*exp(x)^2+(21609*x^4-1764*x^3+3
6*x^2)*exp(x)),x, algorithm="maxima")

[Out]

441*integrate((147*x^3 + 141*x^2 - 2*(2*x - 1)*e^x)/(12*(49*x^2 - 2*x)*e^(2*x) - 9*(2401*x^4 - 196*x^3 + 4*x^2
)*e^x - 4*e^(3*x)), x)

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mupad [B]  time = 7.38, size = 24, normalized size = 0.89 \begin {gather*} -\frac {441\,x}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x-147\,x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(62181*x^2 - exp(x)*(1764*x - 882) + 64827*x^3)/(4*exp(3*x) + exp(2*x)*(24*x - 588*x^2) + exp(x)*(36*x^2
- 1764*x^3 + 21609*x^4)),x)

[Out]

-(441*x)/(2*exp(2*x) + exp(x)*(6*x - 147*x^2))

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sympy [A]  time = 0.21, size = 34, normalized size = 1.26 \begin {gather*} - \frac {7203}{- \frac {352947 x^{3}}{2} + 14406 x^{2} - 294 x + \left (2401 x - 98\right ) e^{x}} + \frac {147 e^{- x}}{49 x - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1764*x-882)*exp(x)-64827*x**3-62181*x**2)/(4*exp(x)**3+(-588*x**2+24*x)*exp(x)**2+(21609*x**4-1764
*x**3+36*x**2)*exp(x)),x)

[Out]

-7203/(-352947*x**3/2 + 14406*x**2 - 294*x + (2401*x - 98)*exp(x)) + 147*exp(-x)/(49*x - 2)

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