3.10.78 \(\int \frac {-364+406 x+189 x^2+e^{2 x} (405+162 x^2)+e^x (270+594 x-108 x^2+162 x^3)}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} (54432-157464 x+118098 x^2)+e^x (-25920+108864 x-157464 x^2+78732 x^3)} \, dx\)

Optimal. Leaf size=31 \[ \frac {-3-x-x^2}{3 \left (4+9 \left (2-3 \left (e^x+x\right )\right )^2\right )} \]

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Rubi [F]  time = 4.79, 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 {-364+406 x+189 x^2+e^{2 x} \left (405+162 x^2\right )+e^x \left (270+594 x-108 x^2+162 x^3\right )}{4800+19683 e^{4 x}-25920 x+54432 x^2-52488 x^3+19683 x^4+e^{3 x} (-52488+78732 x)+e^{2 x} \left (54432-157464 x+118098 x^2\right )+e^x \left (-25920+108864 x-157464 x^2+78732 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-364 + 406*x + 189*x^2 + E^(2*x)*(405 + 162*x^2) + E^x*(270 + 594*x - 108*x^2 + 162*x^3))/(4800 + 19683*E
^(4*x) - 25920*x + 54432*x^2 - 52488*x^3 + 19683*x^4 + E^(3*x)*(-52488 + 78732*x) + E^(2*x)*(54432 - 157464*x
+ 118098*x^2) + E^x*(-25920 + 108864*x - 157464*x^2 + 78732*x^3)),x]

[Out]

-188*Defer[Int][(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^(-2), x] + 270*Defer[Int][E^x/(40 - 1
08*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^2, x] + (946*Defer[Int][x/(40 - 108*E^x + 81*E^(2*x) - 108*x
+ 162*E^x*x + 81*x^2)^2, x])/3 - 72*Defer[Int][(E^x*x)/(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^
2)^2, x] - (296*Defer[Int][x^2/(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^2, x])/3 + 36*Defer[In
t][(E^x*x^2)/(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^2, x] + 72*Defer[Int][x^3/(40 - 108*E^x
+ 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^2, x] - 54*Defer[Int][(E^x*x^3)/(40 - 108*E^x + 81*E^(2*x) - 108*x
+ 162*E^x*x + 81*x^2)^2, x] - 54*Defer[Int][x^4/(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^2, x]
+ (5*Defer[Int][(40 - 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2)^(-1), x])/3 + (2*Defer[Int][x^2/(40
- 108*E^x + 81*E^(2*x) - 108*x + 162*E^x*x + 81*x^2), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {81 e^{2 x} \left (5+2 x^2\right )+7 \left (-52+58 x+27 x^2\right )+54 e^x \left (5+11 x-2 x^2+3 x^3\right )}{3 \left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {81 e^{2 x} \left (5+2 x^2\right )+7 \left (-52+58 x+27 x^2\right )+54 e^x \left (5+11 x-2 x^2+3 x^3\right )}{\left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {2 \left (3+x+x^2\right ) \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {5+2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {5+2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-\frac {2}{3} \int \frac {\left (3+x+x^2\right ) \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {5}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}+\frac {2 x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {3 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {x \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {x^2 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {2}{3} \int \frac {x \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx\right )-\frac {2}{3} \int \frac {x^2 \left (94-135 e^x-189 x+81 e^x x+81 x^2\right )}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+\frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-2 \int \frac {94-135 e^x-189 x+81 e^x x+81 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx\\ &=\frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-\frac {2}{3} \int \left (\frac {94 x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx-\frac {2}{3} \int \left (\frac {94 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^4}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-2 \int \left (\frac {94}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {135 e^x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}-\frac {189 x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}+\frac {81 x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2}\right ) \, dx\\ &=\frac {2}{3} \int \frac {x^2}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx+\frac {5}{3} \int \frac {1}{40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2} \, dx-54 \int \frac {e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {e^x x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-54 \int \frac {x^4}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-\frac {188}{3} \int \frac {x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-\frac {188}{3} \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+90 \int \frac {e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+90 \int \frac {e^x x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+126 \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+126 \int \frac {x^3}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-162 \int \frac {e^x x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-162 \int \frac {x^2}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx-188 \int \frac {1}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+270 \int \frac {e^x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx+378 \int \frac {x}{\left (40-108 e^x+81 e^{2 x}-108 x+162 e^x x+81 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.90, size = 39, normalized size = 1.26 \begin {gather*} -\frac {3+x+x^2}{3 \left (40+81 e^{2 x}-108 x+81 x^2+54 e^x (-2+3 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-364 + 406*x + 189*x^2 + E^(2*x)*(405 + 162*x^2) + E^x*(270 + 594*x - 108*x^2 + 162*x^3))/(4800 + 1
9683*E^(4*x) - 25920*x + 54432*x^2 - 52488*x^3 + 19683*x^4 + E^(3*x)*(-52488 + 78732*x) + E^(2*x)*(54432 - 157
464*x + 118098*x^2) + E^x*(-25920 + 108864*x - 157464*x^2 + 78732*x^3)),x]

[Out]

-1/3*(3 + x + x^2)/(40 + 81*E^(2*x) - 108*x + 81*x^2 + 54*E^x*(-2 + 3*x))

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fricas [A]  time = 0.56, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406*x-364)/(19683*exp(x)^4+(78732
*x-52488)*exp(x)^3+(118098*x^2-157464*x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4
-52488*x^3+54432*x^2-25920*x+4800),x, algorithm="fricas")

[Out]

-1/3*(x^2 + x + 3)/(81*x^2 + 54*(3*x - 2)*e^x - 108*x + 81*e^(2*x) + 40)

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giac [A]  time = 0.80, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 162 \, x e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} - 108 \, e^{x} + 40\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406*x-364)/(19683*exp(x)^4+(78732
*x-52488)*exp(x)^3+(118098*x^2-157464*x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4
-52488*x^3+54432*x^2-25920*x+4800),x, algorithm="giac")

[Out]

-1/3*(x^2 + x + 3)/(81*x^2 + 162*x*e^x - 108*x + 81*e^(2*x) - 108*e^x + 40)

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maple [A]  time = 0.11, size = 36, normalized size = 1.16

method	result	size
risch	\(-\frac {x^{2}+x +3}{3 \left (81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40\right )}\)	\(36\)
norman	\(\frac {-\frac {1}{3} x -\frac {1}{3} x^{2}-1}{81 \,{\mathrm e}^{2 x}+162 \,{\mathrm e}^{x} x +81 x^{2}-108 \,{\mathrm e}^{x}-108 x +40}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406*x-364)/(19683*exp(x)^4+(78732*x-524
88)*exp(x)^3+(118098*x^2-157464*x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4-52488
*x^3+54432*x^2-25920*x+4800),x,method=_RETURNVERBOSE)

[Out]

-1/3*(x^2+x+3)/(81*exp(2*x)+162*exp(x)*x+81*x^2-108*exp(x)-108*x+40)

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maxima [A]  time = 0.48, size = 35, normalized size = 1.13 \begin {gather*} -\frac {x^{2} + x + 3}{3 \, {\left (81 \, x^{2} + 54 \, {\left (3 \, x - 2\right )} e^{x} - 108 \, x + 81 \, e^{\left (2 \, x\right )} + 40\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x^2+405)*exp(x)^2+(162*x^3-108*x^2+594*x+270)*exp(x)+189*x^2+406*x-364)/(19683*exp(x)^4+(78732
*x-52488)*exp(x)^3+(118098*x^2-157464*x+54432)*exp(x)^2+(78732*x^3-157464*x^2+108864*x-25920)*exp(x)+19683*x^4
-52488*x^3+54432*x^2-25920*x+4800),x, algorithm="maxima")

[Out]

-1/3*(x^2 + x + 3)/(81*x^2 + 54*(3*x - 2)*e^x - 108*x + 81*e^(2*x) + 40)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {406\,x+{\mathrm {e}}^{2\,x}\,\left (162\,x^2+405\right )+189\,x^2+{\mathrm {e}}^x\,\left (162\,x^3-108\,x^2+594\,x+270\right )-364}{19683\,{\mathrm {e}}^{4\,x}-25920\,x+{\mathrm {e}}^{2\,x}\,\left (118098\,x^2-157464\,x+54432\right )+{\mathrm {e}}^{3\,x}\,\left (78732\,x-52488\right )+54432\,x^2-52488\,x^3+19683\,x^4+{\mathrm {e}}^x\,\left (78732\,x^3-157464\,x^2+108864\,x-25920\right )+4800} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((406*x + exp(2*x)*(162*x^2 + 405) + 189*x^2 + exp(x)*(594*x - 108*x^2 + 162*x^3 + 270) - 364)/(19683*exp(4
*x) - 25920*x + exp(2*x)*(118098*x^2 - 157464*x + 54432) + exp(3*x)*(78732*x - 52488) + 54432*x^2 - 52488*x^3
+ 19683*x^4 + exp(x)*(108864*x - 157464*x^2 + 78732*x^3 - 25920) + 4800),x)

[Out]

int((406*x + exp(2*x)*(162*x^2 + 405) + 189*x^2 + exp(x)*(594*x - 108*x^2 + 162*x^3 + 270) - 364)/(19683*exp(4
*x) - 25920*x + exp(2*x)*(118098*x^2 - 157464*x + 54432) + exp(3*x)*(78732*x - 52488) + 54432*x^2 - 52488*x^3
+ 19683*x^4 + exp(x)*(108864*x - 157464*x^2 + 78732*x^3 - 25920) + 4800), x)

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sympy [A]  time = 0.22, size = 32, normalized size = 1.03 \begin {gather*} \frac {- x^{2} - x - 3}{243 x^{2} - 324 x + \left (486 x - 324\right ) e^{x} + 243 e^{2 x} + 120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((162*x**2+405)*exp(x)**2+(162*x**3-108*x**2+594*x+270)*exp(x)+189*x**2+406*x-364)/(19683*exp(x)**4+
(78732*x-52488)*exp(x)**3+(118098*x**2-157464*x+54432)*exp(x)**2+(78732*x**3-157464*x**2+108864*x-25920)*exp(x
)+19683*x**4-52488*x**3+54432*x**2-25920*x+4800),x)

[Out]

(-x**2 - x - 3)/(243*x**2 - 324*x + (486*x - 324)*exp(x) + 243*exp(2*x) + 120)

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