3.103.39 \(\int \frac {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x (-9 x^2+7 x^3-x^4-7 x^6+x^7)}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x (-36+20 x-4 x^2)} \, dx\)

Optimal. Leaf size=33 \[ -1+\frac {x^3 \left (3-x+x^4\right )}{2 \left (-e^x+(3-x)^2+x\right )} \]

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Rubi [F]  time = 2.26, 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 {81 x^2-66 x^3+18 x^4-2 x^5+63 x^6-30 x^7+5 x^8+e^x \left (-9 x^2+7 x^3-x^4-7 x^6+x^7\right )}{162+2 e^{2 x}-180 x+86 x^2-20 x^3+2 x^4+e^x \left (-36+20 x-4 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(81*x^2 - 66*x^3 + 18*x^4 - 2*x^5 + 63*x^6 - 30*x^7 + 5*x^8 + E^x*(-9*x^2 + 7*x^3 - x^4 - 7*x^6 + x^7))/(1
62 + 2*E^(2*x) - 180*x + 86*x^2 - 20*x^3 + 2*x^4 + E^x*(-36 + 20*x - 4*x^2)),x]

[Out]

21*Defer[Int][x^3/(9 - E^x - 5*x + x^2)^2, x] - (35*Defer[Int][x^4/(9 - E^x - 5*x + x^2)^2, x])/2 + 5*Defer[In
t][x^5/(9 - E^x - 5*x + x^2)^2, x] - Defer[Int][x^6/(9 - E^x - 5*x + x^2)^2, x]/2 + 7*Defer[Int][x^7/(9 - E^x
- 5*x + x^2)^2, x] - (7*Defer[Int][x^8/(9 - E^x - 5*x + x^2)^2, x])/2 + Defer[Int][x^9/(9 - E^x - 5*x + x^2)^2
, x]/2 + (9*Defer[Int][x^2/(9 - E^x - 5*x + x^2), x])/2 - (7*Defer[Int][x^3/(9 - E^x - 5*x + x^2), x])/2 + Def
er[Int][x^4/(9 - E^x - 5*x + x^2), x]/2 + (7*Defer[Int][x^6/(9 - E^x - 5*x + x^2), x])/2 - Defer[Int][x^7/(9 -
E^x - 5*x + x^2), x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (81-66 x+18 x^2-2 x^3+63 x^4-30 x^5+5 x^6+e^x \left (-9+7 x-x^2-7 x^4+x^5\right )\right )}{2 \left (9-e^x-5 x+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {x^2 \left (81-66 x+18 x^2-2 x^3+63 x^4-30 x^5+5 x^6+e^x \left (-9+7 x-x^2-7 x^4+x^5\right )\right )}{\left (9-e^x-5 x+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (-\frac {x^2 \left (-9+7 x-x^2-7 x^4+x^5\right )}{9-e^x-5 x+x^2}+\frac {x^3 \left (42-35 x+10 x^2-x^3+14 x^4-7 x^5+x^6\right )}{\left (9-e^x-5 x+x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^2 \left (-9+7 x-x^2-7 x^4+x^5\right )}{9-e^x-5 x+x^2} \, dx\right )+\frac {1}{2} \int \frac {x^3 \left (42-35 x+10 x^2-x^3+14 x^4-7 x^5+x^6\right )}{\left (9-e^x-5 x+x^2\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {42 x^3}{\left (9-e^x-5 x+x^2\right )^2}-\frac {35 x^4}{\left (9-e^x-5 x+x^2\right )^2}+\frac {10 x^5}{\left (9-e^x-5 x+x^2\right )^2}-\frac {x^6}{\left (9-e^x-5 x+x^2\right )^2}+\frac {14 x^7}{\left (9-e^x-5 x+x^2\right )^2}-\frac {7 x^8}{\left (9-e^x-5 x+x^2\right )^2}+\frac {x^9}{\left (9-e^x-5 x+x^2\right )^2}\right ) \, dx-\frac {1}{2} \int \left (-\frac {9 x^2}{9-e^x-5 x+x^2}+\frac {7 x^3}{9-e^x-5 x+x^2}-\frac {x^4}{9-e^x-5 x+x^2}-\frac {7 x^6}{9-e^x-5 x+x^2}+\frac {x^7}{9-e^x-5 x+x^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^6}{\left (9-e^x-5 x+x^2\right )^2} \, dx\right )+\frac {1}{2} \int \frac {x^9}{\left (9-e^x-5 x+x^2\right )^2} \, dx+\frac {1}{2} \int \frac {x^4}{9-e^x-5 x+x^2} \, dx-\frac {1}{2} \int \frac {x^7}{9-e^x-5 x+x^2} \, dx-\frac {7}{2} \int \frac {x^8}{\left (9-e^x-5 x+x^2\right )^2} \, dx-\frac {7}{2} \int \frac {x^3}{9-e^x-5 x+x^2} \, dx+\frac {7}{2} \int \frac {x^6}{9-e^x-5 x+x^2} \, dx+\frac {9}{2} \int \frac {x^2}{9-e^x-5 x+x^2} \, dx+5 \int \frac {x^5}{\left (9-e^x-5 x+x^2\right )^2} \, dx+7 \int \frac {x^7}{\left (9-e^x-5 x+x^2\right )^2} \, dx-\frac {35}{2} \int \frac {x^4}{\left (9-e^x-5 x+x^2\right )^2} \, dx+21 \int \frac {x^3}{\left (9-e^x-5 x+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.54, size = 30, normalized size = 0.91 \begin {gather*} \frac {x^3 \left (3-x+x^4\right )}{2 \left (9-e^x-5 x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(81*x^2 - 66*x^3 + 18*x^4 - 2*x^5 + 63*x^6 - 30*x^7 + 5*x^8 + E^x*(-9*x^2 + 7*x^3 - x^4 - 7*x^6 + x^
7))/(162 + 2*E^(2*x) - 180*x + 86*x^2 - 20*x^3 + 2*x^4 + E^x*(-36 + 20*x - 4*x^2)),x]

[Out]

(x^3*(3 - x + x^4))/(2*(9 - E^x - 5*x + x^2))

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fricas [A]  time = 0.69, size = 30, normalized size = 0.91 \begin {gather*} \frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*
x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180*x+162),x, algorithm="fricas")

[Out]

1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)

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giac [A]  time = 0.15, size = 30, normalized size = 0.91 \begin {gather*} \frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*
x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180*x+162),x, algorithm="giac")

[Out]

1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)

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maple [A]  time = 0.08, size = 28, normalized size = 0.85

method	result	size
risch	\(\frac {\left (x^{4}-x +3\right ) x^{3}}{2 x^{2}-10 x -2 \,{\mathrm e}^{x}+18}\)	\(28\)
norman	\(\frac {\frac {3}{2} x^{3}-\frac {1}{2} x^{4}+\frac {1}{2} x^{7}}{x^{2}-5 x -{\mathrm e}^{x}+9}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*x^2+20
*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180*x+162),x,method=_RETURNVERBOSE)

[Out]

1/2*(x^4-x+3)*x^3/(x^2-5*x-exp(x)+9)

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maxima [A]  time = 0.41, size = 30, normalized size = 0.91 \begin {gather*} \frac {x^{7} - x^{4} + 3 \, x^{3}}{2 \, {\left (x^{2} - 5 \, x - e^{x} + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7-7*x^6-x^4+7*x^3-9*x^2)*exp(x)+5*x^8-30*x^7+63*x^6-2*x^5+18*x^4-66*x^3+81*x^2)/(2*exp(x)^2+(-4*
x^2+20*x-36)*exp(x)+2*x^4-20*x^3+86*x^2-180*x+162),x, algorithm="maxima")

[Out]

1/2*(x^7 - x^4 + 3*x^3)/(x^2 - 5*x - e^x + 9)

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mupad [B]  time = 6.14, size = 32, normalized size = 0.97 \begin {gather*} -\frac {x^7-x^4+3\,x^3}{10\,x+2\,{\mathrm {e}}^x-2\,x^2-18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(9*x^2 - 7*x^3 + x^4 + 7*x^6 - x^7) - 81*x^2 + 66*x^3 - 18*x^4 + 2*x^5 - 63*x^6 + 30*x^7 - 5*x^8)
/(2*exp(2*x) - 180*x - exp(x)*(4*x^2 - 20*x + 36) + 86*x^2 - 20*x^3 + 2*x^4 + 162),x)

[Out]

-(3*x^3 - x^4 + x^7)/(10*x + 2*exp(x) - 2*x^2 - 18)

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sympy [A]  time = 0.22, size = 26, normalized size = 0.79 \begin {gather*} \frac {- x^{7} + x^{4} - 3 x^{3}}{- 2 x^{2} + 10 x + 2 e^{x} - 18} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**7-7*x**6-x**4+7*x**3-9*x**2)*exp(x)+5*x**8-30*x**7+63*x**6-2*x**5+18*x**4-66*x**3+81*x**2)/(2*e
xp(x)**2+(-4*x**2+20*x-36)*exp(x)+2*x**4-20*x**3+86*x**2-180*x+162),x)

[Out]

(-x**7 + x**4 - 3*x**3)/(-2*x**2 + 10*x + 2*exp(x) - 18)

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