3.17.65 \(\int \frac {24 x^3-18 e^{2 x} x^3-8 x^5+e^x (-27 x^2-9 x^3+24 x^4)}{9-12 x^2+9 e^{2 x} x^2+4 x^4+e^x (18 x-12 x^3)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2}{-1-\frac {1}{\left (e^x-\frac {2 x}{3}\right ) x}} \]

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Rubi [F]  time = 1.35, 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 {24 x^3-18 e^{2 x} x^3-8 x^5+e^x \left (-27 x^2-9 x^3+24 x^4\right )}{9-12 x^2+9 e^{2 x} x^2+4 x^4+e^x \left (18 x-12 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(24*x^3 - 18*E^(2*x)*x^3 - 8*x^5 + E^x*(-27*x^2 - 9*x^3 + 24*x^4))/(9 - 12*x^2 + 9*E^(2*x)*x^2 + 4*x^4 + E
^x*(18*x - 12*x^3)),x]

[Out]

-x^2 + 9*Defer[Int][x/(-3 - 3*E^x*x + 2*x^2)^2, x] + 9*Defer[Int][x^2/(-3 - 3*E^x*x + 2*x^2)^2, x] + 6*Defer[I
nt][x^3/(-3 - 3*E^x*x + 2*x^2)^2, x] - 6*Defer[Int][x^4/(-3 - 3*E^x*x + 2*x^2)^2, x] - 3*Defer[Int][x/(-3 - 3*
E^x*x + 2*x^2), x] + 3*Defer[Int][x^2/(-3 - 3*E^x*x + 2*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^2 \left (-18 e^{2 x} x-e^x \left (27+9 x-24 x^2\right )-8 x \left (-3+x^2\right )\right )}{\left (3+3 e^x x-2 x^2\right )^2} \, dx\\ &=\int \left (-2 x+\frac {3 (-1+x) x}{-3-3 e^x x+2 x^2}-\frac {3 x \left (-3-3 x-2 x^2+2 x^3\right )}{\left (-3-3 e^x x+2 x^2\right )^2}\right ) \, dx\\ &=-x^2+3 \int \frac {(-1+x) x}{-3-3 e^x x+2 x^2} \, dx-3 \int \frac {x \left (-3-3 x-2 x^2+2 x^3\right )}{\left (-3-3 e^x x+2 x^2\right )^2} \, dx\\ &=-x^2-3 \int \left (-\frac {3 x}{\left (-3-3 e^x x+2 x^2\right )^2}-\frac {3 x^2}{\left (-3-3 e^x x+2 x^2\right )^2}-\frac {2 x^3}{\left (-3-3 e^x x+2 x^2\right )^2}+\frac {2 x^4}{\left (-3-3 e^x x+2 x^2\right )^2}\right ) \, dx+3 \int \left (-\frac {x}{-3-3 e^x x+2 x^2}+\frac {x^2}{-3-3 e^x x+2 x^2}\right ) \, dx\\ &=-x^2-3 \int \frac {x}{-3-3 e^x x+2 x^2} \, dx+3 \int \frac {x^2}{-3-3 e^x x+2 x^2} \, dx+6 \int \frac {x^3}{\left (-3-3 e^x x+2 x^2\right )^2} \, dx-6 \int \frac {x^4}{\left (-3-3 e^x x+2 x^2\right )^2} \, dx+9 \int \frac {x}{\left (-3-3 e^x x+2 x^2\right )^2} \, dx+9 \int \frac {x^2}{\left (-3-3 e^x x+2 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 26, normalized size = 1.08 \begin {gather*} -x^2+\frac {3 x^2}{3+3 e^x x-2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(24*x^3 - 18*E^(2*x)*x^3 - 8*x^5 + E^x*(-27*x^2 - 9*x^3 + 24*x^4))/(9 - 12*x^2 + 9*E^(2*x)*x^2 + 4*x
^4 + E^x*(18*x - 12*x^3)),x]

[Out]

-x^2 + (3*x^2)/(3 + 3*E^x*x - 2*x^2)

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fricas [A]  time = 0.58, size = 29, normalized size = 1.21 \begin {gather*} -\frac {2 \, x^{4} - 3 \, x^{3} e^{x}}{2 \, x^{2} - 3 \, x e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(x)^2*x^3+(24*x^4-9*x^3-27*x^2)*exp(x)-8*x^5+24*x^3)/(9*exp(x)^2*x^2+(-12*x^3+18*x)*exp(x)+4
*x^4-12*x^2+9),x, algorithm="fricas")

[Out]

-(2*x^4 - 3*x^3*e^x)/(2*x^2 - 3*x*e^x - 3)

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giac [A]  time = 0.22, size = 29, normalized size = 1.21 \begin {gather*} -\frac {2 \, x^{4} - 3 \, x^{3} e^{x}}{2 \, x^{2} - 3 \, x e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(x)^2*x^3+(24*x^4-9*x^3-27*x^2)*exp(x)-8*x^5+24*x^3)/(9*exp(x)^2*x^2+(-12*x^3+18*x)*exp(x)+4
*x^4-12*x^2+9),x, algorithm="giac")

[Out]

-(2*x^4 - 3*x^3*e^x)/(2*x^2 - 3*x*e^x - 3)

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maple [A]  time = 0.06, size = 26, normalized size = 1.08

method	result	size
risch	\(-x^{2}-\frac {3 x^{2}}{2 x^{2}-3 \,{\mathrm e}^{x} x -3}\)	\(26\)
norman	\(\frac {-2 x^{4}+3 \,{\mathrm e}^{x} x^{3}}{2 x^{2}-3 \,{\mathrm e}^{x} x -3}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-18*exp(x)^2*x^3+(24*x^4-9*x^3-27*x^2)*exp(x)-8*x^5+24*x^3)/(9*exp(x)^2*x^2+(-12*x^3+18*x)*exp(x)+4*x^4-1
2*x^2+9),x,method=_RETURNVERBOSE)

[Out]

-x^2-3*x^2/(2*x^2-3*exp(x)*x-3)

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maxima [A]  time = 0.60, size = 29, normalized size = 1.21 \begin {gather*} -\frac {2 \, x^{4} - 3 \, x^{3} e^{x}}{2 \, x^{2} - 3 \, x e^{x} - 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(x)^2*x^3+(24*x^4-9*x^3-27*x^2)*exp(x)-8*x^5+24*x^3)/(9*exp(x)^2*x^2+(-12*x^3+18*x)*exp(x)+4
*x^4-12*x^2+9),x, algorithm="maxima")

[Out]

-(2*x^4 - 3*x^3*e^x)/(2*x^2 - 3*x*e^x - 3)

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mupad [B]  time = 1.27, size = 25, normalized size = 1.04 \begin {gather*} \frac {3\,x^2}{3\,x\,{\mathrm {e}}^x-2\,x^2+3}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(27*x^2 + 9*x^3 - 24*x^4) + 18*x^3*exp(2*x) - 24*x^3 + 8*x^5)/(9*x^2*exp(2*x) + exp(x)*(18*x - 12
*x^3) - 12*x^2 + 4*x^4 + 9),x)

[Out]

(3*x^2)/(3*x*exp(x) - 2*x^2 + 3) - x^2

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sympy [A]  time = 0.15, size = 20, normalized size = 0.83 \begin {gather*} - x^{2} + \frac {3 x^{2}}{- 2 x^{2} + 3 x e^{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-18*exp(x)**2*x**3+(24*x**4-9*x**3-27*x**2)*exp(x)-8*x**5+24*x**3)/(9*exp(x)**2*x**2+(-12*x**3+18*x
)*exp(x)+4*x**4-12*x**2+9),x)

[Out]

-x**2 + 3*x**2/(-2*x**2 + 3*x*exp(x) + 3)

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