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

Optimal. Leaf size=24 $$\frac{x^2}{-1-\frac{1}{(e^x-\frac{2x}{3})x}}$$

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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 = {} $$\begin{array}{c}\int \frac{24x^3-18e^{2x}{x}^3-8x^5+e^x(-27x^2-9x^3+24x^4)}{9-12x^2+9e^{2x}{x}^2+4x^4+e^x(18x-12x^3)}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{x^2(-18e^{2x}x-e^x(27+9x-24x^2)-8x(-3+x^2))}{{(3+3e^{x}x-2x^2)}^2}dx\\ & =\int (-2x+\frac{3(-1+x)x}{-3-3e^{x}x+2x^2}-\frac{3x(-3-3x-2x^2+2x^3)}{{(-3-3e^{x}x+2x^2)}^2})dx\\ & =-x^2+3\int \frac{(-1+x)x}{-3-3e^{x}x+2x^2}dx-3\int \frac{x(-3-3x-2x^2+2x^3)}{{(-3-3e^{x}x+2x^2)}^2}dx\\ & =-x^2-3\int (-\frac{3x}{{(-3-3e^{x}x+2x^2)}^2}-\frac{3x^2}{{(-3-3e^{x}x+2x^2)}^2}-\frac{2x^3}{{(-3-3e^{x}x+2x^2)}^2}+\frac{2x^4}{{(-3-3e^{x}x+2x^2)}^2})dx+3\int (-\frac{x}{-3-3e^{x}x+2x^2}+\frac{x^2}{-3-3e^{x}x+2x^2})dx\\ & =-x^2-3\int \frac{x}{-3-3e^{x}x+2x^2}dx+3\int \frac{x^2}{-3-3e^{x}x+2x^2}dx+6\int \frac{x^3}{{(-3-3e^{x}x+2x^2)}^2}dx-6\int \frac{x^4}{{(-3-3e^{x}x+2x^2)}^2}dx+9\int \frac{x}{{(-3-3e^{x}x+2x^2)}^2}dx+9\int \frac{x^2}{{(-3-3e^{x}x+2x^2)}^2}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}-\frac{2x^4-3x^3{e}^x}{2x^2-3x{e}^x-3}\end{array}$$

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 $$\begin{array}{c}-\frac{2x^4-3x^3{e}^x}{2x^2-3x{e}^x-3}\end{array}$$

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	$-x^2-\frac{3x^2}{2x^2-3{\mathrm{e}}^{x}x-3}$	$26$
norman	$\frac{-2x^4+3{\mathrm{e}}^x{x}^3}{2x^2-3{\mathrm{e}}^{x}x-3}$	$29$

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 $$\begin{array}{c}-\frac{2x^4-3x^3{e}^x}{2x^2-3x{e}^x-3}\end{array}$$

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 $$\begin{array}{c}\frac{3x^2}{3x{\mathrm{e}}^x-2x^2+3}-x^2\end{array}$$

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 $$\begin{array}{c}-x^2+\frac{3x^2}{-2x^2+3x{e}^x+3}\end{array}$$

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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