3.55.61 \(\int \frac {e^{2 x^2} (16-72 x-16 x^2+48 x^3)}{-x^5+9 x^6-27 x^7+27 x^8} \, dx\)

Optimal. Leaf size=21 \[ \frac {4 e^{2 x^2}}{x^2 \left (x-3 x^2\right )^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 42, normalized size of antiderivative = 2.00, number of steps used = 1, number of rules used = 1, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.022, Rules used = {2288} \begin {gather*} \frac {4 e^{2 x^2} \left (x^2-3 x^3\right )}{x \left (-27 x^8+27 x^7-9 x^6+x^5\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2*x^2)*(16 - 72*x - 16*x^2 + 48*x^3))/(-x^5 + 9*x^6 - 27*x^7 + 27*x^8),x]

[Out]

(4*E^(2*x^2)*(x^2 - 3*x^3))/(x*(x^5 - 9*x^6 + 27*x^7 - 27*x^8))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {4 e^{2 x^2} \left (x^2-3 x^3\right )}{x \left (x^5-9 x^6+27 x^7-27 x^8\right )}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 19, normalized size = 0.90 \begin {gather*} \frac {4 e^{2 x^2}}{(1-3 x)^2 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x^2)*(16 - 72*x - 16*x^2 + 48*x^3))/(-x^5 + 9*x^6 - 27*x^7 + 27*x^8),x]

[Out]

(4*E^(2*x^2))/((1 - 3*x)^2*x^4)

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 24, normalized size = 1.14 \begin {gather*} \frac {4 \, e^{\left (2 \, x^{2}\right )}}{9 \, x^{6} - 6 \, x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-16*x^2-72*x+16)*exp(x^2)^2/(27*x^8-27*x^7+9*x^6-x^5),x, algorithm="fricas")

[Out]

4*e^(2*x^2)/(9*x^6 - 6*x^5 + x^4)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 24, normalized size = 1.14 \begin {gather*} \frac {4 \, e^{\left (2 \, x^{2}\right )}}{9 \, x^{6} - 6 \, x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-16*x^2-72*x+16)*exp(x^2)^2/(27*x^8-27*x^7+9*x^6-x^5),x, algorithm="giac")

[Out]

4*e^(2*x^2)/(9*x^6 - 6*x^5 + x^4)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 19, normalized size = 0.90

method	result	size
norman	\(\frac {4 \,{\mathrm e}^{2 x^{2}}}{x^{4} \left (3 x -1\right )^{2}}\)	\(19\)
risch	\(\frac {4 \,{\mathrm e}^{2 x^{2}}}{x^{4} \left (3 x -1\right )^{2}}\)	\(19\)
gosper	\(\frac {4 \,{\mathrm e}^{2 x^{2}}}{x^{4} \left (9 x^{2}-6 x +1\right )}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((48*x^3-16*x^2-72*x+16)*exp(x^2)^2/(27*x^8-27*x^7+9*x^6-x^5),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 24, normalized size = 1.14 \begin {gather*} \frac {4 \, e^{\left (2 \, x^{2}\right )}}{9 \, x^{6} - 6 \, x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x^3-16*x^2-72*x+16)*exp(x^2)^2/(27*x^8-27*x^7+9*x^6-x^5),x, algorithm="maxima")

[Out]

4*e^(2*x^2)/(9*x^6 - 6*x^5 + x^4)

________________________________________________________________________________________

mupad [B]  time = 0.27, size = 24, normalized size = 1.14 \begin {gather*} \frac {4\,{\mathrm {e}}^{2\,x^2}}{9\,x^6-6\,x^5+x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x^2)*(72*x + 16*x^2 - 48*x^3 - 16))/(x^5 - 9*x^6 + 27*x^7 - 27*x^8),x)

[Out]

(4*exp(2*x^2))/(x^4 - 6*x^5 + 9*x^6)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 20, normalized size = 0.95 \begin {gather*} \frac {4 e^{2 x^{2}}}{9 x^{6} - 6 x^{5} + x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((48*x**3-16*x**2-72*x+16)*exp(x**2)**2/(27*x**8-27*x**7+9*x**6-x**5),x)

[Out]

4*exp(2*x**2)/(9*x**6 - 6*x**5 + x**4)

________________________________________________________________________________________