3.32.42 \(\int \frac {e^{16-4 x^2} (4 x^3-8 x^5-8 x^6)}{(1+x)^5} \, dx\)

Optimal. Leaf size=18 \[ \frac {e^{16-4 x^2} x^4}{(1+x)^4} \]

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Rubi [A]  time = 0.13, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1594, 2288} \begin {gather*} \frac {e^{16-4 x^2} x^2 \left (x^3+x^2\right )}{(x+1)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(16 - 4*x^2)*(4*x^3 - 8*x^5 - 8*x^6))/(1 + x)^5,x]

[Out]

(E^(16 - 4*x^2)*x^2*(x^2 + x^3))/(1 + x)^5

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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} &=\int \frac {e^{16-4 x^2} x^3 \left (4-8 x^2-8 x^3\right )}{(1+x)^5} \, dx\\ &=\frac {e^{16-4 x^2} x^2 \left (x^2+x^3\right )}{(1+x)^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {e^{16-4 x^2} x^4}{(1+x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(16 - 4*x^2)*(4*x^3 - 8*x^5 - 8*x^6))/(1 + x)^5,x]

[Out]

(E^(16 - 4*x^2)*x^4)/(1 + x)^4

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fricas [A]  time = 0.54, size = 18, normalized size = 1.00 \begin {gather*} x^{4} e^{\left (-4 \, x^{2} - 4 \, \log \left (x + 1\right ) + 16\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-8*x^5+4*x^3)*exp(-log(x+1)-x^2+4)^4/(x+1),x, algorithm="fricas")

[Out]

x^4*e^(-4*x^2 - 4*log(x + 1) + 16)

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giac [A]  time = 0.20, size = 32, normalized size = 1.78 \begin {gather*} \frac {x^{4} e^{\left (-4 \, x^{2} + 16\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-8*x^5+4*x^3)*exp(-log(x+1)-x^2+4)^4/(x+1),x, algorithm="giac")

[Out]

x^4*e^(-4*x^2 + 16)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)

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maple [A]  time = 0.53, size = 19, normalized size = 1.06

method	result	size
risch	\(\frac {x^{4} {\mathrm e}^{-4 \left (x -2\right ) \left (2+x \right )}}{\left (x +1\right )^{4}}\)	\(19\)
gosper	\(\frac {x^{4} {\mathrm e}^{-4 x^{2}+16}}{\left (x +1\right )^{4}}\)	\(21\)
meijerg	\(-\frac {8 \,{\mathrm e}^{16} \left (x +1\right )^{-4+4 \,{\mathrm e}^{-4 x^{2}}} \hypergeom \left (\left [7, 1+4 \,{\mathrm e}^{-4 x^{2}}\right ], \relax [8], -x \right ) x^{7} {\mathrm e}^{-4 x^{2}}}{7}-\frac {4 \,{\mathrm e}^{16} \left (x +1\right )^{-4+4 \,{\mathrm e}^{-4 x^{2}}} \hypergeom \left (\left [6, 1+4 \,{\mathrm e}^{-4 x^{2}}\right ], \relax [7], -x \right ) x^{6} {\mathrm e}^{-4 x^{2}}}{3}+{\mathrm e}^{16} \left (x +1\right )^{-4+4 \,{\mathrm e}^{-4 x^{2}}} \hypergeom \left (\left [4, 1+4 \,{\mathrm e}^{-4 x^{2}}\right ], \relax [5], -x \right ) x^{4} {\mathrm e}^{-4 x^{2}}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^6-8*x^5+4*x^3)*exp(-ln(x+1)-x^2+4)^4/(x+1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.73, size = 32, normalized size = 1.78 \begin {gather*} \frac {x^{4} e^{\left (-4 \, x^{2} + 16\right )}}{x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^6-8*x^5+4*x^3)*exp(-log(x+1)-x^2+4)^4/(x+1),x, algorithm="maxima")

[Out]

x^4*e^(-4*x^2 + 16)/(x^4 + 4*x^3 + 6*x^2 + 4*x + 1)

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mupad [B]  time = 0.15, size = 17, normalized size = 0.94 \begin {gather*} \frac {x^4\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-4\,x^2}}{{\left (x+1\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(16 - 4*x^2 - 4*log(x + 1))*(8*x^5 - 4*x^3 + 8*x^6))/(x + 1),x)

[Out]

(x^4*exp(16)*exp(-4*x^2))/(x + 1)^4

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sympy [A]  time = 0.12, size = 29, normalized size = 1.61 \begin {gather*} \frac {x^{4} e^{16 - 4 x^{2}}}{x^{4} + 4 x^{3} + 6 x^{2} + 4 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**6-8*x**5+4*x**3)*exp(-ln(x+1)-x**2+4)**4/(x+1),x)

[Out]

x**4*exp(16 - 4*x**2)/(x**4 + 4*x**3 + 6*x**2 + 4*x + 1)

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