3.20.8 \(\int (6+e^{\frac {1}{4} (9-8 x-4 x^2+8 x^3+4 x^4)} (1-2 x-2 x^2+6 x^3+4 x^4)) \, dx\)

Optimal. Leaf size=22 \[ \left (6+e^{\frac {5}{4}+\left (1-x-x^2\right )^2}\right ) x \]

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Rubi [B]  time = 0.11, antiderivative size = 61, normalized size of antiderivative = 2.77, number of steps used = 2, number of rules used = 1, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {2288} \begin {gather*} \frac {e^{\frac {1}{4} \left (4 x^4+8 x^3-4 x^2-8 x+9\right )} \left (-2 x^4-3 x^3+x^2+x\right )}{-2 x^3-3 x^2+x+1}+6 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[6 + E^((9 - 8*x - 4*x^2 + 8*x^3 + 4*x^4)/4)*(1 - 2*x - 2*x^2 + 6*x^3 + 4*x^4),x]

[Out]

6*x + (E^((9 - 8*x - 4*x^2 + 8*x^3 + 4*x^4)/4)*(x + x^2 - 3*x^3 - 2*x^4))/(1 + x - 3*x^2 - 2*x^3)

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

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Mathematica [A]  time = 0.08, size = 24, normalized size = 1.09 \begin {gather*} \left (6+e^{\frac {9}{4}+2 x^3+x^4-x (2+x)}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[6 + E^((9 - 8*x - 4*x^2 + 8*x^3 + 4*x^4)/4)*(1 - 2*x - 2*x^2 + 6*x^3 + 4*x^4),x]

[Out]

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

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fricas [A]  time = 0.83, size = 25, normalized size = 1.14 \begin {gather*} x e^{\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + \frac {9}{4}\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+6*x^3-2*x^2-2*x+1)*exp(x^4+2*x^3-x^2-2*x+9/4)+6,x, algorithm="fricas")

[Out]

x*e^(x^4 + 2*x^3 - x^2 - 2*x + 9/4) + 6*x

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giac [A]  time = 0.25, size = 25, normalized size = 1.14 \begin {gather*} x e^{\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + \frac {9}{4}\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+6*x^3-2*x^2-2*x+1)*exp(x^4+2*x^3-x^2-2*x+9/4)+6,x, algorithm="giac")

[Out]

x*e^(x^4 + 2*x^3 - x^2 - 2*x + 9/4) + 6*x

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maple [A]  time = 0.04, size = 26, normalized size = 1.18

method	result	size
default	\(6 x +{\mathrm e}^{x^{4}+2 x^{3}-x^{2}-2 x +\frac {9}{4}} x\)	\(26\)
norman	\(6 x +{\mathrm e}^{x^{4}+2 x^{3}-x^{2}-2 x +\frac {9}{4}} x\)	\(26\)
risch	\(6 x +{\mathrm e}^{x^{4}+2 x^{3}-x^{2}-2 x +\frac {9}{4}} x\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^4+6*x^3-2*x^2-2*x+1)*exp(x^4+2*x^3-x^2-2*x+9/4)+6,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.89, size = 25, normalized size = 1.14 \begin {gather*} x e^{\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + \frac {9}{4}\right )} + 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^4+6*x^3-2*x^2-2*x+1)*exp(x^4+2*x^3-x^2-2*x+9/4)+6,x, algorithm="maxima")

[Out]

x*e^(x^4 + 2*x^3 - x^2 - 2*x + 9/4) + 6*x

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mupad [B]  time = 1.13, size = 27, normalized size = 1.23 \begin {gather*} x\,\left ({\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{9/4}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{2\,x^3}+6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x^3 - x^2 - 2*x + x^4 + 9/4)*(6*x^3 - 2*x^2 - 2*x + 4*x^4 + 1) + 6,x)

[Out]

x*(exp(-2*x)*exp(x^4)*exp(9/4)*exp(-x^2)*exp(2*x^3) + 6)

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sympy [A]  time = 0.11, size = 24, normalized size = 1.09 \begin {gather*} x e^{x^{4} + 2 x^{3} - x^{2} - 2 x + \frac {9}{4}} + 6 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**4+6*x**3-2*x**2-2*x+1)*exp(x**4+2*x**3-x**2-2*x+9/4)+6,x)

[Out]

x*exp(x**4 + 2*x**3 - x**2 - 2*x + 9/4) + 6*x

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