3.38.84 \(\int (-3+2 x+5 x^4+e^{-3 x-x^2} (1-3 x-2 x^2)) \, dx\)

Optimal. Leaf size=17 \[ x \left (-3+e^{(-3-x) x}+x+x^4\right ) \]

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Rubi [B]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 2.24, number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {2288} \begin {gather*} x^5+x^2+\frac {e^{-x^2-3 x} \left (2 x^2+3 x\right )}{2 x+3}-3 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.03, size = 16, normalized size = 0.94 \begin {gather*} x \left (-3+e^{-x (3+x)}+x+x^4\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x*(-3 + E^(-(x*(3 + x))) + x + x^4)

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fricas [A]  time = 0.92, size = 22, normalized size = 1.29 \begin {gather*} x^{5} + x^{2} + x e^{\left (-x^{2} - 3 \, x\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5 + x^2 + x*e^(-x^2 - 3*x) - 3*x

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giac [A]  time = 0.13, size = 22, normalized size = 1.29 \begin {gather*} x^{5} + x^{2} + x e^{\left (-x^{2} - 3 \, x\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5 + x^2 + x*e^(-x^2 - 3*x) - 3*x

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maple [A]  time = 0.03, size = 20, normalized size = 1.18

method	result	size
risch	\(-3 x +{\mathrm e}^{-\left (3+x \right ) x} x +x^{2}+x^{5}\)	\(20\)
default	\(-3 x +{\mathrm e}^{-x^{2}-3 x} x +x^{2}+x^{5}\)	\(23\)
norman	\(-3 x +{\mathrm e}^{-x^{2}-3 x} x +x^{2}+x^{5}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x+exp(-(3+x)*x)*x+x^2+x^5

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maxima [A]  time = 0.49, size = 22, normalized size = 1.29 \begin {gather*} x^{5} + x^{2} + x e^{\left (-x^{2} - 3 \, x\right )} - 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^5 + x^2 + x*e^(-x^2 - 3*x) - 3*x

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mupad [B]  time = 0.07, size = 18, normalized size = 1.06 \begin {gather*} x\,\left (x+{\mathrm {e}}^{-x^2-3\,x}+x^4-3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x - exp(- 3*x - x^2)*(3*x + 2*x^2 - 1) + 5*x^4 - 3,x)

[Out]

x*(x + exp(- 3*x - x^2) + x^4 - 3)

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sympy [A]  time = 0.10, size = 20, normalized size = 1.18 \begin {gather*} x^{5} + x^{2} + x e^{- x^{2} - 3 x} - 3 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**2-3*x+1)*exp(-x**2-3*x)+5*x**4+2*x-3,x)

[Out]

x**5 + x**2 + x*exp(-x**2 - 3*x) - 3*x

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