3.8.24 \(\int e^{1-x+27 x^3} (2 x-x^2+81 x^4) \, dx\)

Optimal. Leaf size=16 \[ e^{1-x+27 x^3} x^2 \]

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Rubi [A]  time = 0.05, antiderivative size = 30, normalized size of antiderivative = 1.88, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1594, 2288} \begin {gather*} \frac {e^{27 x^3-x+1} x \left (x-81 x^3\right )}{1-81 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(1 - x + 27*x^3)*(2*x - x^2 + 81*x^4),x]

[Out]

(E^(1 - x + 27*x^3)*x*(x - 81*x^3))/(1 - 81*x^2)

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 e^{1-x+27 x^3} x \left (2-x+81 x^3\right ) \, dx\\ &=\frac {e^{1-x+27 x^3} x \left (x-81 x^3\right )}{1-81 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} e^{1-x+27 x^3} x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(1 - x + 27*x^3)*(2*x - x^2 + 81*x^4),x]

[Out]

E^(1 - x + 27*x^3)*x^2

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fricas [A]  time = 1.11, size = 15, normalized size = 0.94 \begin {gather*} x^{2} e^{\left (27 \, x^{3} - x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4-x^2+2*x)/exp(-27*x^3+x-1),x, algorithm="fricas")

[Out]

x^2*e^(27*x^3 - x + 1)

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giac [A]  time = 0.70, size = 15, normalized size = 0.94 \begin {gather*} x^{2} e^{\left (27 \, x^{3} - x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4-x^2+2*x)/exp(-27*x^3+x-1),x, algorithm="giac")

[Out]

x^2*e^(27*x^3 - x + 1)

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maple [A]  time = 0.05, size = 16, normalized size = 1.00

method	result	size
gosper	\({\mathrm e}^{27 x^{3}-x +1} x^{2}\)	\(16\)
norman	\({\mathrm e}^{27 x^{3}-x +1} x^{2}\)	\(16\)
risch	\({\mathrm e}^{27 x^{3}-x +1} x^{2}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((81*x^4-x^2+2*x)/exp(-27*x^3+x-1),x,method=_RETURNVERBOSE)

[Out]

x^2/exp(-27*x^3+x-1)

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maxima [A]  time = 0.79, size = 15, normalized size = 0.94 \begin {gather*} x^{2} e^{\left (27 \, x^{3} - x + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x^4-x^2+2*x)/exp(-27*x^3+x-1),x, algorithm="maxima")

[Out]

x^2*e^(27*x^3 - x + 1)

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mupad [B]  time = 0.04, size = 16, normalized size = 1.00 \begin {gather*} x^2\,{\mathrm {e}}^{-x}\,\mathrm {e}\,{\mathrm {e}}^{27\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(27*x^3 - x + 1)*(2*x - x^2 + 81*x^4),x)

[Out]

x^2*exp(-x)*exp(1)*exp(27*x^3)

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sympy [A]  time = 0.10, size = 12, normalized size = 0.75 \begin {gather*} x^{2} e^{27 x^{3} - x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((81*x**4-x**2+2*x)/exp(-27*x**3+x-1),x)

[Out]

x**2*exp(27*x**3 - x + 1)

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