3.18.44 \(\int \frac {1}{7} e^{\frac {1}{4} (-25+4 x+20 x^2-4 x^4)} (1+10 x-4 x^3) \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{7} e^{x-\left (\frac {5}{2}-x^2\right )^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {12, 6706} \begin {gather*} \frac {1}{7} e^{\frac {1}{4} \left (-4 x^4+20 x^2+4 x-25\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((-25 + 4*x + 20*x^2 - 4*x^4)/4)*(1 + 10*x - 4*x^3))/7,x]

[Out]

E^((-25 + 4*x + 20*x^2 - 4*x^4)/4)/7

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{7} \int e^{\frac {1}{4} \left (-25+4 x+20 x^2-4 x^4\right )} \left (1+10 x-4 x^3\right ) \, dx\\ &=\frac {1}{7} e^{\frac {1}{4} \left (-25+4 x+20 x^2-4 x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 21, normalized size = 1.00 \begin {gather*} \frac {1}{7} e^{-\frac {25}{4}+x+5 x^2-x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-25 + 4*x + 20*x^2 - 4*x^4)/4)*(1 + 10*x - 4*x^3))/7,x]

[Out]

E^(-25/4 + x + 5*x^2 - x^4)/7

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fricas [A]  time = 0.84, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, e^{\left (-x^{4} + 5 \, x^{2} + x - \frac {25}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-4*x^3+10*x+1)*exp(-x^4+5*x^2+x-25/4),x, algorithm="fricas")

[Out]

1/7*e^(-x^4 + 5*x^2 + x - 25/4)

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giac [A]  time = 0.21, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, e^{\left (-x^{4} + 5 \, x^{2} + x - \frac {25}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-4*x^3+10*x+1)*exp(-x^4+5*x^2+x-25/4),x, algorithm="giac")

[Out]

1/7*e^(-x^4 + 5*x^2 + x - 25/4)

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maple [A]  time = 0.02, size = 17, normalized size = 0.81

method	result	size
gosper	\(\frac {{\mathrm e}^{-x^{4}+5 x^{2}+x -\frac {25}{4}}}{7}\)	\(17\)
derivativedivides	\(\frac {{\mathrm e}^{-x^{4}+5 x^{2}+x -\frac {25}{4}}}{7}\)	\(17\)
default	\(\frac {{\mathrm e}^{-x^{4}+5 x^{2}+x -\frac {25}{4}}}{7}\)	\(17\)
norman	\(\frac {{\mathrm e}^{-x^{4}+5 x^{2}+x -\frac {25}{4}}}{7}\)	\(17\)
risch	\(\frac {{\mathrm e}^{-x^{4}+5 x^{2}+x -\frac {25}{4}}}{7}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/7*(-4*x^3+10*x+1)*exp(-x^4+5*x^2+x-25/4),x,method=_RETURNVERBOSE)

[Out]

1/7*exp(-x^4+5*x^2+x-25/4)

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maxima [A]  time = 0.49, size = 16, normalized size = 0.76 \begin {gather*} \frac {1}{7} \, e^{\left (-x^{4} + 5 \, x^{2} + x - \frac {25}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-4*x^3+10*x+1)*exp(-x^4+5*x^2+x-25/4),x, algorithm="maxima")

[Out]

1/7*e^(-x^4 + 5*x^2 + x - 25/4)

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mupad [B]  time = 1.07, size = 18, normalized size = 0.86 \begin {gather*} \frac {{\mathrm {e}}^{-\frac {25}{4}}\,{\mathrm {e}}^{-x^4}\,{\mathrm {e}}^{5\,x^2}\,{\mathrm {e}}^x}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + 5*x^2 - x^4 - 25/4)*(10*x - 4*x^3 + 1))/7,x)

[Out]

(exp(-25/4)*exp(-x^4)*exp(5*x^2)*exp(x))/7

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sympy [A]  time = 0.11, size = 15, normalized size = 0.71 \begin {gather*} \frac {e^{- x^{4} + 5 x^{2} + x - \frac {25}{4}}}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/7*(-4*x**3+10*x+1)*exp(-x**4+5*x**2+x-25/4),x)

[Out]

exp(-x**4 + 5*x**2 + x - 25/4)/7

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