3.16.64 \(\int \frac {1}{4} e^{\frac {1}{4} (16+7 x-x^3+x^4)} (7-3 x^2+4 x^3) \, dx\)

Optimal. Leaf size=20 \[ e^{4+x+\frac {1}{4} x \left (3-x^2+x^3\right )} \]

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Rubi [A]  time = 0.05, antiderivative size = 19, normalized size of antiderivative = 0.95, 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*} e^{\frac {1}{4} \left (x^4-x^3+7 x+16\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((16 + 7*x - x^3 + x^4)/4)*(7 - 3*x^2 + 4*x^3))/4,x]

[Out]

E^((16 + 7*x - x^3 + x^4)/4)

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}{4} \int e^{\frac {1}{4} \left (16+7 x-x^3+x^4\right )} \left (7-3 x^2+4 x^3\right ) \, dx\\ &=e^{\frac {1}{4} \left (16+7 x-x^3+x^4\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 19, normalized size = 0.95 \begin {gather*} e^{\frac {1}{4} \left (16+7 x-x^3+x^4\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16 + 7*x - x^3 + x^4)/4)*(7 - 3*x^2 + 4*x^3))/4,x]

[Out]

E^((16 + 7*x - x^3 + x^4)/4)

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fricas [A]  time = 0.88, size = 16, normalized size = 0.80 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} - \frac {1}{4} \, x^{3} + \frac {7}{4} \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(1/4*x^4 - 1/4*x^3 + 7/4*x + 4)

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giac [A]  time = 0.51, size = 16, normalized size = 0.80 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} - \frac {1}{4} \, x^{3} + \frac {7}{4} \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(1/4*x^4 - 1/4*x^3 + 7/4*x + 4)

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maple [A]  time = 0.07, size = 17, normalized size = 0.85

method	result	size
gosper	\({\mathrm e}^{\frac {1}{4} x^{4}-\frac {1}{4} x^{3}+\frac {7}{4} x +4}\)	\(17\)
norman	\({\mathrm e}^{\frac {1}{4} x^{4}-\frac {1}{4} x^{3}+\frac {7}{4} x +4}\)	\(17\)
risch	\({\mathrm e}^{\frac {1}{4} x^{4}-\frac {1}{4} x^{3}+\frac {7}{4} x +4}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(1/4*x^4-1/4*x^3+7/4*x+4)

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maxima [A]  time = 0.38, size = 16, normalized size = 0.80 \begin {gather*} e^{\left (\frac {1}{4} \, x^{4} - \frac {1}{4} \, x^{3} + \frac {7}{4} \, x + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(1/4*x^4 - 1/4*x^3 + 7/4*x + 4)

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mupad [B]  time = 1.02, size = 19, normalized size = 0.95 \begin {gather*} {\mathrm {e}}^{\frac {7\,x}{4}}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-\frac {x^3}{4}}\,{\mathrm {e}}^{\frac {x^4}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((7*x)/4 - x^3/4 + x^4/4 + 4)*(4*x^3 - 3*x^2 + 7))/4,x)

[Out]

exp((7*x)/4)*exp(4)*exp(-x^3/4)*exp(x^4/4)

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sympy [A]  time = 0.11, size = 17, normalized size = 0.85 \begin {gather*} e^{\frac {x^{4}}{4} - \frac {x^{3}}{4} + \frac {7 x}{4} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(x**4/4 - x**3/4 + 7*x/4 + 4)

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