3.89.14 \(\int e^{36 x^2-120 x^3+100 x^4} (72 x-360 x^2+400 x^3) \, dx\)

Optimal. Leaf size=16 \[ e^{(-x+5 x (-1+2 x))^2} \]

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Rubi [A]  time = 0.12, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1594, 6706} \begin {gather*} e^{100 x^4-120 x^3+36 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(36*x^2 - 120*x^3 + 100*x^4)*(72*x - 360*x^2 + 400*x^3),x]

[Out]

E^(36*x^2 - 120*x^3 + 100*x^4)

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 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} &=\int e^{36 x^2-120 x^3+100 x^4} x \left (72-360 x+400 x^2\right ) \, dx\\ &=e^{36 x^2-120 x^3+100 x^4}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} e^{4 (3-5 x)^2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(36*x^2 - 120*x^3 + 100*x^4)*(72*x - 360*x^2 + 400*x^3),x]

[Out]

E^(4*(3 - 5*x)^2*x^2)

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fricas [A]  time = 0.54, size = 17, normalized size = 1.06 \begin {gather*} e^{\left (100 \, x^{4} - 120 \, x^{3} + 36 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((400*x^3-360*x^2+72*x)*exp(100*x^4-120*x^3+36*x^2),x, algorithm="fricas")

[Out]

e^(100*x^4 - 120*x^3 + 36*x^2)

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giac [A]  time = 0.17, size = 17, normalized size = 1.06 \begin {gather*} e^{\left (100 \, x^{4} - 120 \, x^{3} + 36 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((400*x^3-360*x^2+72*x)*exp(100*x^4-120*x^3+36*x^2),x, algorithm="giac")

[Out]

e^(100*x^4 - 120*x^3 + 36*x^2)

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maple [A]  time = 0.05, size = 14, normalized size = 0.88

method	result	size
risch	\({\mathrm e}^{4 x^{2} \left (5 x -3\right )^{2}}\)	\(14\)
gosper	\({\mathrm e}^{100 x^{4}-120 x^{3}+36 x^{2}}\)	\(18\)
derivativedivides	\({\mathrm e}^{100 x^{4}-120 x^{3}+36 x^{2}}\)	\(18\)
norman	\({\mathrm e}^{100 x^{4}-120 x^{3}+36 x^{2}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((400*x^3-360*x^2+72*x)*exp(100*x^4-120*x^3+36*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(4*x^2*(5*x-3)^2)

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maxima [A]  time = 0.36, size = 17, normalized size = 1.06 \begin {gather*} e^{\left (100 \, x^{4} - 120 \, x^{3} + 36 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((400*x^3-360*x^2+72*x)*exp(100*x^4-120*x^3+36*x^2),x, algorithm="maxima")

[Out]

e^(100*x^4 - 120*x^3 + 36*x^2)

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mupad [B]  time = 5.55, size = 19, normalized size = 1.19 \begin {gather*} {\mathrm {e}}^{36\,x^2}\,{\mathrm {e}}^{100\,x^4}\,{\mathrm {e}}^{-120\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(36*x^2 - 120*x^3 + 100*x^4)*(72*x - 360*x^2 + 400*x^3),x)

[Out]

exp(36*x^2)*exp(100*x^4)*exp(-120*x^3)

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sympy [A]  time = 0.10, size = 15, normalized size = 0.94 \begin {gather*} e^{100 x^{4} - 120 x^{3} + 36 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((400*x**3-360*x**2+72*x)*exp(100*x**4-120*x**3+36*x**2),x)

[Out]

exp(100*x**4 - 120*x**3 + 36*x**2)

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