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

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

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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} $$\begin{array}{c}{e}^{100x^4-120x^3+36x^2}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int e^{36x^2-120x^3+100x^4}x(72-360x+400x^2)dx\\ & =e^{36x^2-120x^3+100x^4}\end{array}\end{array}$$

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

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 $$\begin{array}{c}{e}^{(100x^4-120x^3+36x^2)}\end{array}$$

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 $$\begin{array}{c}{e}^{(100x^4-120x^3+36x^2)}\end{array}$$

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	${\mathrm{e}}^{4x^2{(5x-3)}^2}$	$14$
gosper	${\mathrm{e}}^{100x^4-120x^3+36x^2}$	$18$
derivativedivides	${\mathrm{e}}^{100x^4-120x^3+36x^2}$	$18$
norman	${\mathrm{e}}^{100x^4-120x^3+36x^2}$	$18$

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 $$\begin{array}{c}{e}^{(100x^4-120x^3+36x^2)}\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^{36x^2}{\mathrm{e}}^{100x^4}{\mathrm{e}}^{-120x^3}\end{array}$$

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 $$\begin{array}{c}{e}^{100x^4-120x^3+36x^2}\end{array}$$

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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