3.52.59 \(\int e^{100 x-240 x^2-20 e^x x^2} (1600-7680 x+e^x (-640 x-320 x^2)) \, dx\)

Optimal. Leaf size=17 \[ 16 e^{20 x \left (5-\left (12+e^x\right ) x\right )} \]

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Rubi [A]  time = 0.13, antiderivative size = 21, normalized size of antiderivative = 1.24, number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6706} \begin {gather*} 16 e^{-20 e^x x^2-240 x^2+100 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(100*x - 240*x^2 - 20*E^x*x^2)*(1600 - 7680*x + E^x*(-640*x - 320*x^2)),x]

[Out]

16*E^(100*x - 240*x^2 - 20*E^x*x^2)

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} &=16 e^{100 x-240 x^2-20 e^x x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 16, normalized size = 0.94 \begin {gather*} 16 e^{-20 x \left (-5+\left (12+e^x\right ) x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(100*x - 240*x^2 - 20*E^x*x^2)*(1600 - 7680*x + E^x*(-640*x - 320*x^2)),x]

[Out]

16/E^(20*x*(-5 + (12 + E^x)*x))

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fricas [A]  time = 0.66, size = 19, normalized size = 1.12 \begin {gather*} 16 \, e^{\left (-20 \, x^{2} e^{x} - 240 \, x^{2} + 100 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-320*x^2-640*x)*exp(x)-7680*x+1600)*exp(-10*exp(x)*x^2-120*x^2+50*x)^2,x, algorithm="fricas")

[Out]

16*e^(-20*x^2*e^x - 240*x^2 + 100*x)

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giac [A]  time = 0.18, size = 19, normalized size = 1.12 \begin {gather*} 16 \, e^{\left (-20 \, x^{2} e^{x} - 240 \, x^{2} + 100 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-320*x^2-640*x)*exp(x)-7680*x+1600)*exp(-10*exp(x)*x^2-120*x^2+50*x)^2,x, algorithm="giac")

[Out]

16*e^(-20*x^2*e^x - 240*x^2 + 100*x)

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maple [A]  time = 0.06, size = 16, normalized size = 0.94

method	result	size
risch	\(16 \,{\mathrm e}^{-20 x \left ({\mathrm e}^{x} x +12 x -5\right )}\)	\(16\)
norman	\(16 \,{\mathrm e}^{-20 \,{\mathrm e}^{x} x^{2}-240 x^{2}+100 x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-320*x^2-640*x)*exp(x)-7680*x+1600)*exp(-10*exp(x)*x^2-120*x^2+50*x)^2,x,method=_RETURNVERBOSE)

[Out]

16*exp(-20*x*(exp(x)*x+12*x-5))

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maxima [A]  time = 64.81, size = 19, normalized size = 1.12 \begin {gather*} 16 \, e^{\left (-20 \, x^{2} e^{x} - 240 \, x^{2} + 100 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-320*x^2-640*x)*exp(x)-7680*x+1600)*exp(-10*exp(x)*x^2-120*x^2+50*x)^2,x, algorithm="maxima")

[Out]

16*e^(-20*x^2*e^x - 240*x^2 + 100*x)

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mupad [B]  time = 3.24, size = 20, normalized size = 1.18 \begin {gather*} 16\,{\mathrm {e}}^{100\,x}\,{\mathrm {e}}^{-20\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-240\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(100*x - 20*x^2*exp(x) - 240*x^2)*(7680*x + exp(x)*(640*x + 320*x^2) - 1600),x)

[Out]

16*exp(100*x)*exp(-20*x^2*exp(x))*exp(-240*x^2)

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sympy [A]  time = 0.19, size = 19, normalized size = 1.12 \begin {gather*} 16 e^{- 20 x^{2} e^{x} - 240 x^{2} + 100 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-320*x**2-640*x)*exp(x)-7680*x+1600)*exp(-10*exp(x)*x**2-120*x**2+50*x)**2,x)

[Out]

16*exp(-20*x**2*exp(x) - 240*x**2 + 100*x)

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