3.77.66 \(\int e^{-e^{1+x^2}+e (5+2 x)} (7+2 x+e (14 x+2 x^2)+e^{1+x^2} (-14 x^2-2 x^3)) \, dx\)

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

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Rubi [B]  time = 0.17, antiderivative size = 62, normalized size of antiderivative = 2.95, number of steps used = 1, number of rules used = 1, integrand size = 55, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2288} \begin {gather*} \frac {e^{e (2 x+5)-e^{x^2+1}} \left (e \left (x^2+7 x\right )-e^{x^2+1} \left (x^3+7 x^2\right )\right )}{e-e^{x^2+1} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-E^(1 + x^2) + E*(5 + 2*x))*(7 + 2*x + E*(14*x + 2*x^2) + E^(1 + x^2)*(-14*x^2 - 2*x^3)),x]

[Out]

(E^(-E^(1 + x^2) + E*(5 + 2*x))*(E*(7*x + x^2) - E^(1 + x^2)*(7*x^2 + x^3)))/(E - E^(1 + x^2)*x)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {e^{-e^{1+x^2}+e (5+2 x)} \left (e \left (7 x+x^2\right )-e^{1+x^2} \left (7 x^2+x^3\right )\right )}{e-e^{1+x^2} x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.50, size = 20, normalized size = 0.95 \begin {gather*} e^{-e \left (-5+e^{x^2}-2 x\right )} x (7+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-E^(1 + x^2) + E*(5 + 2*x))*(7 + 2*x + E*(14*x + 2*x^2) + E^(1 + x^2)*(-14*x^2 - 2*x^3)),x]

[Out]

(x*(7 + x))/E^(E*(-5 + E^x^2 - 2*x))

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fricas [A]  time = 0.73, size = 26, normalized size = 1.24 \begin {gather*} {\left (x^{2} + 7 \, x\right )} e^{\left ({\left (2 \, x + 5\right )} e - e^{\left (x^{2} + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="fricas")

[Out]

(x^2 + 7*x)*e^((2*x + 5)*e - e^(x^2 + 1))

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giac [B]  time = 0.15, size = 46, normalized size = 2.19 \begin {gather*} x^{2} e^{\left (2 \, x e + 5 \, e - e^{\left (x^{2} + 1\right )}\right )} + 7 \, x e^{\left (2 \, x e + 5 \, e - e^{\left (x^{2} + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="giac")

[Out]

x^2*e^(2*x*e + 5*e - e^(x^2 + 1)) + 7*x*e^(2*x*e + 5*e - e^(x^2 + 1))

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maple [A]  time = 0.10, size = 25, normalized size = 1.19

method	result	size
risch	\(\left (x +7\right ) x \,{\mathrm e}^{2 x \,{\mathrm e}+5 \,{\mathrm e}-{\mathrm e}^{x^{2}+1}}\)	\(25\)
norman	\(x^{2} {\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{x^{2}}+\left (5+2 x \right ) {\mathrm e}}+7 x \,{\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{x^{2}}+\left (5+2 x \right ) {\mathrm e}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,method=
_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.45, size = 34, normalized size = 1.62 \begin {gather*} {\left (x^{2} e^{\left (5 \, e\right )} + 7 \, x e^{\left (5 \, e\right )}\right )} e^{\left (2 \, x e - e^{\left (x^{2} + 1\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="maxima")

[Out]

(x^2*e^(5*e) + 7*x*e^(5*e))*e^(2*x*e - e^(x^2 + 1))

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mupad [B]  time = 0.18, size = 25, normalized size = 1.19 \begin {gather*} x\,{\mathrm {e}}^{5\,\mathrm {e}}\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}\,\mathrm {e}}\,{\mathrm {e}}^{2\,x\,\mathrm {e}}\,\left (x+7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(1)*(2*x + 5) - exp(x^2)*exp(1))*(2*x + exp(1)*(14*x + 2*x^2) - exp(x^2)*exp(1)*(14*x^2 + 2*x^3) +
7),x)

[Out]

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

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sympy [A]  time = 3.03, size = 24, normalized size = 1.14 \begin {gather*} \left (x^{2} + 7 x\right ) e^{e \left (2 x + 5\right ) - e e^{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3-14*x**2)*exp(1)*exp(x**2)+(2*x**2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x**2)+(5+2*x)*exp(1)
),x)

[Out]

(x**2 + 7*x)*exp(E*(2*x + 5) - E*exp(x**2))

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