3.11.67 \(\int e^{e^{23 x^2+5 e^{e^4} x^2+e^x x^2}} (1+e^{23 x^2+5 e^{e^4} x^2+e^x x^2} (46 x^2+10 e^{e^4} x^2+e^x (2 x^2+x^3))) \, dx\)

Optimal. Leaf size=25 \[ e^{e^{x \left (e^x x+\left (23+5 e^{e^4}\right ) x\right )}} x \]

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Rubi [B]  time = 0.20, antiderivative size = 84, normalized size of antiderivative = 3.36, number of steps used = 1, number of rules used = 1, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {2288} \begin {gather*} \frac {e^{e^{e^x x^2+5 e^{e^4} x^2+23 x^2}} \left (10 e^{e^4} x^2+46 x^2+e^x \left (x^3+2 x^2\right )\right )}{e^x x^2+2 e^x x+10 e^{e^4} x+46 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^E^(23*x^2 + 5*E^E^4*x^2 + E^x*x^2)*(1 + E^(23*x^2 + 5*E^E^4*x^2 + E^x*x^2)*(46*x^2 + 10*E^E^4*x^2 + E^x*
(2*x^2 + x^3))),x]

[Out]

(E^E^(23*x^2 + 5*E^E^4*x^2 + E^x*x^2)*(46*x^2 + 10*E^E^4*x^2 + E^x*(2*x^2 + x^3)))/(46*x + 10*E^E^4*x + 2*E^x*
x + E^x*x^2)

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^{23 x^2+5 e^{e^4} x^2+e^x x^2}} \left (46 x^2+10 e^{e^4} x^2+e^x \left (2 x^2+x^3\right )\right )}{46 x+10 e^{e^4} x+2 e^x x+e^x x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 22, normalized size = 0.88 \begin {gather*} e^{e^{\left (23+5 e^{e^4}+e^x\right ) x^2}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^E^(23*x^2 + 5*E^E^4*x^2 + E^x*x^2)*(1 + E^(23*x^2 + 5*E^E^4*x^2 + E^x*x^2)*(46*x^2 + 10*E^E^4*x^2
+ E^x*(2*x^2 + x^3))),x]

[Out]

E^E^((23 + 5*E^E^4 + E^x)*x^2)*x

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fricas [A]  time = 0.58, size = 24, normalized size = 0.96 \begin {gather*} x e^{\left (e^{\left (x^{2} e^{x} + 5 \, x^{2} e^{\left (e^{4}\right )} + 23 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(exp(4))+(x^3+2*x^2)*exp(x)+46*x^2)*exp(5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)+1)*exp(exp(
5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)),x, algorithm="fricas")

[Out]

x*e^(e^(x^2*e^x + 5*x^2*e^(e^4) + 23*x^2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left ({\left (10 \, x^{2} e^{\left (e^{4}\right )} + 46 \, x^{2} + {\left (x^{3} + 2 \, x^{2}\right )} e^{x}\right )} e^{\left (x^{2} e^{x} + 5 \, x^{2} e^{\left (e^{4}\right )} + 23 \, x^{2}\right )} + 1\right )} e^{\left (e^{\left (x^{2} e^{x} + 5 \, x^{2} e^{\left (e^{4}\right )} + 23 \, x^{2}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(exp(4))+(x^3+2*x^2)*exp(x)+46*x^2)*exp(5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)+1)*exp(exp(
5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)),x, algorithm="giac")

[Out]

integrate(((10*x^2*e^(e^4) + 46*x^2 + (x^3 + 2*x^2)*e^x)*e^(x^2*e^x + 5*x^2*e^(e^4) + 23*x^2) + 1)*e^(e^(x^2*e
^x + 5*x^2*e^(e^4) + 23*x^2)), x)

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maple [A]  time = 0.08, size = 18, normalized size = 0.72

method	result	size
risch	\(x \,{\mathrm e}^{{\mathrm e}^{x^{2} \left ({\mathrm e}^{x}+5 \,{\mathrm e}^{{\mathrm e}^{4}}+23\right )}}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x^2*exp(exp(4))+(x^3+2*x^2)*exp(x)+46*x^2)*exp(5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)+1)*exp(exp(5*x^2*
exp(exp(4))+exp(x)*x^2+23*x^2)),x,method=_RETURNVERBOSE)

[Out]

x*exp(exp(x^2*(exp(x)+5*exp(exp(4))+23)))

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maxima [A]  time = 0.56, size = 24, normalized size = 0.96 \begin {gather*} x e^{\left (e^{\left (x^{2} e^{x} + 5 \, x^{2} e^{\left (e^{4}\right )} + 23 \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x^2*exp(exp(4))+(x^3+2*x^2)*exp(x)+46*x^2)*exp(5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)+1)*exp(exp(
5*x^2*exp(exp(4))+exp(x)*x^2+23*x^2)),x, algorithm="maxima")

[Out]

x*e^(e^(x^2*e^x + 5*x^2*e^(e^4) + 23*x^2))

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mupad [B]  time = 0.99, size = 26, normalized size = 1.04 \begin {gather*} x\,{\mathrm {e}}^{{\mathrm {e}}^{5\,x^2\,{\mathrm {e}}^{{\mathrm {e}}^4}}\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{23\,x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(5*x^2*exp(exp(4)) + x^2*exp(x) + 23*x^2))*(exp(5*x^2*exp(exp(4)) + x^2*exp(x) + 23*x^2)*(10*x^2*ex
p(exp(4)) + exp(x)*(2*x^2 + x^3) + 46*x^2) + 1),x)

[Out]

x*exp(exp(5*x^2*exp(exp(4)))*exp(x^2*exp(x))*exp(23*x^2))

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sympy [A]  time = 4.76, size = 26, normalized size = 1.04 \begin {gather*} x e^{e^{x^{2} e^{x} + 23 x^{2} + 5 x^{2} e^{e^{4}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x**2*exp(exp(4))+(x**3+2*x**2)*exp(x)+46*x**2)*exp(5*x**2*exp(exp(4))+exp(x)*x**2+23*x**2)+1)*e
xp(exp(5*x**2*exp(exp(4))+exp(x)*x**2+23*x**2)),x)

[Out]

x*exp(exp(x**2*exp(x) + 23*x**2 + 5*x**2*exp(exp(4))))

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