3.56.75 \(\int e^{2 e x+4 x^2} (1+2 e x+8 x^2) \, dx\)

Optimal. Leaf size=12 \[ e^{2 x (e+2 x)} x \]

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Rubi [B]  time = 0.03, antiderivative size = 29, normalized size of antiderivative = 2.42, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2288} \begin {gather*} \frac {e^{4 x^2+2 e x} \left (4 x^2+e x\right )}{4 x+e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(2*E*x + 4*x^2)*(1 + 2*E*x + 8*x^2),x]

[Out]

(E^(2*E*x + 4*x^2)*(E*x + 4*x^2))/(E + 4*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^{2 e x+4 x^2} \left (e x+4 x^2\right )}{e+4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} e^{2 x (e+2 x)} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*E*x + 4*x^2)*(1 + 2*E*x + 8*x^2),x]

[Out]

E^(2*x*(E + 2*x))*x

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fricas [A]  time = 0.81, size = 14, normalized size = 1.17 \begin {gather*} x e^{\left (4 \, x^{2} + 2 \, x e\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)+8*x^2+1)*exp(2*x*exp(1)+4*x^2),x, algorithm="fricas")

[Out]

x*e^(4*x^2 + 2*x*e)

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giac [B]  time = 1.00, size = 38, normalized size = 3.17 \begin {gather*} \frac {1}{4} \, {\left (4 \, x - e\right )} e^{\left (4 \, x^{2} + 2 \, x e\right )} + \frac {1}{4} \, e^{\left (4 \, x^{2} + 2 \, x e + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)+8*x^2+1)*exp(2*x*exp(1)+4*x^2),x, algorithm="giac")

[Out]

1/4*(4*x - e)*e^(4*x^2 + 2*x*e) + 1/4*e^(4*x^2 + 2*x*e + 1)

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maple [A]  time = 0.06, size = 13, normalized size = 1.08

method	result	size
risch	\({\mathrm e}^{2 \left ({\mathrm e}+2 x \right ) x} x\)	\(13\)
gosper	\(x \,{\mathrm e}^{2 x \,{\mathrm e}+4 x^{2}}\)	\(15\)
norman	\(x \,{\mathrm e}^{2 x \,{\mathrm e}+4 x^{2}}\)	\(15\)
default	\(x \,{\mathrm e}^{2 x \,{\mathrm e}+4 x^{2}}-2 \,{\mathrm e} \left (\frac {{\mathrm e}^{2 x \,{\mathrm e}+4 x^{2}}}{8}+\frac {i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {{\mathrm e}^{2}}{4}} \erf \left (2 i x +\frac {i {\mathrm e}}{2}\right )}{16}\right )+\frac {{\mathrm e}^{2 x \,{\mathrm e}+4 x^{2}+1}}{4}+\frac {i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{1-\frac {{\mathrm e}^{2}}{4}} \erf \left (2 i x +\frac {i {\mathrm e}}{2}\right )}{8}\)	\(104\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(1)+8*x^2+1)*exp(2*x*exp(1)+4*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(2*(exp(1)+2*x)*x)*x

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maxima [C]  time = 0.52, size = 182, normalized size = 15.17 \begin {gather*} -\frac {1}{4} i \, \sqrt {\pi } \operatorname {erf}\left (2 i \, x + \frac {1}{2} i \, e\right ) e^{\left (-\frac {1}{4} \, e^{2}\right )} + \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (4 \, x + e\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (4 \, x + e\right )}^{2}}\right ) - 1\right )} e^{2}}{\sqrt {-{\left (4 \, x + e\right )}^{2}}} - \frac {4 \, {\left (4 \, x + e\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {1}{4} \, {\left (4 \, x + e\right )}^{2}\right )}{\left (-{\left (4 \, x + e\right )}^{2}\right )^{\frac {3}{2}}} - 4 \, e^{\left (\frac {1}{4} \, {\left (4 \, x + e\right )}^{2} + 1\right )}\right )} e^{\left (-\frac {1}{4} \, e^{2}\right )} - \frac {1}{8} \, {\left (\frac {\sqrt {\pi } {\left (4 \, x + e\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-{\left (4 \, x + e\right )}^{2}}\right ) - 1\right )} e}{\sqrt {-{\left (4 \, x + e\right )}^{2}}} - 2 \, e^{\left (\frac {1}{4} \, {\left (4 \, x + e\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{4} \, e^{2} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)+8*x^2+1)*exp(2*x*exp(1)+4*x^2),x, algorithm="maxima")

[Out]

-1/4*I*sqrt(pi)*erf(2*I*x + 1/2*I*e)*e^(-1/4*e^2) + 1/8*(sqrt(pi)*(4*x + e)*(erf(1/2*sqrt(-(4*x + e)^2)) - 1)*
e^2/sqrt(-(4*x + e)^2) - 4*(4*x + e)^3*gamma(3/2, -1/4*(4*x + e)^2)/(-(4*x + e)^2)^(3/2) - 4*e^(1/4*(4*x + e)^
2 + 1))*e^(-1/4*e^2) - 1/8*(sqrt(pi)*(4*x + e)*(erf(1/2*sqrt(-(4*x + e)^2)) - 1)*e/sqrt(-(4*x + e)^2) - 2*e^(1
/4*(4*x + e)^2))*e^(-1/4*e^2 + 1)

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mupad [B]  time = 0.09, size = 14, normalized size = 1.17 \begin {gather*} x\,{\mathrm {e}}^{4\,x^2+2\,\mathrm {e}\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x*exp(1) + 4*x^2)*(2*x*exp(1) + 8*x^2 + 1),x)

[Out]

x*exp(2*x*exp(1) + 4*x^2)

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sympy [A]  time = 0.10, size = 14, normalized size = 1.17 \begin {gather*} x e^{4 x^{2} + 2 e x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x*exp(1)+8*x**2+1)*exp(2*x*exp(1)+4*x**2),x)

[Out]

x*exp(4*x**2 + 2*E*x)

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