3.6.8 \(\int (2 e^{x^2} x+e^{4 e^x} (6 x^5+4 e^x x^6)) \, dx\)

Optimal. Leaf size=19 \[ e^{x^2}+e^{4 e^x} x^6+\log (2) \]

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 0.89, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2209, 2288} \begin {gather*} e^{4 e^x} x^6+e^{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[2*E^x^2*x + E^(4*E^x)*(6*x^5 + 4*E^x*x^6),x]

[Out]

E^x^2 + E^(4*E^x)*x^6

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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} &=2 \int e^{x^2} x \, dx+\int e^{4 e^x} \left (6 x^5+4 e^x x^6\right ) \, dx\\ &=e^{x^2}+e^{4 e^x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 17, normalized size = 0.89 \begin {gather*} e^{x^2}+e^{4 e^x} x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[2*E^x^2*x + E^(4*E^x)*(6*x^5 + 4*E^x*x^6),x]

[Out]

E^x^2 + E^(4*E^x)*x^6

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fricas [A]  time = 0.78, size = 14, normalized size = 0.74 \begin {gather*} x^{6} e^{\left (4 \, e^{x}\right )} + e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6*exp(x)+6*x^5)*exp(exp(x))^4+2*exp(x^2)*x,x, algorithm="fricas")

[Out]

x^6*e^(4*e^x) + e^(x^2)

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giac [A]  time = 0.35, size = 14, normalized size = 0.74 \begin {gather*} x^{6} e^{\left (4 \, e^{x}\right )} + e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6*exp(x)+6*x^5)*exp(exp(x))^4+2*exp(x^2)*x,x, algorithm="giac")

[Out]

x^6*e^(4*e^x) + e^(x^2)

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maple [A]  time = 0.07, size = 15, normalized size = 0.79

method	result	size
risch	\({\mathrm e}^{4 \,{\mathrm e}^{x}} x^{6}+{\mathrm e}^{x^{2}}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^6*exp(x)+6*x^5)*exp(exp(x))^4+2*exp(x^2)*x,x,method=_RETURNVERBOSE)

[Out]

exp(4*exp(x))*x^6+exp(x^2)

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maxima [A]  time = 0.43, size = 14, normalized size = 0.74 \begin {gather*} x^{6} e^{\left (4 \, e^{x}\right )} + e^{\left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^6*exp(x)+6*x^5)*exp(exp(x))^4+2*exp(x^2)*x,x, algorithm="maxima")

[Out]

x^6*e^(4*e^x) + e^(x^2)

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mupad [B]  time = 0.52, size = 14, normalized size = 0.74 \begin {gather*} {\mathrm {e}}^{x^2}+x^6\,{\mathrm {e}}^{4\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*exp(x^2) + exp(4*exp(x))*(4*x^6*exp(x) + 6*x^5),x)

[Out]

exp(x^2) + x^6*exp(4*exp(x))

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sympy [A]  time = 0.23, size = 14, normalized size = 0.74 \begin {gather*} x^{6} e^{4 e^{x}} + e^{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**6*exp(x)+6*x**5)*exp(exp(x))**4+2*exp(x**2)*x,x)

[Out]

x**6*exp(4*exp(x)) + exp(x**2)

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