3.215 \(\int e^{\sqrt{5+3 x}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{3} e^{\sqrt{3 x+5}} \sqrt{3 x+5}-\frac{2}{3} e^{\sqrt{3 x+5}} \]

[Out]

(-2*E^Sqrt[5 + 3*x])/3 + (2*E^Sqrt[5 + 3*x]*Sqrt[5 + 3*x])/3

________________________________________________________________________________________

Rubi [A]  time = 0.0123375, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2207, 2176, 2194} \[ \frac{2}{3} e^{\sqrt{3 x+5}} \sqrt{3 x+5}-\frac{2}{3} e^{\sqrt{3 x+5}} \]

Antiderivative was successfully verified.

[In]

Int[E^Sqrt[5 + 3*x],x]

[Out]

(-2*E^Sqrt[5 + 3*x])/3 + (2*E^Sqrt[5 + 3*x]*Sqrt[5 + 3*x])/3

Rule 2207

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> With[{k = Denominator[n]}, Dist[k/d, Subst[In
t[x^(k - 1)*F^(a + b*x^(k*n)), x], x, (c + d*x)^(1/k)], x]] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n] &&
!IntegerQ[n]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int e^{\sqrt{5+3 x}} \, dx &=\frac{2}{3} \operatorname{Subst}\left (\int e^x x \, dx,x,\sqrt{5+3 x}\right )\\ &=\frac{2}{3} e^{\sqrt{5+3 x}} \sqrt{5+3 x}-\frac{2}{3} \operatorname{Subst}\left (\int e^x \, dx,x,\sqrt{5+3 x}\right )\\ &=-\frac{2}{3} e^{\sqrt{5+3 x}}+\frac{2}{3} e^{\sqrt{5+3 x}} \sqrt{5+3 x}\\ \end{align*}

Mathematica [A]  time = 0.0117546, size = 26, normalized size = 0.65 \[ \frac{2}{3} e^{\sqrt{3 x+5}} \left (\sqrt{3 x+5}-1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^Sqrt[5 + 3*x],x]

[Out]

(2*E^Sqrt[5 + 3*x]*(-1 + Sqrt[5 + 3*x]))/3

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 29, normalized size = 0.7 \begin{align*} -{\frac{2}{3}{{\rm e}^{\sqrt{5+3\,x}}}}+{\frac{2}{3}{{\rm e}^{\sqrt{5+3\,x}}}\sqrt{5+3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp((5+3*x)^(1/2)),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.05085, size = 26, normalized size = 0.65 \begin{align*} \frac{2}{3} \,{\left (\sqrt{3 \, x + 5} - 1\right )} e^{\left (\sqrt{3 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(3*x + 5) - 1)*e^(sqrt(3*x + 5))

________________________________________________________________________________________

Fricas [A]  time = 1.45192, size = 58, normalized size = 1.45 \begin{align*} \frac{2}{3} \,{\left (\sqrt{3 \, x + 5} - 1\right )} e^{\left (\sqrt{3 \, x + 5}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(3*x + 5) - 1)*e^(sqrt(3*x + 5))

________________________________________________________________________________________

Sympy [A]  time = 0.181676, size = 34, normalized size = 0.85 \begin{align*} \frac{2 \sqrt{3 x + 5} e^{\sqrt{3 x + 5}}}{3} - \frac{2 e^{\sqrt{3 x + 5}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp((5+3*x)**(1/2)),x)

[Out]

2*sqrt(3*x + 5)*exp(sqrt(3*x + 5))/3 - 2*exp(sqrt(3*x + 5))/3

________________________________________________________________________________________

Giac [A]  time = 1.26458, size = 28, normalized size = 0.7 \begin{align*} \frac{2}{3} \,{\left (\sqrt{3 \, x + 5} - 1\right )} e^{\left (\sqrt{3 \, x + 5}\right )} + \frac{2}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(sqrt(3*x + 5) - 1)*e^(sqrt(3*x + 5)) + 2/3