3.112 \(\int \sqrt{-2+5 x+3 x^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{1}{12} (6 x+5) \sqrt{3 x^2+5 x-2}-\frac{49 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{24 \sqrt{3}} \]

[Out]

((5 + 6*x)*Sqrt[-2 + 5*x + 3*x^2])/12 - (49*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[-2 + 5*x + 3*x^2])])/(24*Sqrt[3]
)

________________________________________________________________________________________

Rubi [A]  time = 0.0132514, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {612, 621, 206} \[ \frac{1}{12} (6 x+5) \sqrt{3 x^2+5 x-2}-\frac{49 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x-2}}\right )}{24 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

((5 + 6*x)*Sqrt[-2 + 5*x + 3*x^2])/12 - (49*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[-2 + 5*x + 3*x^2])])/(24*Sqrt[3]
)

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \sqrt{-2+5 x+3 x^2} \, dx &=\frac{1}{12} (5+6 x) \sqrt{-2+5 x+3 x^2}-\frac{49}{24} \int \frac{1}{\sqrt{-2+5 x+3 x^2}} \, dx\\ &=\frac{1}{12} (5+6 x) \sqrt{-2+5 x+3 x^2}-\frac{49}{12} \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{-2+5 x+3 x^2}}\right )\\ &=\frac{1}{12} (5+6 x) \sqrt{-2+5 x+3 x^2}-\frac{49 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{-2+5 x+3 x^2}}\right )}{24 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0269587, size = 55, normalized size = 0.89 \[ \frac{1}{72} \left (6 (6 x+5) \sqrt{3 x^2+5 x-2}-49 \sqrt{3} \log \left (2 \sqrt{9 x^2+15 x-6}+6 x+5\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-2 + 5*x + 3*x^2],x]

[Out]

(6*(5 + 6*x)*Sqrt[-2 + 5*x + 3*x^2] - 49*Sqrt[3]*Log[5 + 6*x + 2*Sqrt[-6 + 15*x + 9*x^2]])/72

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 50, normalized size = 0.8 \begin{align*}{\frac{6\,x+5}{12}\sqrt{3\,{x}^{2}+5\,x-2}}-{\frac{49\,\sqrt{3}}{72}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x-2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(6*x+5)*(3*x^2+5*x-2)^(1/2)-49/72*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x-2)^(1/2))*3^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.66813, size = 78, normalized size = 1.26 \begin{align*} \frac{1}{2} \, \sqrt{3 \, x^{2} + 5 \, x - 2} x - \frac{49}{72} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2} + 6 \, x + 5\right ) + \frac{5}{12} \, \sqrt{3 \, x^{2} + 5 \, x - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(3*x^2 + 5*x - 2)*x - 49/72*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x - 2) + 6*x + 5) + 5/12*sqrt(3*x^2 +
5*x - 2)

________________________________________________________________________________________

Fricas [A]  time = 2.04633, size = 167, normalized size = 2.69 \begin{align*} \frac{1}{12} \, \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )} + \frac{49}{144} \, \sqrt{3} \log \left (-4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 49/144*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 72*x^2
+ 120*x + 1)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{3 x^{2} + 5 x - 2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(3*x**2 + 5*x - 2), x)

________________________________________________________________________________________

Giac [A]  time = 1.20793, size = 73, normalized size = 1.18 \begin{align*} \frac{1}{12} \, \sqrt{3 \, x^{2} + 5 \, x - 2}{\left (6 \, x + 5\right )} + \frac{49}{72} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x - 2}\right )} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*sqrt(3*x^2 + 5*x - 2)*(6*x + 5) + 49/72*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x - 2)) -
5))