3.1539 \(\int \frac{1}{\sqrt{-b x} \sqrt{2+b x}} \, dx\)

Optimal. Leaf size=10 \[ \frac{\sin ^{-1}(b x+1)}{b} \]

[Out]

ArcSin[1 + b*x]/b

________________________________________________________________________________________

Rubi [A]  time = 0.009602, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {53, 619, 216} \[ \frac{\sin ^{-1}(b x+1)}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[-(b*x)]*Sqrt[2 + b*x]),x]

[Out]

ArcSin[1 + b*x]/b

Rule 53

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

Rule 619

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

Rule 216

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-b x} \sqrt{2+b x}} \, dx &=\int \frac{1}{\sqrt{-2 b x-b^2 x^2}} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{4 b^2}}} \, dx,x,-2 b-2 b^2 x\right )}{2 b^2}\\ &=\frac{\sin ^{-1}(1+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 0.0121286, size = 51, normalized size = 5.1 \[ \frac{2 \sqrt{x} \sqrt{b x+2} \sinh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b} \sqrt{-b x (b x+2)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[-(b*x)]*Sqrt[2 + b*x]),x]

[Out]

(2*Sqrt[x]*Sqrt[2 + b*x]*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/(Sqrt[b]*Sqrt[-(b*x*(2 + b*x))])

________________________________________________________________________________________

Maple [B]  time = 0.005, size = 58, normalized size = 5.8 \begin{align*}{\sqrt{-bx \left ( bx+2 \right ) }\arctan \left ({(x+{b}^{-1})\sqrt{{b}^{2}}{\frac{1}{\sqrt{-{b}^{2}{x}^{2}-2\,bx}}}} \right ){\frac{1}{\sqrt{-bx}}}{\frac{1}{\sqrt{bx+2}}}{\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*x*(b*x+2))^(1/2)/(-b*x)^(1/2)/(b*x+2)^(1/2)/(b^2)^(1/2)*arctan((b^2)^(1/2)*(x+1/b)/(-b^2*x^2-2*b*x)^(1/2))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.00147, size = 61, normalized size = 6.1 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b x + 2} \sqrt{-b x}}{b x}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctan(sqrt(b*x + 2)*sqrt(-b*x)/(b*x))/b

________________________________________________________________________________________

Sympy [C]  time = 1.41673, size = 24, normalized size = 2.4 \begin{align*} - \frac{2 i \operatorname{asinh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-b*x)**(1/2)/(b*x+2)**(1/2),x)

[Out]

-2*I*asinh(sqrt(2)*sqrt(b)*sqrt(x)/2)/b

________________________________________________________________________________________

Giac [A]  time = 1.06147, size = 24, normalized size = 2.4 \begin{align*} \frac{2 \, \arcsin \left (\frac{1}{2} \, \sqrt{2} \sqrt{b x + 2}\right )}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arcsin(1/2*sqrt(2)*sqrt(b*x + 2))/b