3.649 \(\int \frac{1}{\sqrt{x} \sqrt{1-b x}} \, dx\)

Optimal. Leaf size=19 \[ \frac{2 \sin ^{-1}\left (\sqrt{b} \sqrt{x}\right )}{\sqrt{b}} \]

[Out]

(2*ArcSin[Sqrt[b]*Sqrt[x]])/Sqrt[b]

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Rubi [A]  time = 0.004793, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {54, 216} \[ \frac{2 \sin ^{-1}\left (\sqrt{b} \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*Sqrt[1 - b*x]),x]

[Out]

(2*ArcSin[Sqrt[b]*Sqrt[x]])/Sqrt[b]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 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{x} \sqrt{1-b x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-b x^2}} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \sin ^{-1}\left (\sqrt{b} \sqrt{x}\right )}{\sqrt{b}}\\ \end{align*}

Mathematica [A]  time = 0.0066255, size = 19, normalized size = 1. \[ \frac{2 \sin ^{-1}\left (\sqrt{b} \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcSin[Sqrt[b]*Sqrt[x]])/Sqrt[b]

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Maple [B]  time = 0.007, size = 48, normalized size = 2.5 \begin{align*}{\sqrt{x \left ( -bx+1 \right ) }\arctan \left ({\sqrt{b} \left ( x-{\frac{1}{2\,b}} \right ){\frac{1}{\sqrt{-b{x}^{2}+x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+1}}}{\frac{1}{\sqrt{b}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53397, size = 165, normalized size = 8.68 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (-2 \, b x + 2 \, \sqrt{-b x + 1} \sqrt{-b} \sqrt{x} + 1\right )}{b}, -\frac{2 \, \arctan \left (\frac{\sqrt{-b x + 1}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-sqrt(-b)*log(-2*b*x + 2*sqrt(-b*x + 1)*sqrt(-b)*sqrt(x) + 1)/b, -2*arctan(sqrt(-b*x + 1)/(sqrt(b)*sqrt(x)))/
sqrt(b)]

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Sympy [A]  time = 1.15522, size = 42, normalized size = 2.21 \begin{align*} \begin{cases} - \frac{2 i \operatorname{acosh}{\left (\sqrt{b} \sqrt{x} \right )}}{\sqrt{b}} & \text{for}\: \left |{b x}\right | > 1 \\\frac{2 \operatorname{asin}{\left (\sqrt{b} \sqrt{x} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*I*acosh(sqrt(b)*sqrt(x))/sqrt(b), Abs(b*x) > 1), (2*asin(sqrt(b)*sqrt(x))/sqrt(b), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError