3.631 \(\int \frac{\sqrt{x}}{\sqrt{2-b x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{\sqrt{x} \sqrt{2-b x}}{b} \]

[Out]

-((Sqrt[x]*Sqrt[2 - b*x])/b) + (2*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Rubi [A]  time = 0.0344775, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{\sqrt{x} \sqrt{2-b x}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[2 - b*x],x]

[Out]

-((Sqrt[x]*Sqrt[2 - b*x])/b) + (2*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Rubi in Sympy [A]  time = 5.91702, size = 39, normalized size = 0.87 \[ - \frac{\sqrt{x} \sqrt{- b x + 2}}{b} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(-b*x+2)**(1/2),x)

[Out]

-sqrt(x)*sqrt(-b*x + 2)/b + 2*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2)

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Mathematica [A]  time = 0.0317318, size = 45, normalized size = 1. \[ \frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}-\frac{\sqrt{x} \sqrt{2-b x}}{b} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/Sqrt[2 - b*x],x]

[Out]

-((Sqrt[x]*Sqrt[2 - b*x])/b) + (2*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]])/b^(3/2)

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Maple [A]  time = 0.007, size = 67, normalized size = 1.5 \[ -{\frac{1}{b}\sqrt{x}\sqrt{-bx+2}}+{1\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(-b*x+2)^(1/2),x)

[Out]

-x^(1/2)*(-b*x+2)^(1/2)/b+1/b^(3/2)*((-b*x+2)*x)^(1/2)/(-b*x+2)^(1/2)/x^(1/2)*ar
ctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt(-b*x + 2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221633, size = 1, normalized size = 0.02 \[ \left [-\frac{\sqrt{-b x + 2} \sqrt{-b} \sqrt{x} - \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{\sqrt{-b} b}, -\frac{\sqrt{-b x + 2} \sqrt{b} \sqrt{x} + 2 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{b^{\frac{3}{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt(-b*x + 2),x, algorithm="fricas")

[Out]

[-(sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) - log(-sqrt(-b*x + 2)*b*sqrt(x) - (b*x - 1)*s
qrt(-b)))/(sqrt(-b)*b), -(sqrt(-b*x + 2)*sqrt(b)*sqrt(x) + 2*arctan(sqrt(-b*x +
2)/(sqrt(b)*sqrt(x))))/b^(3/2)]

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Sympy [A]  time = 7.02129, size = 121, normalized size = 2.69 \[ \begin{cases} - \frac{i x^{\frac{3}{2}}}{\sqrt{b x - 2}} + \frac{2 i \sqrt{x}}{b \sqrt{b x - 2}} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{x^{\frac{3}{2}}}{\sqrt{- b x + 2}} - \frac{2 \sqrt{x}}{b \sqrt{- b x + 2}} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-I*x**(3/2)/sqrt(b*x - 2) + 2*I*sqrt(x)/(b*sqrt(b*x - 2)) - 2*I*acosh
(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2), Abs(b*x)/2 > 1), (x**(3/2)/sqrt(-b*x + 2)
- 2*sqrt(x)/(b*sqrt(-b*x + 2)) + 2*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/b**(3/2), Tru
e))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt(-b*x + 2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError