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

Optimal. Leaf size=41 \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0295683, antiderivative size = 41, 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 \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - b*x]/Sqrt[x],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.00571, size = 37, normalized size = 0.9 \[ \sqrt{x} \sqrt{- b x + 2} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0213109, size = 41, normalized size = 1. \[ \sqrt{x} \sqrt{2-b x}+\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 63, normalized size = 1.5 \[ \sqrt{x}\sqrt{-bx+2}+{1\sqrt{ \left ( -bx+2 \right ) x}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(1/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218371, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + \log \left (-\sqrt{-b x + 2} b \sqrt{x} -{\left (b x - 1\right )} \sqrt{-b}\right )}{\sqrt{-b}}, \frac{\sqrt{-b x + 2} \sqrt{b} \sqrt{x} - 2 \, \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right )}{\sqrt{b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.05513, size = 121, normalized size = 2.95 \[ \begin{cases} \frac{i b x^{\frac{3}{2}}}{\sqrt{b x - 2}} - \frac{2 i \sqrt{x}}{\sqrt{b x - 2}} - \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{b x^{\frac{3}{2}}}{\sqrt{- b x + 2}} + \frac{2 \sqrt{x}}{\sqrt{- b x + 2}} + \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{\sqrt{b}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((I*b*x**(3/2)/sqrt(b*x - 2) - 2*I*sqrt(x)/sqrt(b*x - 2) - 2*I*acosh(sq
rt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), Abs(b*x)/2 > 1), (-b*x**(3/2)/sqrt(-b*x + 2) +
2*sqrt(x)/sqrt(-b*x + 2) + 2*asin(sqrt(2)*sqrt(b)*sqrt(x)/2)/sqrt(b), True))

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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(-b*x + 2)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError