3.517 \(\int \frac{\sqrt{2-b x}}{x^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0330862, antiderivative size = 42, 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 \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2 - b*x]/x^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.43805, size = 41, normalized size = 0.98 \[ - 2 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} - \frac{2 \sqrt{- b x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x+2)**(1/2)/x**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0251087, size = 42, normalized size = 1. \[ -\frac{2 \sqrt{2-b x}}{\sqrt{x}}-2 \sqrt{b} \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2 - b*x]/x^(3/2),x]

[Out]

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

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Maple [B]  time = 0.042, size = 90, normalized size = 2.1 \[ 2\,{\frac{ \left ( bx-2 \right ) \sqrt{ \left ( -bx+2 \right ) x}}{\sqrt{-x \left ( bx-2 \right ) }\sqrt{x}\sqrt{-bx+2}}}-{1\sqrt{b}\arctan \left ({1\sqrt{b} \left ( x-{b}^{-1} \right ){\frac{1}{\sqrt{-b{x}^{2}+2\,x}}}} \right ) \sqrt{ \left ( -bx+2 \right ) x}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x+2)^(1/2)/x^(3/2),x)

[Out]

2*(b*x-2)/(-x*(b*x-2))^(1/2)*((-b*x+2)*x)^(1/2)/x^(1/2)/(-b*x+2)^(1/2)-b^(1/2)*a
rctan(b^(1/2)*(x-1/b)/(-b*x^2+2*x)^(1/2))*((-b*x+2)*x)^(1/2)/x^(1/2)/(-b*x+2)^(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)/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220783, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-b} x \log \left (-b x + \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) - 2 \, \sqrt{-b x + 2} \sqrt{x}}{x}, \frac{2 \,{\left (\sqrt{b} x \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) - \sqrt{-b x + 2} \sqrt{x}\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x + 2)/x^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.03104, size = 138, normalized size = 3.29 \[ \begin{cases} - 2 \sqrt{b} \sqrt{-1 + \frac{2}{b x}} - i \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 i \sqrt{b} \log{\left (\frac{1}{\sqrt{b} \sqrt{x}} \right )} - 2 \sqrt{b} \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\- 2 i \sqrt{b} \sqrt{1 - \frac{2}{b x}} - i \sqrt{b} \log{\left (\frac{1}{b x} \right )} + 2 i \sqrt{b} \log{\left (\sqrt{1 - \frac{2}{b x}} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x+2)**(1/2)/x**(3/2),x)

[Out]

Piecewise((-2*sqrt(b)*sqrt(-1 + 2/(b*x)) - I*sqrt(b)*log(1/(b*x)) + 2*I*sqrt(b)*
log(1/(sqrt(b)*sqrt(x))) - 2*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2), 2*Abs(1/(b
*x)) > 1), (-2*I*sqrt(b)*sqrt(1 - 2/(b*x)) - I*sqrt(b)*log(1/(b*x)) + 2*I*sqrt(b
)*log(sqrt(1 - 2/(b*x)) + 1), 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)/x^(3/2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError