3.633 \(\int \frac{1}{x^{3/2} \sqrt{2-b x}} \, dx\)

Optimal. Leaf size=17 \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Rubi [A]  time = 0.0116922, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Rubi in Sympy [A]  time = 2.55384, size = 14, normalized size = 0.82 \[ - \frac{\sqrt{- b x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Mathematica [A]  time = 0.011696, size = 17, normalized size = 1. \[ -\frac{\sqrt{2-b x}}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^(3/2)*Sqrt[2 - b*x]),x]

[Out]

-(Sqrt[2 - b*x]/Sqrt[x])

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Maple [A]  time = 0.005, size = 14, normalized size = 0.8 \[ -{1\sqrt{-bx+2}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(-b*x+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.32228, size = 18, normalized size = 1.06 \[ -\frac{\sqrt{-b x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x + 2)*x^(3/2)),x, algorithm="maxima")

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Fricas [A]  time = 0.20879, size = 18, normalized size = 1.06 \[ -\frac{\sqrt{-b x + 2}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x + 2)*x^(3/2)),x, algorithm="fricas")

[Out]

-sqrt(-b*x + 2)/sqrt(x)

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Sympy [A]  time = 4.63751, size = 41, normalized size = 2.41 \[ \begin{cases} - \sqrt{b} \sqrt{-1 + \frac{2}{b x}} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\- i \sqrt{b} \sqrt{1 - \frac{2}{b x}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224528, size = 41, normalized size = 2.41 \[ -\frac{\sqrt{-b x + 2} b^{2}}{\sqrt{{\left (b x - 2\right )} b + 2 \, b}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-b*x + 2)*x^(3/2)),x, algorithm="giac")

[Out]

-sqrt(-b*x + 2)*b^2/(sqrt((b*x - 2)*b + 2*b)*abs(b))