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

Optimal. Leaf size=16 \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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Rubi [A]  time = 0.0114071, antiderivative size = 16, 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 \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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Rubi in Sympy [A]  time = 2.60775, size = 12, normalized size = 0.75 \[ \frac{\sqrt{x}}{\sqrt{- b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/sqrt(-b*x + 2)

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Mathematica [A]  time = 0.0143253, size = 16, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{2-b x}} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x]/Sqrt[2 - b*x]

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Maple [A]  time = 0.005, size = 13, normalized size = 0.8 \[{1\sqrt{x}{\frac{1}{\sqrt{-bx+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x+2)^(3/2)/x^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.35518, size = 16, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{-b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/sqrt(-b*x + 2)

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Fricas [A]  time = 0.210902, size = 16, normalized size = 1. \[ \frac{\sqrt{x}}{\sqrt{-b x + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)/sqrt(-b*x + 2)

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Sympy [A]  time = 4.59417, size = 41, normalized size = 2.56 \[ \begin{cases} \frac{1}{\sqrt{b} \sqrt{-1 + \frac{2}{b x}}} & \text{for}\: 2 \left |{\frac{1}{b x}}\right | > 1 \\- \frac{i}{\sqrt{b} \sqrt{1 - \frac{2}{b x}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206826, size = 68, normalized size = 4.25 \[ -\frac{4 \, \sqrt{-b} b}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*sqrt(-b)*b/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*abs
(b))