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

Optimal. Leaf size=20 \[ -\frac{2 \sqrt{a-b x}}{a \sqrt{x}} \]

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

_______________________________________________________________________________________

Rubi [A]  time = 0.0131849, antiderivative size = 20, 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 \[ -\frac{2 \sqrt{a-b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.65643, size = 17, normalized size = 0.85 \[ - \frac{2 \sqrt{a - b x}}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a - b*x)/(a*sqrt(x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.014332, size = 20, normalized size = 1. \[ -\frac{2 \sqrt{a-b x}}{a \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a - b*x])/(a*Sqrt[x])

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 17, normalized size = 0.9 \[ -2\,{\frac{\sqrt{-bx+a}}{a\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^(3/2)/(-b*x+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38745, size = 22, normalized size = 1.1 \[ -\frac{2 \, \sqrt{-b x + a}}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

_______________________________________________________________________________________

Fricas [A]  time = 0.211522, size = 22, normalized size = 1.1 \[ -\frac{2 \, \sqrt{-b x + a}}{a \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(-b*x + a)/(a*sqrt(x))

_______________________________________________________________________________________

Sympy [A]  time = 4.64539, size = 46, normalized size = 2.3 \[ \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{a} & \text{for}\: \left |{\frac{a}{b x}}\right | > 1 \\- \frac{2 i \sqrt{b} \sqrt{- \frac{a}{b x} + 1}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230662, size = 47, normalized size = 2.35 \[ -\frac{2 \, \sqrt{-b x + a} b^{2}}{\sqrt{{\left (b x - a\right )} b + a b} a{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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