3.806 \(\int \frac{\sqrt{a+b x}}{x^2 \sqrt{-a-b x}} \, dx\)

Optimal. Leaf size=26 \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

[Out]

-(Sqrt[a + b*x]/(x*Sqrt[-a - b*x]))

_______________________________________________________________________________________

Rubi [A]  time = 0.0112768, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]/(x*Sqrt[-a - b*x]))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.52962, size = 20, normalized size = 0.77 \[ - \frac{\sqrt{a + b x}}{x \sqrt{- a - b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a + b*x)/(x*sqrt(-a - b*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.00767703, size = 26, normalized size = 1. \[ -\frac{\sqrt{a+b x}}{x \sqrt{-a-b x}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + b*x]/(x*Sqrt[-a - b*x]))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 23, normalized size = 0.9 \[ -{\frac{1}{x}\sqrt{bx+a}{\frac{1}{\sqrt{-bx-a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^2/(-b*x-a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238346, size = 1, normalized size = 0.04 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

0

_______________________________________________________________________________________

Sympy [A]  time = 5.07113, size = 20, normalized size = 0.77 \[ \frac{i b^{2} \left (\frac{a}{b} + x\right )}{- a^{2} + a b \left (\frac{a}{b} + x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*b**2*(a/b + x)/(-a**2 + a*b*(a/b + x))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219432, size = 24, normalized size = 0.92 \[ \frac{i \,{\left (\frac{b^{2}}{a} + \frac{b}{x}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*(b^2/a + b/x)/b