3.2815 \(\int \frac{1}{\sqrt{\frac{c}{a+b x}}} \, dx\)

Optimal. Leaf size=25 \[ \frac{2 (a+b x)}{3 b \sqrt{\frac{c}{a+b x}}} \]

[Out]

(2*(a + b*x))/(3*b*Sqrt[c/(a + b*x)])

_______________________________________________________________________________________

Rubi [A]  time = 0.0203903, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 (a+b x)}{3 b \sqrt{\frac{c}{a+b x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[c/(a + b*x)],x]

[Out]

(2*(a + b*x))/(3*b*Sqrt[c/(a + b*x)])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.6183, size = 22, normalized size = 0.88 \[ \frac{2 \sqrt{\frac{c}{a + b x}} \left (a + b x\right )^{2}}{3 b c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c/(a + b*x))*(a + b*x)**2/(3*b*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0203679, size = 21, normalized size = 0.84 \[ \frac{2 c}{3 b \left (\frac{c}{a+b x}\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[c/(a + b*x)],x]

[Out]

(2*c)/(3*b*(c/(a + b*x))^(3/2))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 22, normalized size = 0.9 \[{\frac{2\,bx+2\,a}{3\,b}{\frac{1}{\sqrt{{\frac{c}{bx+a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c/(b*x+a))^(1/2),x)

[Out]

2/3*(b*x+a)/b/(c/(b*x+a))^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.48133, size = 23, normalized size = 0.92 \[ \frac{2 \, c}{3 \, b \left (\frac{c}{b x + a}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)),x, algorithm="maxima")

[Out]

2/3*c/(b*(c/(b*x + a))^(3/2))

_______________________________________________________________________________________

Fricas [A]  time = 0.216505, size = 28, normalized size = 1.12 \[ \frac{2 \,{\left (b x + a\right )}}{3 \, b \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)),x, algorithm="fricas")

[Out]

2/3*(b*x + a)/(b*sqrt(c/(b*x + a)))

_______________________________________________________________________________________

Sympy [A]  time = 3.31804, size = 49, normalized size = 1.96 \[ \begin{cases} \frac{2 a}{3 b \sqrt{c} \sqrt{\frac{1}{a + b x}}} + \frac{2 x}{3 \sqrt{c} \sqrt{\frac{1}{a + b x}}} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt{\frac{c}{a}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*a/(3*b*sqrt(c)*sqrt(1/(a + b*x))) + 2*x/(3*sqrt(c)*sqrt(1/(a + b*x)
)), Ne(b, 0)), (x/sqrt(c/a), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216532, size = 84, normalized size = 3.36 \[ \frac{2 \,{\left (3 \, \sqrt{b c x + a c} a - \frac{3 \, \sqrt{b c x + a c} a c -{\left (b c x + a c\right )}^{\frac{3}{2}}}{c}\right )}}{3 \, b c{\rm sign}\left (b x + a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(c/(b*x + a)),x, algorithm="giac")

[Out]

2/3*(3*sqrt(b*c*x + a*c)*a - (3*sqrt(b*c*x + a*c)*a*c - (b*c*x + a*c)^(3/2))/c)/
(b*c*sign(b*x + a))