3.2814 \(\int \sqrt{\frac{c}{a+b x}} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 (a+b x) \sqrt{\frac{c}{a+b x}}}{b} \]

[Out]

(2*Sqrt[c/(a + b*x)]*(a + b*x))/b

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Rubi [A]  time = 0.020987, antiderivative size = 23, 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 \[ \frac{2 (a+b x) \sqrt{\frac{c}{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c/(a + b*x)],x]

[Out]

(2*Sqrt[c/(a + b*x)]*(a + b*x))/b

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Rubi in Sympy [A]  time = 1.93603, size = 17, normalized size = 0.74 \[ \frac{2 \sqrt{\frac{c}{a + b x}} \left (a + b x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c/(a + b*x))*(a + b*x)/b

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Mathematica [A]  time = 0.0156059, size = 19, normalized size = 0.83 \[ \frac{2 c}{b \sqrt{\frac{c}{a+b x}}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c/(a + b*x)],x]

[Out]

(2*c)/(b*Sqrt[c/(a + b*x)])

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Maple [A]  time = 0.005, size = 22, normalized size = 1. \[ 2\,{\frac{bx+a}{b}\sqrt{{\frac{c}{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c/(b*x+a))^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.37526, size = 23, normalized size = 1. \[ \frac{2 \, c}{b \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c/(b*sqrt(c/(b*x + a)))

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Fricas [A]  time = 0.217475, size = 23, normalized size = 1. \[ \frac{2 \, c}{b \sqrt{\frac{c}{b x + a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c/(b*sqrt(c/(b*x + a)))

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Sympy [A]  time = 0.71416, size = 46, normalized size = 2. \[ \begin{cases} \frac{2 a \sqrt{c} \sqrt{\frac{1}{a + b x}}}{b} + 2 \sqrt{c} x \sqrt{\frac{1}{a + b x}} & \text{for}\: b \neq 0 \\x \sqrt{\frac{c}{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c/(b*x+a))**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.217833, size = 28, normalized size = 1.22 \[ \frac{2 \, \sqrt{b c x + a c}{\rm sign}\left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*c*x + a*c)*sign(b*x + a)/b