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