Optimal. Leaf size=23 \[ \frac{2 (a+b x) \sqrt{\frac{c}{a+b x}}}{b} \]
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Rubi [A] time = 0.0063936, 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, Rules used = {247, 15, 30} \[ \frac{2 (a+b x) \sqrt{\frac{c}{a+b x}}}{b} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \sqrt{\frac{c}{a+b x}} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\frac{c}{x}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (\sqrt{\frac{c}{a+b x}} \sqrt{a+b x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x}} \, dx,x,a+b x\right )}{b}\\ &=\frac{2 \sqrt{\frac{c}{a+b x}} (a+b x)}{b}\\ \end{align*}
Mathematica [A] time = 0.0097345, size = 19, normalized size = 0.83 \[ \frac{2 c}{b \sqrt{\frac{c}{a+b x}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 22, normalized size = 1. \begin{align*} 2\,{\frac{bx+a}{b}\sqrt{{\frac{c}{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.964305, size = 23, normalized size = 1. \begin{align*} \frac{2 \, c}{b \sqrt{\frac{c}{b x + a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27387, size = 45, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (b x + a\right )} \sqrt{\frac{c}{b x + a}}}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.239331, size = 46, normalized size = 2. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0956, size = 28, normalized size = 1.22 \begin{align*} \frac{2 \, \sqrt{b c x + a c} \mathrm{sgn}\left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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