Optimal. Leaf size=25 \[ \frac{2 (a+b x)}{3 b \sqrt{\frac{c}{a+b x}}} \]
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Rubi [A] time = 0.006398, 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, Rules used = {247, 15, 30} \[ \frac{2 (a+b x)}{3 b \sqrt{\frac{c}{a+b x}}} \]
Antiderivative was successfully verified.
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Rule 247
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\frac{c}{a+b x}}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\frac{c}{x}}} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{x} \, dx,x,a+b x\right )}{b \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}}\\ &=\frac{2 (a+b x)}{3 b \sqrt{\frac{c}{a+b x}}}\\ \end{align*}
Mathematica [A] time = 0.0120305, size = 21, normalized size = 0.84 \[ \frac{2 c}{3 b \left (\frac{c}{a+b x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 22, normalized size = 0.9 \begin{align*}{\frac{2\,bx+2\,a}{3\,b}{\frac{1}{\sqrt{{\frac{c}{bx+a}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2842, size = 23, normalized size = 0.92 \begin{align*} \frac{2 \, c}{3 \, b \left (\frac{c}{b x + a}\right )^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.31992, size = 74, normalized size = 2.96 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{\frac{c}{b x + a}}}{3 \, b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.914531, size = 49, normalized size = 1.96 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14069, size = 84, normalized size = 3.36 \begin{align*} \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 \mathrm{sgn}\left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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