Optimal. Leaf size=23 \[ \frac {2 (a+b x) \sqrt {\frac {c}{a+b x}}}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 15
Rule 30
Rule 247
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 0.83 \[ \frac {2 c}{b \sqrt {\frac {c}{a+b x}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 21, normalized size = 0.91 \[ \frac {2 \, {\left (b x + a\right )} \sqrt {\frac {c}{b x + a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.91 \[ \frac {2 \, \sqrt {b c x + a c} \mathrm {sgn}\left (b x + a\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.96 \[ \frac {2 \left (b x +a \right ) \sqrt {\frac {c}{b x +a}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 17, normalized size = 0.74 \[ \frac {2 \, c}{b \sqrt {\frac {c}{b x + a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 22, normalized size = 0.96 \[ \sqrt {\frac {c}{a+b\,x}}\,\left (2\,x+\frac {2\,a}{b}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.34, size = 46, normalized size = 2.00 \[ \begin {cases} \frac {2 a \sqrt {c} \sqrt {\frac {1}{a + b x}}}{b} + 2 \sqrt {c} x \sqrt {\frac {1}{a + b x}} & \text {for}\: b \neq 0 \\x \sqrt {\frac {c}{a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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