Optimal. Leaf size=25 \[ \frac {2 (a+b x)}{3 b \sqrt {\frac {c}{a+b x}}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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} \[ \frac {2 (a+b x)}{3 b \sqrt {\frac {c}{a+b x}}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 247
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.76, size = 35, normalized size = 1.40 \[ \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {\frac {c}{b x + a}}}{3 \, b c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 62, normalized size = 2.48 \[ \frac {2 \, {\left (3 \, \sqrt {b c x + a c} a - \frac {3 \, \sqrt {b c x + a c} a c - {\left (b c x + a c\right )}^{\frac {3}{2}}}{c}\right )}}{3 \, b c \mathrm {sgn}\left (b x + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.88 \[ \frac {\frac {2 b x}{3}+\frac {2 a}{3}}{\sqrt {\frac {c}{b x +a}}\, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 17, normalized size = 0.68 \[ \frac {2 \, c}{3 \, b \left (\frac {c}{b x + a}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 40, normalized size = 1.60 \[ \sqrt {\frac {c}{a+b\,x}}\,\left (\frac {2\,b\,x^2}{3\,c}+\frac {2\,a^2}{3\,b\,c}+\frac {4\,a\,x}{3\,c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.62, size = 49, normalized size = 1.96 \[ \begin {cases} \frac {2 a}{3 b \sqrt {c} \sqrt {\frac {1}{a + b x}}} + \frac {2 x}{3 \sqrt {c} \sqrt {\frac {1}{a + b x}}} & \text {for}\: b \neq 0 \\\frac {x}{\sqrt {\frac {c}{a}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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