Optimal. Leaf size=23 \[ \frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \]
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Rubi [A] time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {648} \begin {gather*} \frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt {x}}{\sqrt {b x+c x^2}} \, dx &=\frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \sqrt {x (b+c x)}}{c \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b x+c x^2}}{c \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x}}{c \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {c x + b}}{c} - \frac {2 \, \sqrt {b}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 25, normalized size = 1.09 \begin {gather*} \frac {2 \left (c x +b \right ) \sqrt {x}}{\sqrt {c \,x^{2}+b x}\, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 12, normalized size = 0.52 \begin {gather*} \frac {2 \, \sqrt {c x + b}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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