Optimal. Leaf size=25 \[ \frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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Rubi [A] time = 0.01, antiderivative size = 25, 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 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx &=\frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 23, normalized size = 0.92 \begin {gather*} \frac {2 (x (b+c x))^{3/2}}{3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 24, normalized size = 0.96 \begin {gather*} \frac {2 \, \sqrt {c x^{2} + b x} {\left (c x + b\right )}}{3 \, c \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 21, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, c} - \frac {2 \, b^{\frac {3}{2}}}{3 \, 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 = 1.00 \begin {gather*} \frac {2 \left (c x +b \right ) \sqrt {c \,x^{2}+b x}}{3 c \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.46, size = 12, normalized size = 0.48 \begin {gather*} \frac {2 \, {\left (c x + b\right )}^{\frac {3}{2}}}{3 \, 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 {c\,x^2+b\,x}}{\sqrt {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 \left (b + c x\right )}}{\sqrt {x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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