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 = 0.0062485, antiderivative size = 25, normalized size of antiderivative = 1., 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} \[ \frac{2 \left (b x+c x^2\right )^{3/2}}{3 c x^{3/2}} \]
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*}
Mathematica [A] time = 0.0108567, size = 23, normalized size = 0.92 \[ \frac{2 (x (b+c x))^{3/2}}{3 c x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 25, normalized size = 1. \begin{align*}{\frac{2\,cx+2\,b}{3\,c}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16604, size = 16, normalized size = 0.64 \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21826, size = 61, normalized size = 2.44 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x + b\right )}}{3 \, c \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x \left (b + c x\right )}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19745, size = 28, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}}}{3 \, c} - \frac{2 \, b^{\frac{3}{2}}}{3 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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