Optimal. Leaf size=52 \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]
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Rubi [A] time = 0.0157122, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {656, 648} \[ \frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx &=\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{(2 b) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{3 c}\\ &=-\frac{4 b \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}+\frac{2 \sqrt{x} \sqrt{b x+c x^2}}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0166868, size = 30, normalized size = 0.58 \[ \frac{2 (c x-2 b) \sqrt{x (b+c x)}}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.055, size = 33, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -cx+2\,b \right ) }{3\,{c}^{2}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12455, size = 41, normalized size = 0.79 \begin{align*} \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )}}{3 \, \sqrt{c x + b} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.909, size = 66, normalized size = 1.27 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (c x - 2 \, b\right )}}{3 \, c^{2} \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{x^{\frac{3}{2}}}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25214, size = 43, normalized size = 0.83 \begin{align*} \frac{4 \, b^{\frac{3}{2}}}{3 \, c^{2}} + \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b\right )}}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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