Optimal. Leaf size=23 \[ -\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0070982, antiderivative size = 23, 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} \[ -\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 648
Rubi steps
\begin{align*} \int \frac{x^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 \sqrt{x}}{c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0067679, size = 21, normalized size = 0.91 \[ -\frac{2 \sqrt{x}}{c \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 25, normalized size = 1.1 \begin{align*} -2\,{\frac{ \left ( cx+b \right ){x}^{3/2}}{c \left ( c{x}^{2}+bx \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08504, size = 65, normalized size = 2.83 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x} \sqrt{x}}{c^{2} x^{2} + b c 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}}}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21502, size = 28, normalized size = 1.22 \begin{align*} -\frac{2}{\sqrt{c x + b} c} + \frac{2}{\sqrt{b} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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