Optimal. Leaf size=24 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0033446, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {613} \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 613
Rubi steps
\begin{align*} \int \frac{1}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b+2 c x)}{b^2 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0072823, size = 22, normalized size = 0.92 \[ -\frac{2 (b+2 c x)}{b^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 29, normalized size = 1.2 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( 2\,cx+b \right ) }{{b}^{2} \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 [A] time = 1.06922, size = 47, normalized size = 1.96 \begin{align*} -\frac{4 \, c x}{\sqrt{c x^{2} + b x} b^{2}} - \frac{2}{\sqrt{c x^{2} + b x} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90418, size = 73, normalized size = 3.04 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x + b\right )}}{b^{2} c x^{2} + b^{3} 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{1}{\left (b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.30407, size = 32, normalized size = 1.33 \begin{align*} -\frac{2 \,{\left (\frac{2 \, c x}{b^{2}} + \frac{1}{b}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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