Optimal. Leaf size=19 \[ \frac{2 x}{b \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0053672, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {636} \[ \frac{2 x}{b \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{x}{\left (b x+c x^2\right )^{3/2}} \, dx &=\frac{2 x}{b \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.006217, size = 17, normalized size = 0.89 \[ \frac{2 x}{b \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 25, normalized size = 1.3 \begin{align*} 2\,{\frac{{x}^{2} \left ( cx+b \right ) }{b \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 = 1.09251, size = 23, normalized size = 1.21 \begin{align*} \frac{2 \, x}{\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 = 1.96491, size = 47, normalized size = 2.47 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}}{b c x + b^{2}} \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}{\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 = 1.20963, size = 43, normalized size = 2.26 \begin{align*} \frac{2}{{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )} \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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