Optimal. Leaf size=33 \[ -\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0103688, antiderivative size = 33, 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 = {636} \[ -\frac{2 (x (2 c d-b e)+b d)}{b^2 \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 636
Rubi steps
\begin{align*} \int \frac{d+e x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b d+(2 c d-b e) x)}{b^2 \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0118774, size = 30, normalized size = 0.91 \[ \frac{2 (-b d+b e x-2 c d x)}{b^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 37, normalized size = 1.1 \begin{align*} -2\,{\frac{x \left ( cx+b \right ) \left ( -bxe+2\,cdx+bd \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 = 1.05328, size = 74, normalized size = 2.24 \begin{align*} -\frac{4 \, c d x}{\sqrt{c x^{2} + b x} b^{2}} + \frac{2 \, e x}{\sqrt{c x^{2} + b x} b} - \frac{2 \, d}{\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 = 1.96475, size = 89, normalized size = 2.7 \begin{align*} -\frac{2 \, \sqrt{c x^{2} + b x}{\left (b d +{\left (2 \, c d - b e\right )} x\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{d + e 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 = 1.32997, size = 46, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (\frac{d}{b} + \frac{{\left (2 \, c d - b e\right )} x}{b^{2}}\right )}}{\sqrt{c x^{2} + b x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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