Optimal. Leaf size=61 \[ \frac{2 B \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{2 \sqrt{b x+c x^2} (2 b B-3 A c)}{3 c^2 \sqrt{x}} \]
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Rubi [A] time = 0.0407592, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {794, 648} \[ \frac{2 B \sqrt{x} \sqrt{b x+c x^2}}{3 c}-\frac{2 \sqrt{b x+c x^2} (2 b B-3 A c)}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{\sqrt{x} (A+B x)}{\sqrt{b x+c x^2}} \, dx &=\frac{2 B \sqrt{x} \sqrt{b x+c x^2}}{3 c}+\frac{\left (2 \left (\frac{1}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{3 c}\\ &=-\frac{2 (2 b B-3 A c) \sqrt{b x+c x^2}}{3 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \sqrt{b x+c x^2}}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0255501, size = 36, normalized size = 0.59 \[ \frac{2 \sqrt{x (b+c x)} (3 A c-2 b B+B c x)}{3 c^2 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 38, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( Bcx+3\,Ac-2\,bB \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 = 1.08302, size = 61, normalized size = 1. \begin{align*} \frac{2 \, \sqrt{c x + b} A}{c} + \frac{2 \,{\left (c^{2} x^{2} - b c x - 2 \, b^{2}\right )} B}{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 = 1.56172, size = 82, normalized size = 1.34 \begin{align*} \frac{2 \,{\left (B c x - 2 \, B b + 3 \, A c\right )} \sqrt{c x^{2} + b x}}{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{\sqrt{x} \left (A + B x\right )}{\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 = 1.14612, size = 77, normalized size = 1.26 \begin{align*} \frac{2 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} B - 3 \, \sqrt{c x + b} B b + 3 \, \sqrt{c x + b} A c\right )}}{3 \, c^{2}} + \frac{2 \,{\left (2 \, B b^{\frac{3}{2}} - 3 \, A \sqrt{b} c\right )}}{3 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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