Optimal. Leaf size=61 \[ \frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \]
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Rubi [A] time = 0.042698, 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} \[ \frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}-\frac{2 \left (b x+c x^2\right )^{3/2} (2 b B-5 A c)}{15 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Rule 794
Rule 648
Rubi steps
\begin{align*} \int \frac{(A+B x) \sqrt{b x+c x^2}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}+\frac{\left (2 \left (\frac{1}{2} (b B-A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{5 c}\\ &=-\frac{2 (2 b B-5 A c) \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{3/2}}{5 c \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.026309, size = 37, normalized size = 0.61 \[ \frac{2 (x (b+c x))^{3/2} (5 A c-2 b B+3 B c x)}{15 c^2 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3\,Bcx+5\,Ac-2\,bB \right ) }{15\,{c}^{2}}\sqrt{c{x}^{2}+bx}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17729, size = 61, normalized size = 1. \begin{align*} \frac{2 \,{\left (c x + b\right )}^{\frac{3}{2}} A}{3 \, c} + \frac{2 \,{\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt{c x + b} B}{15 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55229, size = 127, normalized size = 2.08 \begin{align*} \frac{2 \,{\left (3 \, B c^{2} x^{2} - 2 \, B b^{2} + 5 \, A b c +{\left (B b c + 5 \, A c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{15 \, 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 (b + c x\right )} \left (A + B x\right )}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14927, size = 81, normalized size = 1.33 \begin{align*} \frac{2}{15} \, B{\left (\frac{2 \, b^{\frac{5}{2}}}{c^{2}} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} - 5 \,{\left (c x + b\right )}^{\frac{3}{2}} b}{c^{2}}\right )} + \frac{2}{3} \, A{\left (\frac{{\left (c x + b\right )}^{\frac{3}{2}}}{c} - \frac{b^{\frac{3}{2}}}{c}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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