Optimal. Leaf size=70 \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.0450948, antiderivative size = 70, 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 = {788, 648} \[ \frac{2 \sqrt{b x+c x^2} (2 b B-A c)}{b c^2 \sqrt{x}}-\frac{2 x^{3/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{3/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b B-A c) x^{3/2}}{b c \sqrt{b x+c x^2}}-\frac{\left (2 \left (\frac{1}{2} (b B-2 A c)+\frac{3}{2} (-b B+A c)\right )\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{b c}\\ &=-\frac{2 (b B-A c) x^{3/2}}{b c \sqrt{b x+c x^2}}+\frac{2 (2 b B-A c) \sqrt{b x+c x^2}}{b c^2 \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0208443, size = 34, normalized size = 0.49 \[ \frac{2 \sqrt{x} (-A c+2 b B+B c x)}{c^2 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 38, normalized size = 0.5 \begin{align*} -2\,{\frac{ \left ( cx+b \right ) \left ( -Bcx+Ac-2\,bB \right ){x}^{3/2}}{{c}^{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B x + A\right )} x^{\frac{3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82694, size = 96, normalized size = 1.37 \begin{align*} \frac{2 \,{\left (B c x + 2 \, B b - A c\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{c^{3} x^{2} + b c^{2} 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{x^{\frac{3}{2}} \left (A + B x\right )}{\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 = 1.16047, size = 66, normalized size = 0.94 \begin{align*} \frac{2 \,{\left (\sqrt{c x + b} B + \frac{B b - A c}{\sqrt{c x + b}}\right )}}{c^{2}} - \frac{2 \,{\left (2 \, B b - A c\right )}}{\sqrt{b} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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