Optimal. Leaf size=67 \[ \frac{2 x (2 A c+b B)}{3 b^2 c \sqrt{b x+c x^2}}-\frac{2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0542218, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {788, 636} \[ \frac{2 x (2 A c+b B)}{3 b^2 c \sqrt{b x+c x^2}}-\frac{2 x^2 (b B-A c)}{3 b c \left (b x+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 636
Rubi steps
\begin{align*} \int \frac{x^2 (A+B x)}{\left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{(b B+2 A c) \int \frac{x}{\left (b x+c x^2\right )^{3/2}} \, dx}{3 b c}\\ &=-\frac{2 (b B-A c) x^2}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac{2 (b B+2 A c) x}{3 b^2 c \sqrt{b x+c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0155197, size = 35, normalized size = 0.52 \[ \frac{2 x^2 (3 A b+2 A c x+b B x)}{3 b^2 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 0.6 \begin{align*}{\frac{2\,{x}^{3} \left ( cx+b \right ) \left ( 2\,Acx+bBx+3\,Ab \right ) }{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.02029, size = 181, normalized size = 2.7 \begin{align*} -\frac{B x^{2}}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} + \frac{4 \, A x}{3 \, \sqrt{c x^{2} + b x} b^{2}} - \frac{B b x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c^{2}} - \frac{2 \, A x}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}} c} + \frac{2 \, B x}{3 \, \sqrt{c x^{2} + b x} b c} + \frac{B}{3 \, \sqrt{c x^{2} + b x} c^{2}} + \frac{2 \, A}{3 \, \sqrt{c x^{2} + b x} b c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94466, size = 109, normalized size = 1.63 \begin{align*} \frac{2 \, \sqrt{c x^{2} + b x}{\left (3 \, A b +{\left (B b + 2 \, A c\right )} x\right )}}{3 \,{\left (b^{2} c^{2} x^{2} + 2 \, b^{3} c x + b^{4}\right )}} \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^{2} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1674, size = 161, normalized size = 2.4 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )}^{2} B c + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} B b \sqrt{c} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} A c^{\frac{3}{2}} + B b^{2} + 2 \, A b c\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} + b\right )}^{3} c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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