Optimal. Leaf size=170 \[ \frac{32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt{x}}+\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
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Rubi [A] time = 0.158624, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ \frac{32 b^3 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{3465 c^5 x^{3/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{1155 c^4 \sqrt{x}}+\frac{4 b \sqrt{x} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{231 c^3}-\frac{2 x^{3/2} \left (b x+c x^2\right )^{3/2} (8 b B-11 A c)}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{5/2} (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac{\left (2 \left (\frac{5}{2} (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int x^{5/2} \sqrt{b x+c x^2} \, dx}{11 c}\\ &=-\frac{2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac{(2 b (8 b B-11 A c)) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{33 c^2}\\ &=\frac{4 b (8 b B-11 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac{\left (8 b^2 (8 b B-11 A c)\right ) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{231 c^3}\\ &=-\frac{16 b^2 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{4 b (8 b B-11 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}+\frac{\left (16 b^3 (8 b B-11 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{1155 c^4}\\ &=\frac{32 b^3 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{3465 c^5 x^{3/2}}-\frac{16 b^2 (8 b B-11 A c) \left (b x+c x^2\right )^{3/2}}{1155 c^4 \sqrt{x}}+\frac{4 b (8 b B-11 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{231 c^3}-\frac{2 (8 b B-11 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{99 c^2}+\frac{2 B x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}\\ \end{align*}
Mathematica [A] time = 0.0748335, size = 94, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{3/2} \left (24 b^2 c^2 x (11 A+10 B x)-16 b^3 c (11 A+12 B x)-10 b c^3 x^2 (33 A+28 B x)+35 c^4 x^3 (11 A+9 B x)+128 b^4 B\right )}{3465 c^5 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 107, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -315\,B{x}^{4}{c}^{4}-385\,A{c}^{4}{x}^{3}+280\,Bb{c}^{3}{x}^{3}+330\,Ab{c}^{3}{x}^{2}-240\,B{b}^{2}{c}^{2}{x}^{2}-264\,A{b}^{2}{c}^{2}x+192\,B{b}^{3}cx+176\,A{b}^{3}c-128\,{b}^{4}B \right ) }{3465\,{c}^{5}}\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 = 1.16849, size = 162, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt{c x + b} A}{315 \, c^{4}} + \frac{2 \,{\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt{c x + b} B}{3465 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6402, size = 293, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (315 \, B c^{5} x^{5} + 128 \, B b^{5} - 176 \, A b^{4} c + 35 \,{\left (B b c^{4} + 11 \, A c^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{2} c^{3} - 11 \, A b c^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, c^{5} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15083, size = 181, normalized size = 1.06 \begin{align*} -\frac{2}{3465} \, B{\left (\frac{128 \, b^{\frac{11}{2}}}{c^{5}} - \frac{315 \,{\left (c x + b\right )}^{\frac{11}{2}} - 1540 \,{\left (c x + b\right )}^{\frac{9}{2}} b + 2970 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{2} - 2772 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{3} + 1155 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{4}}{c^{5}}\right )} + \frac{2}{315} \, A{\left (\frac{16 \, b^{\frac{9}{2}}}{c^{4}} + \frac{35 \,{\left (c x + b\right )}^{\frac{9}{2}} - 135 \,{\left (c x + b\right )}^{\frac{7}{2}} b + 189 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{2} - 105 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{3}}{c^{4}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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