Optimal. Leaf size=133 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{3465 c^4 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{99 c^2 \sqrt{x}}+\frac{8 b \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{693 c^3 x^{3/2}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]
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Rubi [A] time = 0.110975, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{3465 c^4 x^{5/2}}-\frac{2 \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{99 c^2 \sqrt{x}}+\frac{8 b \left (b x+c x^2\right )^{5/2} (6 b B-11 A c)}{693 c^3 x^{3/2}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \sqrt{x} (A+B x) \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac{\left (2 \left (\frac{1}{2} (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right )\right ) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{11 c}\\ &=-\frac{2 (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{99 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}+\frac{(4 b (6 b B-11 A c)) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{99 c^2}\\ &=\frac{8 b (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{693 c^3 x^{3/2}}-\frac{2 (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{99 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 b^2 (6 b B-11 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{693 c^3}\\ &=-\frac{16 b^2 (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{3465 c^4 x^{5/2}}+\frac{8 b (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{693 c^3 x^{3/2}}-\frac{2 (6 b B-11 A c) \left (b x+c x^2\right )^{5/2}}{99 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{5/2}}{11 c}\\ \end{align*}
Mathematica [A] time = 0.0589493, size = 75, normalized size = 0.56 \[ \frac{2 (x (b+c x))^{5/2} \left (8 b^2 c (11 A+15 B x)-10 b c^2 x (22 A+21 B x)+35 c^3 x^2 (11 A+9 B x)-48 b^3 B\right )}{3465 c^4 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 83, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 315\,B{c}^{3}{x}^{3}+385\,A{x}^{2}{c}^{3}-210\,B{x}^{2}b{c}^{2}-220\,Ab{c}^{2}x+120\,B{b}^{2}cx+88\,A{b}^{2}c-48\,{b}^{3}B \right ) }{3465\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.13368, size = 309, normalized size = 2.32 \begin{align*} \frac{2 \,{\left ({\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 )} x^{3} + 3 \,{\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2}\right )} \sqrt{c x + b} A}{315 \, c^{3} x^{3}} + \frac{2 \,{\left ({\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 )} x^{4} + 11 \,{\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3}\right )} \sqrt{c x + b} B}{3465 \, c^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49021, size = 296, normalized size = 2.23 \begin{align*} \frac{2 \,{\left (315 \, B c^{5} x^{5} - 48 \, B b^{5} + 88 \, A b^{4} c + 35 \,{\left (12 \, B b c^{4} + 11 \, A c^{5}\right )} x^{4} + 5 \,{\left (3 \, B b^{2} c^{3} + 110 \, A b c^{4}\right )} x^{3} - 3 \,{\left (6 \, B b^{3} c^{2} - 11 \, A b^{2} c^{3}\right )} x^{2} + 4 \,{\left (6 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{3465 \, c^{4} \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 \sqrt{x} \left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.2178, size = 333, normalized size = 2.5 \begin{align*} -\frac{2}{3465} \, B c{\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} \, B b{\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 )} + \frac{2}{315} \, A c{\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 )} - \frac{2}{105} \, A b{\left (\frac{8 \, b^{\frac{7}{2}}}{c^{3}} - \frac{15 \,{\left (c x + b\right )}^{\frac{7}{2}} - 42 \,{\left (c x + b\right )}^{\frac{5}{2}} b + 35 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{2}}{c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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