Optimal. Leaf size=133 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^{3/2}}+\frac{8 b \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{105 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.0948934, 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} \[ -\frac{16 b^2 \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^{3/2}}+\frac{8 b \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{105 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{3/2} (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac{\left (2 \left (\frac{3}{2} (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{9 c}\\ &=-\frac{2 (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac{(4 b (2 b B-3 A c)) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{21 c^2}\\ &=\frac{8 b (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{2 (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac{\left (8 b^2 (2 b B-3 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{105 c^3}\\ &=-\frac{16 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac{8 b (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt{x}}-\frac{2 (2 b B-3 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}\\ \end{align*}
Mathematica [A] time = 0.0555318, size = 72, normalized size = 0.54 \[ \frac{2 (x (b+c x))^{3/2} \left (24 b^2 c (A+B x)-6 b c^2 x (6 A+5 B x)+5 c^3 x^2 (9 A+7 B x)-16 b^3 B\right )}{315 c^4 x^{3/2}} \]
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 ( 35\,B{c}^{3}{x}^{3}+45\,A{x}^{2}{c}^{3}-30\,B{x}^{2}b{c}^{2}-36\,Ab{c}^{2}x+24\,B{b}^{2}cx+24\,A{b}^{2}c-16\,{b}^{3}B \right ) }{315\,{c}^{4}}\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 = 1.17248, size = 132, normalized size = 0.99 \begin{align*} \frac{2 \,{\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt{c x + b} A}{105 \, c^{3}} + \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} B}{315 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55302, size = 232, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (35 \, B c^{4} x^{4} - 16 \, B b^{4} + 24 \, A b^{3} c + 5 \,{\left (B b c^{3} + 9 \, A c^{4}\right )} x^{3} - 3 \,{\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{2} + 4 \,{\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, 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 x^{\frac{3}{2}} \sqrt{x \left (b + c x\right )} \left (A + B x\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14463, size = 149, normalized size = 1.12 \begin{align*} \frac{2}{315} \, 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}{105} \, A{\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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