Optimal. Leaf size=170 \[ \frac{32 b^3 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{15015 c^5 x^{5/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{3003 c^4 x^{3/2}}+\frac{4 b \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{429 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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Rubi [A] time = 0.16146, 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} \[ \frac{32 b^3 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{15015 c^5 x^{5/2}}-\frac{16 b^2 \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{3003 c^4 x^{3/2}}+\frac{4 b \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{429 c^3 \sqrt{x}}-\frac{2 \sqrt{x} \left (b x+c x^2\right )^{5/2} (8 b B-13 A c)}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c} \]
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) \left (b x+c x^2\right )^{3/2} \, dx &=\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac{\left (2 \left (\frac{3}{2} (-b B+A c)+\frac{5}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \left (b x+c x^2\right )^{3/2} \, dx}{13 c}\\ &=-\frac{2 (8 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac{(6 b (8 b B-13 A c)) \int \sqrt{x} \left (b x+c x^2\right )^{3/2} \, dx}{143 c^2}\\ &=\frac{4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{2 (8 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}-\frac{\left (8 b^2 (8 b B-13 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{\sqrt{x}} \, dx}{429 c^3}\\ &=-\frac{16 b^2 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{2 (8 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}+\frac{\left (16 b^3 (8 b B-13 A c)\right ) \int \frac{\left (b x+c x^2\right )^{3/2}}{x^{3/2}} \, dx}{3003 c^4}\\ &=\frac{32 b^3 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{15015 c^5 x^{5/2}}-\frac{16 b^2 (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{3003 c^4 x^{3/2}}+\frac{4 b (8 b B-13 A c) \left (b x+c x^2\right )^{5/2}}{429 c^3 \sqrt{x}}-\frac{2 (8 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{5/2}}{143 c^2}+\frac{2 B x^{3/2} \left (b x+c x^2\right )^{5/2}}{13 c}\\ \end{align*}
Mathematica [A] time = 0.0787966, size = 94, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{5/2} \left (40 b^2 c^2 x (13 A+14 B x)-16 b^3 c (13 A+20 B x)-70 b c^3 x^2 (13 A+12 B x)+105 c^4 x^3 (13 A+11 B x)+128 b^4 B\right )}{15015 c^5 x^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 107, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -1155\,B{x}^{4}{c}^{4}-1365\,A{c}^{4}{x}^{3}+840\,Bb{c}^{3}{x}^{3}+910\,Ab{c}^{3}{x}^{2}-560\,B{b}^{2}{c}^{2}{x}^{2}-520\,A{b}^{2}{c}^{2}x+320\,B{b}^{3}cx+208\,A{b}^{3}c-128\,{b}^{4}B \right ) }{15015\,{c}^{5}} \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 [A] time = 1.22939, size = 370, normalized size = 2.18 \begin{align*} \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} A}{3465 \, c^{4} x^{4}} + \frac{2 \,{\left (5 \,{\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} x^{5} + 13 \,{\left (315 \, b c^{5} x^{6} + 35 \, b^{2} c^{4} x^{5} - 40 \, b^{3} c^{3} x^{4} + 48 \, b^{4} c^{2} x^{3} - 64 \, b^{5} c x^{2} + 128 \, b^{6} x\right )} x^{4}\right )} \sqrt{c x + b} B}{45045 \, c^{5} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49967, size = 350, normalized size = 2.06 \begin{align*} \frac{2 \,{\left (1155 \, B c^{6} x^{6} + 128 \, B b^{6} - 208 \, A b^{5} c + 105 \,{\left (14 \, B b c^{5} + 13 \, A c^{6}\right )} x^{5} + 35 \,{\left (B b^{2} c^{4} + 52 \, A b c^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{15015 \, 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 [B] time = 1.23521, size = 398, normalized size = 2.34 \begin{align*} \frac{2}{9009} \, B c{\left (\frac{256 \, b^{\frac{13}{2}}}{c^{6}} + \frac{693 \,{\left (c x + b\right )}^{\frac{13}{2}} - 4095 \,{\left (c x + b\right )}^{\frac{11}{2}} b + 10010 \,{\left (c x + b\right )}^{\frac{9}{2}} b^{2} - 12870 \,{\left (c x + b\right )}^{\frac{7}{2}} b^{3} + 9009 \,{\left (c x + b\right )}^{\frac{5}{2}} b^{4} - 3003 \,{\left (c x + b\right )}^{\frac{3}{2}} b^{5}}{c^{6}}\right )} - \frac{2}{3465} \, B 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}{3465} \, A 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} \, A 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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