Optimal. Leaf size=207 \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]
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Rubi [A] time = 0.194197, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac{256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac{128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt{x}}-\frac{32 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac{16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac{2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int x^{7/2} (A+B x) \sqrt{b x+c x^2} \, dx &=\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{\left (2 \left (\frac{7}{2} (-b B+A c)+\frac{3}{2} (-b B+2 A c)\right )\right ) \int x^{7/2} \sqrt{b x+c x^2} \, dx}{13 c}\\ &=-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{(8 b (10 b B-13 A c)) \int x^{5/2} \sqrt{b x+c x^2} \, dx}{143 c^2}\\ &=\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac{\left (16 b^2 (10 b B-13 A c)\right ) \int x^{3/2} \sqrt{b x+c x^2} \, dx}{429 c^3}\\ &=-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac{\left (64 b^3 (10 b B-13 A c)\right ) \int \sqrt{x} \sqrt{b x+c x^2} \, dx}{3003 c^4}\\ &=\frac{128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt{x}}-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac{\left (128 b^4 (10 b B-13 A c)\right ) \int \frac{\sqrt{b x+c x^2}}{\sqrt{x}} \, dx}{15015 c^5}\\ &=-\frac{256 b^4 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{45045 c^6 x^{3/2}}+\frac{128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt{x}}-\frac{32 b^2 (10 b B-13 A c) \sqrt{x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac{16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac{2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac{2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}\\ \end{align*}
Mathematica [A] time = 0.0952725, size = 113, normalized size = 0.55 \[ \frac{2 (x (b+c x))^{3/2} \left (80 b^2 c^3 x^2 (39 A+35 B x)-96 b^3 c^2 x (26 A+25 B x)+128 b^4 c (13 A+15 B x)-70 b c^4 x^3 (52 A+45 B x)+315 c^5 x^4 (13 A+11 B x)-1280 b^5 B\right )}{45045 c^6 x^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 131, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 3465\,B{x}^{5}{c}^{5}+4095\,A{c}^{5}{x}^{4}-3150\,Bb{c}^{4}{x}^{4}-3640\,Ab{c}^{4}{x}^{3}+2800\,B{b}^{2}{c}^{3}{x}^{3}+3120\,A{b}^{2}{c}^{3}{x}^{2}-2400\,B{b}^{3}{c}^{2}{x}^{2}-2496\,A{b}^{3}{c}^{2}x+1920\,B{b}^{4}cx+1664\,A{b}^{4}c-1280\,B{b}^{5} \right ) }{45045\,{c}^{6}}\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 = 1.30937, size = 192, normalized size = 0.93 \begin{align*} \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} A}{3465 \, c^{5}} + \frac{2 \,{\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 )} \sqrt{c x + b} B}{9009 \, c^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62563, size = 360, normalized size = 1.74 \begin{align*} \frac{2 \,{\left (3465 \, B c^{6} x^{6} - 1280 \, B b^{6} + 1664 \, A b^{5} c + 315 \,{\left (B b c^{5} + 13 \, A c^{6}\right )} x^{5} - 35 \,{\left (10 \, B b^{2} c^{4} - 13 \, A b c^{5}\right )} x^{4} + 40 \,{\left (10 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} - 48 \,{\left (10 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} + 64 \,{\left (10 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, c^{6} \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 = 1.1508, size = 213, normalized size = 1.03 \begin{align*} \frac{2}{9009} \, B{\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} \, A{\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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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