Optimal. Leaf size=133 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{9009 c^4 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{143 c^2 x^{3/2}}+\frac{8 b \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{1287 c^3 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.111131, 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} \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{9009 c^4 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{143 c^2 x^{3/2}}+\frac{8 b \left (b x+c x^2\right )^{7/2} (6 b B-13 A c)}{1287 c^3 x^{5/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{\sqrt{x}} \, dx &=\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}}+\frac{\left (2 \left (\frac{1}{2} (b B-A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{\sqrt{x}} \, dx}{13 c}\\ &=-\frac{2 (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{143 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}}+\frac{(4 b (6 b B-13 A c)) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx}{143 c^2}\\ &=\frac{8 b (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{1287 c^3 x^{5/2}}-\frac{2 (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{143 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}}-\frac{\left (8 b^2 (6 b B-13 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx}{1287 c^3}\\ &=-\frac{16 b^2 (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{9009 c^4 x^{7/2}}+\frac{8 b (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{1287 c^3 x^{5/2}}-\frac{2 (6 b B-13 A c) \left (b x+c x^2\right )^{7/2}}{143 c^2 x^{3/2}}+\frac{2 B \left (b x+c x^2\right )^{7/2}}{13 c \sqrt{x}}\\ \end{align*}
Mathematica [A] time = 0.0623201, size = 82, normalized size = 0.62 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (8 b^2 c (13 A+21 B x)-14 b c^2 x (26 A+27 B x)+63 c^3 x^2 (13 A+11 B x)-48 b^3 B\right )}{9009 c^4 \sqrt{x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.007, size = 83, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 693\,B{c}^{3}{x}^{3}+819\,A{x}^{2}{c}^{3}-378\,B{x}^{2}b{c}^{2}-364\,Ab{c}^{2}x+168\,B{b}^{2}cx+104\,A{b}^{2}c-48\,{b}^{3}B \right ) }{9009\,{c}^{4}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}{x}^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.18043, size = 506, normalized size = 3.8 \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} + 22 \,{\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} + 33 \,{\left (15 \, b^{2} c^{3} x^{5} + 3 \, b^{3} c^{2} x^{4} - 4 \, b^{4} c x^{3} + 8 \, b^{5} x^{2}\right )} x^{2}\right )} \sqrt{c x + b} A}{3465 \, c^{3} 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} + 26 \,{\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} + 143 \,{\left (35 \, b^{2} c^{4} x^{6} + 5 \, b^{3} c^{3} x^{5} - 6 \, b^{4} c^{2} x^{4} + 8 \, b^{5} c x^{3} - 16 \, b^{6} x^{2}\right )} x^{3}\right )} \sqrt{c x + b} B}{45045 \, c^{4} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.59431, size = 351, normalized size = 2.64 \begin{align*} \frac{2 \,{\left (693 \, B c^{6} x^{6} - 48 \, B b^{6} + 104 \, A b^{5} c + 63 \,{\left (27 \, B b c^{5} + 13 \, A c^{6}\right )} x^{5} + 7 \,{\left (159 \, B b^{2} c^{4} + 299 \, A b c^{5}\right )} x^{4} +{\left (15 \, B b^{3} c^{3} + 1469 \, A b^{2} c^{4}\right )} x^{3} - 3 \,{\left (6 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} + 4 \,{\left (6 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{9009 \, c^{4} \sqrt{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2163, size = 562, normalized size = 4.23 \begin{align*} \frac{2}{9009} \, B c^{2}{\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{4}{3465} \, B 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}{3465} \, A c^{2}{\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^{2}{\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{4}{315} \, A b 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^{2}{\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.
[In]
[Out]