Optimal. Leaf size=170 \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{6435 c^4 x^{5/2}}+\frac{32 b^3 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{45045 c^5 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{195 c^2 \sqrt{x}}+\frac{4 b \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{715 c^3 x^{3/2}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c} \]
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Rubi [A] time = 0.144262, 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} \[ -\frac{16 b^2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{6435 c^4 x^{5/2}}+\frac{32 b^3 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{45045 c^5 x^{7/2}}-\frac{2 \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{195 c^2 \sqrt{x}}+\frac{4 b \left (b x+c x^2\right )^{7/2} (8 b B-15 A c)}{715 c^3 x^{3/2}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c} \]
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 )^{5/2} \, dx &=\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c}+\frac{\left (2 \left (\frac{1}{2} (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \sqrt{x} \left (b x+c x^2\right )^{5/2} \, dx}{15 c}\\ &=-\frac{2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c}+\frac{(2 b (8 b B-15 A c)) \int \frac{\left (b x+c x^2\right )^{5/2}}{\sqrt{x}} \, dx}{65 c^2}\\ &=\frac{4 b (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{715 c^3 x^{3/2}}-\frac{2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c}-\frac{\left (8 b^2 (8 b B-15 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx}{715 c^3}\\ &=-\frac{16 b^2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{6435 c^4 x^{5/2}}+\frac{4 b (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{715 c^3 x^{3/2}}-\frac{2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c}+\frac{\left (16 b^3 (8 b B-15 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx}{6435 c^4}\\ &=\frac{32 b^3 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{45045 c^5 x^{7/2}}-\frac{16 b^2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{6435 c^4 x^{5/2}}+\frac{4 b (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{715 c^3 x^{3/2}}-\frac{2 (8 b B-15 A c) \left (b x+c x^2\right )^{7/2}}{195 c^2 \sqrt{x}}+\frac{2 B \sqrt{x} \left (b x+c x^2\right )^{7/2}}{15 c}\\ \end{align*}
Mathematica [A] time = 0.077316, size = 101, normalized size = 0.59 \[ \frac{2 (b+c x)^3 \sqrt{x (b+c x)} \left (168 b^2 c^2 x (5 A+6 B x)-16 b^3 c (15 A+28 B x)-42 b c^3 x^2 (45 A+44 B x)+231 c^4 x^3 (15 A+13 B x)+128 b^4 B\right )}{45045 c^5 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 107, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cx+2\,b \right ) \left ( -3003\,B{x}^{4}{c}^{4}-3465\,A{c}^{4}{x}^{3}+1848\,Bb{c}^{3}{x}^{3}+1890\,Ab{c}^{3}{x}^{2}-1008\,B{b}^{2}{c}^{2}{x}^{2}-840\,A{b}^{2}{c}^{2}x+448\,B{b}^{3}cx+240\,A{b}^{3}c-128\,{b}^{4}B \right ) }{45045\,{c}^{5}} \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.
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Maxima [B] time = 1.1772, size = 595, normalized size = 3.5 \begin{align*} \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} A}{45045 \, c^{4} x^{5}} + \frac{2 \,{\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 10 \,{\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5} + 13 \,{\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt{c x + b} B}{45045 \, c^{5} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57179, size = 405, normalized size = 2.38 \begin{align*} \frac{2 \,{\left (3003 \, B c^{7} x^{7} + 128 \, B b^{7} - 240 \, A b^{6} c + 231 \,{\left (31 \, B b c^{6} + 15 \, A c^{7}\right )} x^{6} + 63 \,{\left (71 \, B b^{2} c^{5} + 135 \, A b c^{6}\right )} x^{5} + 35 \,{\left (B b^{3} c^{4} + 159 \, A b^{2} c^{5}\right )} x^{4} - 5 \,{\left (8 \, B b^{4} c^{3} - 15 \, A b^{3} c^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{5} c^{2} - 15 \, A b^{4} c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{6} c - 15 \, A b^{5} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{45045 \, c^{5} \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{5}{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 = 1.21308, size = 659, normalized size = 3.88 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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