Optimal. Leaf size=170 \[ -\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac{32 b^3 \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt{x}}-\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac{4 b x^{3/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{105 c^3}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c} \]
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Rubi [A] time = 0.147583, 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} \[ -\frac{16 b^2 \sqrt{x} \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^4}+\frac{32 b^3 \sqrt{b x+c x^2} (8 b B-9 A c)}{315 c^5 \sqrt{x}}-\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{63 c^2}+\frac{4 b x^{3/2} \sqrt{b x+c x^2} (8 b B-9 A c)}{105 c^3}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{7/2} (A+B x)}{\sqrt{b x+c x^2}} \, dx &=\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c}+\frac{\left (2 \left (\frac{7}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{x^{7/2}}{\sqrt{b x+c x^2}} \, dx}{9 c}\\ &=-\frac{2 (8 b B-9 A c) x^{5/2} \sqrt{b x+c x^2}}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c}+\frac{(2 b (8 b B-9 A c)) \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx}{21 c^2}\\ &=\frac{4 b (8 b B-9 A c) x^{3/2} \sqrt{b x+c x^2}}{105 c^3}-\frac{2 (8 b B-9 A c) x^{5/2} \sqrt{b x+c x^2}}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c}-\frac{\left (8 b^2 (8 b B-9 A c)\right ) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{105 c^3}\\ &=-\frac{16 b^2 (8 b B-9 A c) \sqrt{x} \sqrt{b x+c x^2}}{315 c^4}+\frac{4 b (8 b B-9 A c) x^{3/2} \sqrt{b x+c x^2}}{105 c^3}-\frac{2 (8 b B-9 A c) x^{5/2} \sqrt{b x+c x^2}}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c}+\frac{\left (16 b^3 (8 b B-9 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{315 c^4}\\ &=\frac{32 b^3 (8 b B-9 A c) \sqrt{b x+c x^2}}{315 c^5 \sqrt{x}}-\frac{16 b^2 (8 b B-9 A c) \sqrt{x} \sqrt{b x+c x^2}}{315 c^4}+\frac{4 b (8 b B-9 A c) x^{3/2} \sqrt{b x+c x^2}}{105 c^3}-\frac{2 (8 b B-9 A c) x^{5/2} \sqrt{b x+c x^2}}{63 c^2}+\frac{2 B x^{7/2} \sqrt{b x+c x^2}}{9 c}\\ \end{align*}
Mathematica [A] time = 0.0762626, size = 94, normalized size = 0.55 \[ \frac{2 \sqrt{x (b+c x)} \left (24 b^2 c^2 x (3 A+2 B x)-16 b^3 c (9 A+4 B x)-2 b c^3 x^2 (27 A+20 B x)+5 c^4 x^3 (9 A+7 B x)+128 b^4 B\right )}{315 c^5 \sqrt{x}} \]
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 ( -35\,B{x}^{4}{c}^{4}-45\,A{c}^{4}{x}^{3}+40\,Bb{c}^{3}{x}^{3}+54\,Ab{c}^{3}{x}^{2}-48\,B{b}^{2}{c}^{2}{x}^{2}-72\,A{b}^{2}{c}^{2}x+64\,B{b}^{3}cx+144\,A{b}^{3}c-128\,{b}^{4}B \right ) }{315\,{c}^{5}}\sqrt{x}{\frac{1}{\sqrt{c{x}^{2}+bx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01821, size = 162, normalized size = 0.95 \begin{align*} \frac{2 \,{\left (5 \, c^{4} x^{4} - b c^{3} x^{3} + 2 \, b^{2} c^{2} x^{2} - 8 \, b^{3} c x - 16 \, b^{4}\right )} A}{35 \, \sqrt{c x + b} c^{4}} + \frac{2 \,{\left (35 \, c^{5} x^{5} - 5 \, b c^{4} x^{4} + 8 \, b^{2} c^{3} x^{3} - 16 \, b^{3} c^{2} x^{2} + 64 \, b^{4} c x + 128 \, b^{5}\right )} B}{315 \, \sqrt{c x + b} c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58888, size = 238, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (35 \, B c^{4} x^{4} + 128 \, B b^{4} - 144 \, A b^{3} c - 5 \,{\left (8 \, B b c^{3} - 9 \, A c^{4}\right )} x^{3} + 6 \,{\left (8 \, B b^{2} c^{2} - 9 \, A b c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{3} c - 9 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{315 \, 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 [A] time = 1.14968, size = 185, normalized size = 1.09 \begin{align*} \frac{2 \,{\left (35 \,{\left (c x + b\right )}^{\frac{9}{2}} B - 180 \,{\left (c x + b\right )}^{\frac{7}{2}} B b + 378 \,{\left (c x + b\right )}^{\frac{5}{2}} B b^{2} - 420 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{3} + 315 \, \sqrt{c x + b} B b^{4} + 45 \,{\left (c x + b\right )}^{\frac{7}{2}} A c - 189 \,{\left (c x + b\right )}^{\frac{5}{2}} A b c + 315 \,{\left (c x + b\right )}^{\frac{3}{2}} A b^{2} c - 315 \, \sqrt{c x + b} A b^{3} c\right )}}{315 \, c^{5}} - \frac{32 \,{\left (8 \, B b^{\frac{9}{2}} - 9 \, A b^{\frac{7}{2}} c\right )}}{315 \, c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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