Optimal. Leaf size=178 \[ -\frac{32 b^2 \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt{x}}+\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac{12 x^{3/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac{16 b \sqrt{x} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac{2 x^{9/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.149217, antiderivative size = 178, 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 = {788, 656, 648} \[ -\frac{32 b^2 \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^5 \sqrt{x}}+\frac{2 x^{5/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{7 b c^2}-\frac{12 x^{3/2} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^3}+\frac{16 b \sqrt{x} \sqrt{b x+c x^2} (8 b B-7 A c)}{35 c^4}-\frac{2 x^{9/2} (b B-A c)}{b c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 788
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{9/2} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\left (\frac{7 A}{b}-\frac{8 B}{c}\right ) \int \frac{x^{7/2}}{\sqrt{b x+c x^2}} \, dx\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}-\frac{(6 (8 b B-7 A c)) \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx}{7 c^2}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}+\frac{(24 b (8 b B-7 A c)) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{35 c^3}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}+\frac{16 b (8 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{35 c^4}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}-\frac{\left (16 b^2 (8 b B-7 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{35 c^4}\\ &=-\frac{2 (b B-A c) x^{9/2}}{b c \sqrt{b x+c x^2}}-\frac{32 b^2 (8 b B-7 A c) \sqrt{b x+c x^2}}{35 c^5 \sqrt{x}}+\frac{16 b (8 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{35 c^4}-\frac{12 (8 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^3}+\frac{2 (8 b B-7 A c) x^{5/2} \sqrt{b x+c x^2}}{7 b c^2}\\ \end{align*}
Mathematica [A] time = 0.0663672, size = 93, normalized size = 0.52 \[ \frac{2 \sqrt{x} \left (8 b^2 c^2 x (7 A+2 B x)+16 b^3 c (7 A-4 B x)-2 b c^3 x^2 (7 A+4 B x)+c^4 x^3 (7 A+5 B x)-128 b^4 B\right )}{35 c^5 \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 107, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 5\,B{x}^{4}{c}^{4}+7\,A{c}^{4}{x}^{3}-8\,Bb{c}^{3}{x}^{3}-14\,Ab{c}^{3}{x}^{2}+16\,B{b}^{2}{c}^{2}{x}^{2}+56\,A{b}^{2}{c}^{2}x-64\,B{b}^{3}cx+112\,A{b}^{3}c-128\,{b}^{4}B \right ) }{35\,{c}^{5}}{x}^{{\frac{3}{2}}} \left ( c{x}^{2}+bx \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \,{\left ({\left (15 \, B c^{5} x^{3} + 3 \, B b c^{4} x^{2} - 4 \, B b^{2} c^{3} x + 8 \, B b^{3} c^{2}\right )} x^{4} +{\left (16 \, B b^{4} c - 3 \,{\left (4 \, B b c^{4} - 7 \, A c^{5}\right )} x^{3} -{\left (8 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{2} + 2 \,{\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x\right )} x^{3} + 4 \,{\left (2 \, B b^{5} +{\left (9 \, B b^{2} c^{3} - 7 \, A b c^{4}\right )} x^{3} + 2 \,{\left (10 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} +{\left (13 \, B b^{4} c - 7 \, A b^{3} c^{2}\right )} x\right )} x^{2}\right )} \sqrt{c x + b}}{105 \,{\left (c^{7} x^{4} + 2 \, b c^{6} x^{3} + b^{2} c^{5} x^{2}\right )}} + \int -\frac{4 \,{\left (4 \, B b^{5} - 2 \, A b^{4} c +{\left (9 \, B b^{3} c^{2} - 7 \, A b^{2} c^{3}\right )} x^{2} +{\left (13 \, B b^{4} c - 9 \, A b^{3} c^{2}\right )} x\right )} \sqrt{c x + b} x^{2}}{15 \,{\left (c^{7} x^{5} + 3 \, b c^{6} x^{4} + 3 \, b^{2} c^{5} x^{3} + b^{3} c^{4} x^{2}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84342, size = 251, normalized size = 1.41 \begin{align*} \frac{2 \,{\left (5 \, B c^{4} x^{4} - 128 \, B b^{4} + 112 \, A b^{3} c -{\left (8 \, B b c^{3} - 7 \, A c^{4}\right )} x^{3} + 2 \,{\left (8 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{2} - 8 \,{\left (8 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{35 \,{\left (c^{6} x^{2} + b c^{5} x\right )}} \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 = 1.17118, size = 182, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (5 \,{\left (c x + b\right )}^{\frac{7}{2}} B - 28 \,{\left (c x + b\right )}^{\frac{5}{2}} B b + 70 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} - 140 \, \sqrt{c x + b} B b^{3} + 7 \,{\left (c x + b\right )}^{\frac{5}{2}} A c - 35 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c + 105 \, \sqrt{c x + b} A b^{2} c - \frac{35 \,{\left (B b^{4} - A b^{3} c\right )}}{\sqrt{c x + b}}\right )}}{35 \, c^{5}} + \frac{32 \,{\left (8 \, B b^{4} - 7 \, A b^{3} c\right )}}{35 \, \sqrt{b} c^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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