Optimal. Leaf size=133 \[ -\frac{16 b^2 \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac{8 b \sqrt{x} \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^3}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
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Rubi [A] time = 0.107513, 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} \[ -\frac{16 b^2 \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^4 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x+c x^2} (6 b B-7 A c)}{35 c^2}+\frac{8 b \sqrt{x} \sqrt{b x+c x^2} (6 b B-7 A c)}{105 c^3}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c} \]
Antiderivative was successfully verified.
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Rule 794
Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{x^{5/2} (A+B x)}{\sqrt{b x+c x^2}} \, dx &=\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{\left (2 \left (\frac{5}{2} (-b B+A c)+\frac{1}{2} (-b B+2 A c)\right )\right ) \int \frac{x^{5/2}}{\sqrt{b x+c x^2}} \, dx}{7 c}\\ &=-\frac{2 (6 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c}+\frac{(4 b (6 b B-7 A c)) \int \frac{x^{3/2}}{\sqrt{b x+c x^2}} \, dx}{35 c^2}\\ &=\frac{8 b (6 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{105 c^3}-\frac{2 (6 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c}-\frac{\left (8 b^2 (6 b B-7 A c)\right ) \int \frac{\sqrt{x}}{\sqrt{b x+c x^2}} \, dx}{105 c^3}\\ &=-\frac{16 b^2 (6 b B-7 A c) \sqrt{b x+c x^2}}{105 c^4 \sqrt{x}}+\frac{8 b (6 b B-7 A c) \sqrt{x} \sqrt{b x+c x^2}}{105 c^3}-\frac{2 (6 b B-7 A c) x^{3/2} \sqrt{b x+c x^2}}{35 c^2}+\frac{2 B x^{5/2} \sqrt{b x+c x^2}}{7 c}\\ \end{align*}
Mathematica [A] time = 0.0577603, size = 75, normalized size = 0.56 \[ \frac{2 \sqrt{x (b+c x)} \left (8 b^2 c (7 A+3 B x)-2 b c^2 x (14 A+9 B x)+3 c^3 x^2 (7 A+5 B x)-48 b^3 B\right )}{105 c^4 \sqrt{x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 83, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2\,cx+2\,b \right ) \left ( 15\,B{c}^{3}{x}^{3}+21\,A{x}^{2}{c}^{3}-18\,B{x}^{2}b{c}^{2}-28\,Ab{c}^{2}x+24\,B{b}^{2}cx+56\,A{b}^{2}c-48\,{b}^{3}B \right ) }{105\,{c}^{4}}\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 = 1.11578, size = 132, normalized size = 0.99 \begin{align*} \frac{2 \,{\left (3 \, c^{3} x^{3} - b c^{2} x^{2} + 4 \, b^{2} c x + 8 \, b^{3}\right )} A}{15 \, \sqrt{c x + b} c^{3}} + \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 )} B}{35 \, \sqrt{c x + b} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62079, size = 186, normalized size = 1.4 \begin{align*} \frac{2 \,{\left (15 \, B c^{3} x^{3} - 48 \, B b^{3} + 56 \, A b^{2} c - 3 \,{\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} x^{2} + 4 \,{\left (6 \, B b^{2} c - 7 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x}}{105 \, c^{4} \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 \frac{x^{\frac{5}{2}} \left (A + B x\right )}{\sqrt{x \left (b + c x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14396, size = 149, normalized size = 1.12 \begin{align*} \frac{2 \,{\left (15 \,{\left (c x + b\right )}^{\frac{7}{2}} B - 63 \,{\left (c x + b\right )}^{\frac{5}{2}} B b + 105 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} - 105 \, \sqrt{c x + b} B b^{3} + 21 \,{\left (c x + b\right )}^{\frac{5}{2}} A c - 70 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c + 105 \, \sqrt{c x + b} A b^{2} c\right )}}{105 \, c^{4}} + \frac{16 \,{\left (6 \, B b^{\frac{7}{2}} - 7 \, A b^{\frac{5}{2}} c\right )}}{105 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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