Optimal. Leaf size=179 \[ \frac{5 c^2 \sqrt{x} (6 b B-7 A c)}{8 b^4 \sqrt{b x+c x^2}}-\frac{5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{9/2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}} \]
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Rubi [A] time = 0.152476, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {792, 672, 666, 660, 207} \[ \frac{5 c^2 \sqrt{x} (6 b B-7 A c)}{8 b^4 \sqrt{b x+c x^2}}-\frac{5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{9/2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
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Rule 792
Rule 672
Rule 666
Rule 660
Rule 207
Rubi steps
\begin{align*} \int \frac{A+B x}{x^{5/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}+\frac{\left (\frac{1}{2} (b B-2 A c)-\frac{5}{2} (-b B+A c)\right ) \int \frac{1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx}{3 b}\\ &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}-\frac{(5 c (6 b B-7 A c)) \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx}{24 b^2}\\ &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\left (5 c^2 (6 b B-7 A c)\right ) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{16 b^3}\\ &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{5 c^2 (6 b B-7 A c) \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}+\frac{\left (5 c^2 (6 b B-7 A c)\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b^4}\\ &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{5 c^2 (6 b B-7 A c) \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}+\frac{\left (5 c^2 (6 b B-7 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{-b+x^2} \, dx,x,\frac{\sqrt{b x+c x^2}}{\sqrt{x}}\right )}{8 b^4}\\ &=-\frac{A}{3 b x^{5/2} \sqrt{b x+c x^2}}-\frac{6 b B-7 A c}{12 b^2 x^{3/2} \sqrt{b x+c x^2}}+\frac{5 c (6 b B-7 A c)}{24 b^3 \sqrt{x} \sqrt{b x+c x^2}}+\frac{5 c^2 (6 b B-7 A c) \sqrt{x}}{8 b^4 \sqrt{b x+c x^2}}-\frac{5 c^2 (6 b B-7 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{9/2}}\\ \end{align*}
Mathematica [C] time = 0.0240292, size = 62, normalized size = 0.35 \[ \frac{c^2 x^3 (6 b B-7 A c) \, _2F_1\left (-\frac{1}{2},3;\frac{1}{2};\frac{c x}{b}+1\right )-A b^3}{3 b^4 x^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 150, normalized size = 0.8 \begin{align*}{\frac{1}{24\,cx+24\,b}\sqrt{x \left ( cx+b \right ) } \left ( 105\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}{c}^{3}-90\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{3}b{c}^{2}-105\,A\sqrt{b}{x}^{3}{c}^{3}+90\,B{b}^{3/2}{x}^{3}{c}^{2}-35\,A{b}^{3/2}{x}^{2}{c}^{2}+30\,B{b}^{5/2}{x}^{2}c+14\,A{b}^{5/2}xc-12\,B{b}^{7/2}x-8\,A{b}^{7/2} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{9}{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*} \int \frac{B x + A}{{\left (c x^{2} + b x\right )}^{\frac{3}{2}} x^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88999, size = 792, normalized size = 4.42 \begin{align*} \left [-\frac{15 \,{\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} +{\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt{b} \log \left (-\frac{c x^{2} + 2 \, b x + 2 \, \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{x^{2}}\right ) + 2 \,{\left (8 \, A b^{4} - 15 \,{\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \,{\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \,{\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}, \frac{15 \,{\left ({\left (6 \, B b c^{3} - 7 \, A c^{4}\right )} x^{5} +{\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{4}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (8 \, A b^{4} - 15 \,{\left (6 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{3} - 5 \,{\left (6 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{4} - 7 \, A b^{3} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \,{\left (b^{5} c x^{5} + b^{6} x^{4}\right )}}\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 = 1.28195, size = 223, normalized size = 1.25 \begin{align*} \frac{5 \,{\left (6 \, B b c^{2} - 7 \, A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{8 \, \sqrt{-b} b^{4}} + \frac{2 \,{\left (B b c^{2} - A c^{3}\right )}}{\sqrt{c x + b} b^{4}} + \frac{42 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{2} - 96 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{2} + 54 \, \sqrt{c x + b} B b^{3} c^{2} - 57 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{3} + 136 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{3} - 87 \, \sqrt{c x + b} A b^{2} c^{3}}{24 \, b^{4} c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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