Optimal. Leaf size=214 \[ \frac{35 c^2 \sqrt{x} (2 b B-3 A c)}{8 b^5 \sqrt{b x+c x^2}}-\frac{35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{11/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}-\frac{7 c \sqrt{x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.18316, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{35 c^2 \sqrt{x} (2 b B-3 A c)}{8 b^5 \sqrt{b x+c x^2}}-\frac{35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{11/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}-\frac{7 c \sqrt{x} (2 b B-3 A c)}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}} \]
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^{3/2} \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}--\frac{\left (-\frac{3}{2} (-b B+A c)-\frac{3}{2} (-b B+2 A c)\right ) \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{5/2}} \, dx}{3 b}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{(7 c (2 b B-3 A c)) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{5/2}} \, dx}{8 b^2}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{7 c (2 b B-3 A c) \sqrt{x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}-\frac{(35 c (2 b B-3 A c)) \int \frac{1}{\sqrt{x} \left (b x+c x^2\right )^{3/2}} \, dx}{24 b^3}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{7 c (2 b B-3 A c) \sqrt{x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{\left (35 c^2 (2 b B-3 A c)\right ) \int \frac{\sqrt{x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{16 b^4}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{7 c (2 b B-3 A c) \sqrt{x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{35 c^2 (2 b B-3 A c) \sqrt{x}}{8 b^5 \sqrt{b x+c x^2}}+\frac{\left (35 c^2 (2 b B-3 A c)\right ) \int \frac{1}{\sqrt{x} \sqrt{b x+c x^2}} \, dx}{16 b^5}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{7 c (2 b B-3 A c) \sqrt{x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{35 c^2 (2 b B-3 A c) \sqrt{x}}{8 b^5 \sqrt{b x+c x^2}}+\frac{\left (35 c^2 (2 b B-3 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^5}\\ &=-\frac{A}{3 b x^{3/2} \left (b x+c x^2\right )^{3/2}}-\frac{2 b B-3 A c}{4 b^2 \sqrt{x} \left (b x+c x^2\right )^{3/2}}-\frac{7 c (2 b B-3 A c) \sqrt{x}}{12 b^3 \left (b x+c x^2\right )^{3/2}}+\frac{35 c (2 b B-3 A c)}{24 b^4 \sqrt{x} \sqrt{b x+c x^2}}+\frac{35 c^2 (2 b B-3 A c) \sqrt{x}}{8 b^5 \sqrt{b x+c x^2}}-\frac{35 c^2 (2 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{11/2}}\\ \end{align*}
Mathematica [C] time = 0.0276966, size = 62, normalized size = 0.29 \[ \frac{c^2 x^3 (2 b B-3 A c) \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{c x}{b}+1\right )-A b^3}{3 b^4 x^{3/2} (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 234, normalized size = 1.1 \begin{align*}{\frac{1}{24\, \left ( cx+b \right ) ^{2}}\sqrt{x \left ( cx+b \right ) } \left ( 315\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}{c}^{4}-210\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ) \sqrt{cx+b}{x}^{4}b{c}^{3}+315\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{3}\sqrt{cx+b}-315\,A\sqrt{b}{x}^{4}{c}^{4}-210\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{b}^{2}{c}^{2}\sqrt{cx+b}+210\,B{b}^{3/2}{x}^{4}{c}^{3}-420\,A{b}^{3/2}{x}^{3}{c}^{3}+280\,B{b}^{5/2}{x}^{3}{c}^{2}-63\,A{b}^{5/2}{x}^{2}{c}^{2}+42\,B{b}^{7/2}{x}^{2}c+18\,A{b}^{7/2}xc-12\,B{b}^{9/2}x-8\,A{b}^{9/2} \right ){x}^{-{\frac{7}{2}}}{b}^{-{\frac{11}{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{5}{2}} x^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9262, size = 1044, normalized size = 4.88 \begin{align*} \left [-\frac{105 \,{\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} +{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} 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^{5} - 105 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} - 140 \,{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{3} - 21 \,{\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} x^{2} + 6 \,{\left (2 \, B b^{5} - 3 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{48 \,{\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} x^{4}\right )}}, \frac{105 \,{\left ({\left (2 \, B b c^{4} - 3 \, A c^{5}\right )} x^{6} + 2 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{5} +{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} 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^{5} - 105 \,{\left (2 \, B b^{2} c^{3} - 3 \, A b c^{4}\right )} x^{4} - 140 \,{\left (2 \, B b^{3} c^{2} - 3 \, A b^{2} c^{3}\right )} x^{3} - 21 \,{\left (2 \, B b^{4} c - 3 \, A b^{3} c^{2}\right )} x^{2} + 6 \,{\left (2 \, B b^{5} - 3 \, A b^{4} c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{x}}{24 \,{\left (b^{6} c^{2} x^{6} + 2 \, b^{7} c x^{5} + b^{8} 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 = 1.31237, size = 270, normalized size = 1.26 \begin{align*} \frac{35 \,{\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{8 \, \sqrt{-b} b^{5}} + \frac{210 \,{\left (c x + b\right )}^{4} B b c^{2} - 560 \,{\left (c x + b\right )}^{3} B b^{2} c^{2} + 462 \,{\left (c x + b\right )}^{2} B b^{3} c^{2} - 96 \,{\left (c x + b\right )} B b^{4} c^{2} - 16 \, B b^{5} c^{2} - 315 \,{\left (c x + b\right )}^{4} A c^{3} + 840 \,{\left (c x + b\right )}^{3} A b c^{3} - 693 \,{\left (c x + b\right )}^{2} A b^{2} c^{3} + 144 \,{\left (c x + b\right )} A b^{3} c^{3} + 16 \, A b^{4} c^{3}}{24 \,{\left ({\left (c x + b\right )}^{\frac{3}{2}} - \sqrt{c x + b} b\right )}^{3} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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