Optimal. Leaf size=155 \[ -\frac{5 b^3 (b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (b B-8 A c)}{4 b x}+\frac{5}{24} \left (b x+c x^2\right )^{3/2} (b B-8 A c)+\frac{5 b (b+2 c x) \sqrt{b x+c x^2} (b B-8 A c)}{64 c}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3} \]
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Rubi [A] time = 0.172502, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {792, 664, 612, 620, 206} \[ -\frac{5 b^3 (b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2}}+\frac{\left (b x+c x^2\right )^{5/2} (b B-8 A c)}{4 b x}+\frac{5}{24} \left (b x+c x^2\right )^{3/2} (b B-8 A c)+\frac{5 b (b+2 c x) \sqrt{b x+c x^2} (b B-8 A c)}{64 c}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3} \]
Antiderivative was successfully verified.
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Rule 792
Rule 664
Rule 612
Rule 620
Rule 206
Rubi steps
\begin{align*} \int \frac{(A+B x) \left (b x+c x^2\right )^{5/2}}{x^3} \, dx &=\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}-\frac{\left (2 \left (-3 (-b B+A c)+\frac{7}{2} (-b B+2 A c)\right )\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x^2} \, dx}{b}\\ &=\frac{(b B-8 A c) \left (b x+c x^2\right )^{5/2}}{4 b x}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}+\frac{1}{8} (5 (b B-8 A c)) \int \frac{\left (b x+c x^2\right )^{3/2}}{x} \, dx\\ &=\frac{5}{24} (b B-8 A c) \left (b x+c x^2\right )^{3/2}+\frac{(b B-8 A c) \left (b x+c x^2\right )^{5/2}}{4 b x}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}+\frac{1}{16} (5 b (b B-8 A c)) \int \sqrt{b x+c x^2} \, dx\\ &=\frac{5 b (b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c}+\frac{5}{24} (b B-8 A c) \left (b x+c x^2\right )^{3/2}+\frac{(b B-8 A c) \left (b x+c x^2\right )^{5/2}}{4 b x}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}-\frac{\left (5 b^3 (b B-8 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{128 c}\\ &=\frac{5 b (b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c}+\frac{5}{24} (b B-8 A c) \left (b x+c x^2\right )^{3/2}+\frac{(b B-8 A c) \left (b x+c x^2\right )^{5/2}}{4 b x}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}-\frac{\left (5 b^3 (b B-8 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{64 c}\\ &=\frac{5 b (b B-8 A c) (b+2 c x) \sqrt{b x+c x^2}}{64 c}+\frac{5}{24} (b B-8 A c) \left (b x+c x^2\right )^{3/2}+\frac{(b B-8 A c) \left (b x+c x^2\right )^{5/2}}{4 b x}+\frac{2 A \left (b x+c x^2\right )^{7/2}}{b x^3}-\frac{5 b^3 (b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.235362, size = 128, normalized size = 0.83 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (2 b^2 c (132 A+59 B x)+8 b c^2 x (26 A+17 B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )-\frac{15 b^{5/2} (b B-8 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{192 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.011, size = 306, normalized size = 2. \begin{align*} 2\,{\frac{A \left ( c{x}^{2}+bx \right ) ^{7/2}}{b{x}^{3}}}-{\frac{16\,Ac}{3\,{b}^{2}{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}+{\frac{16\,A{c}^{2}}{3\,{b}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{10\,A{c}^{2}x}{3\,b} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ac}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Abcx}{4}\sqrt{c{x}^{2}+bx}}-{\frac{5\,A{b}^{2}}{8}\sqrt{c{x}^{2}+bx}}+{\frac{5\,A{b}^{3}}{16}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}}+{\frac{2\,B}{3\,b{x}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{7}{2}}}}-{\frac{2\,Bc}{3\,b} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,Bcx}{12} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,bB}{24} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32}\sqrt{c{x}^{2}+bx}}+{\frac{5\,{b}^{3}B}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{5\,{b}^{4}B}{128}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13837, size = 597, normalized size = 3.85 \begin{align*} \left [-\frac{15 \,{\left (B b^{4} - 8 \, A b^{3} c\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (48 \, B c^{4} x^{3} + 15 \, B b^{3} c + 264 \, A b^{2} c^{2} + 8 \,{\left (17 \, B b c^{3} + 8 \, A c^{4}\right )} x^{2} + 2 \,{\left (59 \, B b^{2} c^{2} + 104 \, A b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{384 \, c^{2}}, \frac{15 \,{\left (B b^{4} - 8 \, A b^{3} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (48 \, B c^{4} x^{3} + 15 \, B b^{3} c + 264 \, A b^{2} c^{2} + 8 \,{\left (17 \, B b c^{3} + 8 \, A c^{4}\right )} x^{2} + 2 \,{\left (59 \, B b^{2} c^{2} + 104 \, A b c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{192 \, c^{2}}\right ] \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{\left (x \left (b + c x\right )\right )^{\frac{5}{2}} \left (A + B x\right )}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1994, size = 190, normalized size = 1.23 \begin{align*} \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B c^{2} x + \frac{17 \, B b c^{4} + 8 \, A c^{5}}{c^{3}}\right )} x + \frac{59 \, B b^{2} c^{3} + 104 \, A b c^{4}}{c^{3}}\right )} x + \frac{3 \,{\left (5 \, B b^{3} c^{2} + 88 \, A b^{2} c^{3}\right )}}{c^{3}}\right )} + \frac{5 \,{\left (B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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