Optimal. Leaf size=170 \[ \frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac{b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac{\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x} \]
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Rubi [A] time = 0.129192, antiderivative size = 170, 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 = {794, 664, 612, 620, 206} \[ \frac{b^3 (b+2 c x) \sqrt{b x+c x^2} (5 b B-12 A c)}{512 c^3}-\frac{b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}-\frac{b (b+2 c x) \left (b x+c x^2\right )^{3/2} (5 b B-12 A c)}{192 c^2}-\frac{\left (b x+c x^2\right )^{5/2} (5 b B-12 A c)}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x} \]
Antiderivative was successfully verified.
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Rule 794
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} \, dx &=\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}+\frac{\left (b B-A c+\frac{7}{2} (-b B+2 A c)\right ) \int \frac{\left (b x+c x^2\right )^{5/2}}{x} \, dx}{6 c}\\ &=-\frac{(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac{(b (5 b B-12 A c)) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c}\\ &=-\frac{b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}+\frac{\left (b^3 (5 b B-12 A c)\right ) \int \sqrt{b x+c x^2} \, dx}{128 c^2}\\ &=\frac{b^3 (5 b B-12 A c) (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac{\left (b^5 (5 b B-12 A c)\right ) \int \frac{1}{\sqrt{b x+c x^2}} \, dx}{1024 c^3}\\ &=\frac{b^3 (5 b B-12 A c) (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac{\left (b^5 (5 b B-12 A c)\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{b x+c x^2}}\right )}{512 c^3}\\ &=\frac{b^3 (5 b B-12 A c) (b+2 c x) \sqrt{b x+c x^2}}{512 c^3}-\frac{b (5 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^2}-\frac{(5 b B-12 A c) \left (b x+c x^2\right )^{5/2}}{60 c}+\frac{B \left (b x+c x^2\right )^{7/2}}{6 c x}-\frac{b^5 (5 b B-12 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{512 c^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.305516, size = 166, normalized size = 0.98 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} \left (48 b^2 c^3 x^2 (62 A+45 B x)+40 b^3 c^2 x (3 A+B x)-10 b^4 c (18 A+5 B x)+64 b c^4 x^3 (63 A+50 B x)+256 c^5 x^4 (6 A+5 B x)+75 b^5 B\right )-\frac{15 b^{9/2} (5 b B-12 A c) \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{x}}{\sqrt{b}}\right )}{\sqrt{x} \sqrt{\frac{c x}{b}+1}}\right )}{7680 c^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 274, normalized size = 1.6 \begin{align*}{\frac{Bx}{6} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{bB}{12\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{b}^{2}Bx}{96\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{3}B}{192\,{c}^{2}} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{4}Bx}{256\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{5\,B{b}^{5}}{512\,{c}^{3}}\sqrt{c{x}^{2}+bx}}-{\frac{5\,B{b}^{6}}{1024}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{A}{5} \left ( c{x}^{2}+bx \right ) ^{{\frac{5}{2}}}}+{\frac{Abx}{8} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{A{b}^{2}}{16\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,A{b}^{3}x}{64\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,A{b}^{4}}{128\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,A{b}^{5}}{256}\ln \left ({ \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{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 = 1.84039, size = 830, normalized size = 4.88 \begin{align*} \left [-\frac{15 \,{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt{c} \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \,{\left (1280 \, B c^{6} x^{5} + 75 \, B b^{5} c - 180 \, A b^{4} c^{2} + 128 \,{\left (25 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 144 \,{\left (15 \, B b^{2} c^{4} + 28 \, A b c^{5}\right )} x^{3} + 8 \,{\left (5 \, B b^{3} c^{3} + 372 \, A b^{2} c^{4}\right )} x^{2} - 10 \,{\left (5 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{15360 \, c^{4}}, \frac{15 \,{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) +{\left (1280 \, B c^{6} x^{5} + 75 \, B b^{5} c - 180 \, A b^{4} c^{2} + 128 \,{\left (25 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 144 \,{\left (15 \, B b^{2} c^{4} + 28 \, A b c^{5}\right )} x^{3} + 8 \,{\left (5 \, B b^{3} c^{3} + 372 \, A b^{2} c^{4}\right )} x^{2} - 10 \,{\left (5 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt{c x^{2} + b x}}{7680 \, c^{4}}\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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15702, size = 267, normalized size = 1.57 \begin{align*} \frac{1}{7680} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, B c^{2} x + \frac{25 \, B b c^{6} + 12 \, A c^{7}}{c^{5}}\right )} x + \frac{9 \,{\left (15 \, B b^{2} c^{5} + 28 \, A b c^{6}\right )}}{c^{5}}\right )} x + \frac{5 \, B b^{3} c^{4} + 372 \, A b^{2} c^{5}}{c^{5}}\right )} x - \frac{5 \,{\left (5 \, B b^{4} c^{3} - 12 \, A b^{3} c^{4}\right )}}{c^{5}}\right )} x + \frac{15 \,{\left (5 \, B b^{5} c^{2} - 12 \, A b^{4} c^{3}\right )}}{c^{5}}\right )} + \frac{{\left (5 \, B b^{6} - 12 \, A b^{5} c\right )} \log \left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{1024 \, c^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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