Optimal. Leaf size=132 \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]
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Rubi [A] time = 0.240908, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{b^3 (3 b B-8 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{64 c^{5/2}}-\frac{b (b+2 c x) \sqrt{b x+c x^2} (3 b B-8 A c)}{64 c^2}-\frac{\left (b x+c x^2\right )^{3/2} (3 b B-8 A c)}{24 c}+\frac{B \left (b x+c x^2\right )^{5/2}}{4 c x} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x,x]
[Out]
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Rubi in Sympy [A] time = 15.2428, size = 119, normalized size = 0.9 \[ \frac{B \left (b x + c x^{2}\right )^{\frac{5}{2}}}{4 c x} - \frac{b^{3} \left (8 A c - 3 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{64 c^{\frac{5}{2}}} + \frac{b \left (b + 2 c x\right ) \left (8 A c - 3 B b\right ) \sqrt{b x + c x^{2}}}{64 c^{2}} + \frac{\left (8 A c - 3 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{24 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.228852, size = 130, normalized size = 0.98 \[ \frac{\sqrt{x (b+c x)} \left (\frac{3 b^3 (3 b B-8 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}+\sqrt{c} \left (6 b^2 c (4 A+B x)+8 b c^2 x (14 A+9 B x)+16 c^3 x^2 (4 A+3 B x)-9 b^3 B\right )\right )}{192 c^{5/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x,x]
[Out]
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Maple [A] time = 0.01, size = 192, normalized size = 1.5 \[{\frac{Bx}{4} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{8\,c} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{b}^{2}Bx}{32\,c}\sqrt{c{x}^{2}+bx}}-{\frac{3\,B{b}^{3}}{64\,{c}^{2}}\sqrt{c{x}^{2}+bx}}+{\frac{3\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{5}{2}}}}+{\frac{A}{3} \left ( c{x}^{2}+bx \right ) ^{{\frac{3}{2}}}}+{\frac{Abx}{4}\sqrt{c{x}^{2}+bx}}+{\frac{{b}^{2}A}{8\,c}\sqrt{c{x}^{2}+bx}}-{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(c*x^2+b*x)^(3/2)/x,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284719, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (48 \, B c^{3} x^{3} - 9 \, B b^{3} + 24 \, A b^{2} c + 8 \,{\left (9 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + 56 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{c} - 3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right )}{384 \, c^{\frac{5}{2}}}, \frac{{\left (48 \, B c^{3} x^{3} - 9 \, B b^{3} + 24 \, A b^{2} c + 8 \,{\left (9 \, B b c^{2} + 8 \, A c^{3}\right )} x^{2} + 2 \,{\left (3 \, B b^{2} c + 56 \, A b c^{2}\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-c} + 3 \,{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right )}{192 \, \sqrt{-c} c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x \left (b + c x\right )\right )^{\frac{3}{2}} \left (A + B x\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.285228, size = 186, normalized size = 1.41 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x}{\left (2 \,{\left (4 \,{\left (6 \, B c x + \frac{9 \, B b c^{3} + 8 \, A c^{4}}{c^{3}}\right )} x + \frac{3 \, B b^{2} c^{2} + 56 \, A b c^{3}}{c^{3}}\right )} x - \frac{3 \,{\left (3 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )}}{c^{3}}\right )} - \frac{{\left (3 \, B b^{4} - 8 \, A b^{3} c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x,x, algorithm="giac")
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