Optimal. Leaf size=170 \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{9/2}}+\frac{5 b \sqrt{b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac{5 x \sqrt{b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac{x^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac{2 x^4 (b B-A c)}{b c \sqrt{b x+c x^2}} \]
[Out]
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Rubi [A] time = 0.378901, 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 \[ -\frac{5 b^2 (7 b B-6 A c) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{8 c^{9/2}}+\frac{5 b \sqrt{b x+c x^2} (7 b B-6 A c)}{8 c^4}-\frac{5 x \sqrt{b x+c x^2} (7 b B-6 A c)}{12 c^3}+\frac{x^2 \sqrt{b x+c x^2} (7 b B-6 A c)}{3 b c^2}-\frac{2 x^4 (b B-A c)}{b c \sqrt{b x+c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 25.2197, size = 162, normalized size = 0.95 \[ \frac{5 b^{2} \left (6 A c - 7 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{8 c^{\frac{9}{2}}} - \frac{5 b \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{8 c^{4}} + \frac{5 x \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{12 c^{3}} + \frac{2 x^{4} \left (A c - B b\right )}{b c \sqrt{b x + c x^{2}}} - \frac{x^{2} \left (6 A c - 7 B b\right ) \sqrt{b x + c x^{2}}}{3 b c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(B*x+A)/(c*x**2+b*x)**(3/2),x)
[Out]
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Mathematica [A] time = 0.209109, size = 132, normalized size = 0.78 \[ \frac{\sqrt{c} x \left (b^2 (35 B c x-90 A c)-2 b c^2 x (15 A+7 B x)+4 c^3 x^2 (3 A+2 B x)+105 b^3 B\right )-15 b^2 \sqrt{x} \sqrt{b+c x} (7 b B-6 A c) \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{24 c^{9/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(A + B*x))/(b*x + c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.012, size = 215, normalized size = 1.3 \[{\frac{A{x}^{3}}{2\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{5\,Ab{x}^{2}}{4\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{15\,Ax{b}^{2}}{4\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{15\,{b}^{2}A}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{7}{2}}}}+{\frac{{x}^{4}B}{3\,c}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{7\,Bb{x}^{3}}{12\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{35\,{b}^{2}B{x}^{2}}{24\,{c}^{3}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}+{\frac{35\,Bx{b}^{3}}{8\,{c}^{4}}{\frac{1}{\sqrt{c{x}^{2}+bx}}}}-{\frac{35\,B{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(B*x+A)/(c*x^2+b*x)^(3/2),x)
[Out]
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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((B*x + A)*x^4/(c*x^2 + b*x)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.300072, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} \sqrt{c x^{2} + b x} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) - 2 \,{\left (8 \, B c^{3} x^{4} - 2 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} + 5 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt{c}}{48 \, \sqrt{c x^{2} + b x} c^{\frac{9}{2}}}, -\frac{15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} \sqrt{c x^{2} + b x} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) -{\left (8 \, B c^{3} x^{4} - 2 \,{\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} + 5 \,{\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 15 \,{\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt{-c}}{24 \, \sqrt{c x^{2} + b x} \sqrt{-c} c^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^4/(c*x^2 + b*x)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4*(B*x+A)/(c*x**2+b*x)**(3/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*x^4/(c*x^2 + b*x)^(3/2),x, algorithm="giac")
[Out]