Optimal. Leaf size=103 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]
[Out]
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Rubi [A] time = 0.288223, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{c (4 b B-3 A c) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{8 b^{5/2}}-\frac{\sqrt{b x^2+c x^4} (4 b B-3 A c)}{8 b^2 x^3}-\frac{A \sqrt{b x^2+c x^4}}{4 b x^5} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^4*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 21.3809, size = 94, normalized size = 0.91 \[ - \frac{A \sqrt{b x^{2} + c x^{4}}}{4 b x^{5}} + \frac{\left (3 A c - 4 B b\right ) \sqrt{b x^{2} + c x^{4}}}{8 b^{2} x^{3}} - \frac{c \left (3 A c - 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{8 b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**4/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.172625, size = 133, normalized size = 1.29 \[ \frac{-\sqrt{b} \left (b+c x^2\right ) \left (2 A b-3 A c x^2+4 b B x^2\right )+c x^4 \log (x) \sqrt{b+c x^2} (3 A c-4 b B)+c x^4 \sqrt{b+c x^2} (4 b B-3 A c) \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{8 b^{5/2} x^3 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^4*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.021, size = 148, normalized size = 1.4 \[ -{\frac{1}{8\,{x}^{3}}\sqrt{c{x}^{2}+b} \left ( 4\,B\sqrt{c{x}^{2}+b}{x}^{2}{b}^{7/2}-3\,Ac\sqrt{c{x}^{2}+b}{x}^{2}{b}^{5/2}+3\,A{c}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{2}-4\,Bc\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{4}{b}^{3}+2\,A\sqrt{c{x}^{2}+b}{b}^{7/2} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^4/(c*x^4+b*x^2)^(1/2),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^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.239479, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{b} x^{5} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{16 \, b^{3} x^{5}}, -\frac{{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt{-b} x^{5} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}}{\left (2 \, A b^{2} +{\left (4 \, B b^{2} - 3 \, A b c\right )} x^{2}\right )}}{8 \, b^{3} x^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{4} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**4/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^4),x, algorithm="giac")
[Out]