Optimal. Leaf size=80 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.232452, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{(A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{4 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^3*Sqrt[a + b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 21.8612, size = 70, normalized size = 0.88 \[ - \frac{A \sqrt{a + b x^{2} + c x^{4}}}{2 a x^{2}} + \frac{\left (A b - 2 B a\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**3/(c*x**4+b*x**2+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.221566, size = 87, normalized size = 1.09 \[ \frac{(2 a B-A b) \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+x^2 \left (b+c x^2\right )}+2 a+b x^2\right )\right )}{4 a^{3/2}}-\frac{A \sqrt{a+b x^2+c x^4}}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^3*Sqrt[a + b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.013, size = 104, normalized size = 1.3 \[ -{\frac{A}{2\,{x}^{2}a}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{Ab}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{2}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^3/(c*x^4+b*x^2+a)^(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 + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.32187, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (2 \, B a - A b\right )} x^{2} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} + a} A \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{2}}, -\frac{{\left (2 \, B a - A b\right )} x^{2} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} + a} A \sqrt{-a}}{4 \, \sqrt{-a} a x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x^{2}}{x^{3} \sqrt{a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**3/(c*x**4+b*x**2+a)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2} + a} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2 + a)*x^3),x, algorithm="giac")
[Out]