Optimal. Leaf size=19 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]
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Rubi [A] time = 0.0152488, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(a*g - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
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Rubi in Sympy [A] time = 13.0074, size = 17, normalized size = 0.89 \[ \frac{g x}{\sqrt{a + b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-c*g*x**4+a*g)/(c*x**4+b*x**2+a)**(3/2),x)
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Mathematica [A] time = 0.0451867, size = 19, normalized size = 1. \[ \frac{g x}{\sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*g - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.007, size = 18, normalized size = 1. \[{gx{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-c*g*x^4+a*g)/(c*x^4+b*x^2+a)^(3/2),x)
[Out]
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Maxima [A] time = 0.768361, size = 23, normalized size = 1.21 \[ \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.259511, size = 23, normalized size = 1.21 \[ \frac{g x}{\sqrt{c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - g \left (\int \left (- \frac{a}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx + \int \frac{c x^{4}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*g*x**4+a*g)/(c*x**4+b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.329233, size = 95, normalized size = 5. \[ \frac{{\left (b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g\right )} x}{32 \,{\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}\right )} \sqrt{c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]