Optimal. Leaf size=57 \[ \frac{f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{g x}{\sqrt{a+b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.132408, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ \frac{f \left (2 a+b x^2\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}}+\frac{g x}{\sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[(a*g + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 18.3178, size = 46, normalized size = 0.81 \[ \frac{2 a f + b f x^{2} + g x \left (- 4 a c + b^{2}\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-c*g*x**4+f*x**3+a*g)/(c*x**4+b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0751125, size = 48, normalized size = 0.84 \[ \frac{2 a (f-2 c g x)+b x (b g+f x)}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a*g + f*x^3 - c*g*x^4)/(a + b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 53, normalized size = 0.9 \[{\frac{4\,acgx-{b}^{2}gx-bf{x}^{2}-2\,fa}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-c*g*x^4+f*x^3+a*g)/(c*x^4+b*x^2+a)^(3/2),x)
[Out]
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Maxima [A] time = 0.771128, size = 66, normalized size = 1.16 \[ \frac{b f x^{2} + 2 \, a f +{\left (b^{2} g - 4 \, a c g\right )} x}{\sqrt{c x^{4} + b x^{2} + a}{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - f*x^3 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.281872, size = 108, normalized size = 1.89 \[ \frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b f x^{2} +{\left (b^{2} - 4 \, a c\right )} g x + 2 \, a f\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - f*x^3 - 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. \[ - \int \left (- \frac{a g}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx - \int \left (- \frac{f x^{3}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\right )\, dx - \int \frac{c g x^{4}}{a \sqrt{a + b x^{2} + c x^{4}} + b x^{2} \sqrt{a + b x^{2} + c x^{4}} + c x^{4} \sqrt{a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-c*g*x**4+f*x**3+a*g)/(c*x**4+b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.314071, size = 221, normalized size = 3.88 \[ \frac{{\left (\frac{{\left (b^{3} f - 4 \, a b c f\right )} x}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}} + \frac{b^{4} g - 8 \, a b^{2} c g + 16 \, a^{2} c^{2} g}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}\right )} x + \frac{2 \,{\left (a b^{2} f - 4 \, a^{2} c f\right )}}{a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4}}}{16 \, \sqrt{c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(c*g*x^4 - f*x^3 - a*g)/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]