Optimal. Leaf size=36 \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
[Out]
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Rubi [A] time = 0.0684588, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a + b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 9.65654, size = 34, normalized size = 0.94 \[ \frac{4 a + 2 b x^{2}}{2 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**4+b*x**2+a)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0346583, size = 36, normalized size = 1. \[ \frac{2 a+b x^2}{\left (b^2-4 a c\right ) \sqrt{a+b x^2+c x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a + b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 38, normalized size = 1.1 \[ -{\frac{b{x}^{2}+2\,a}{4\,ac-{b}^{2}}{\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(c*x^4+b*x^2+a)^(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(x^3/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.291994, size = 90, normalized size = 2.5 \[ \frac{\sqrt{c x^{4} + b x^{2} + a}{\left (b x^{2} + 2 \, a\right )}}{{\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(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. \[ \int \frac{x^{3}}{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(c*x**4+b*x**2+a)**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.312015, size = 59, normalized size = 1.64 \[ \frac{\frac{b x^{2}}{b^{2} - 4 \, a c} + \frac{2 \, a}{b^{2} - 4 \, a c}}{\sqrt{c x^{4} + b x^{2} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^4 + b*x^2 + a)^(3/2),x, algorithm="giac")
[Out]