Optimal. Leaf size=31 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]
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Rubi [A] time = 0.039516, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[x/Sqrt[a - b*x^4],x]
[Out]
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Rubi in Sympy [A] time = 4.47612, size = 26, normalized size = 0.84 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} \right )}}{2 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(-b*x**4+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0121114, size = 31, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}\right )}{2 \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Integrate[x/Sqrt[a - b*x^4],x]
[Out]
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Maple [A] time = 0.008, size = 24, normalized size = 0.8 \[{\frac{1}{2}\arctan \left ({{x}^{2}\sqrt{b}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \right ){\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(-b*x^4+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(x/sqrt(-b*x^4 + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247086, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (2 \, \sqrt{-b x^{4} + a} b x^{2} +{\left (2 \, b x^{4} - a\right )} \sqrt{-b}\right )}{4 \, \sqrt{-b}}, \frac{\arctan \left (\frac{\sqrt{b} x^{2}}{\sqrt{-b x^{4} + a}}\right )}{2 \, \sqrt{b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(-b*x^4 + a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.75775, size = 53, normalized size = 1.71 \[ \begin{cases} - \frac{i \operatorname{acosh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{for}\: \left |{\frac{b x^{4}}{a}}\right | > 1 \\\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(-b*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.223236, size = 41, normalized size = 1.32 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{-b} x^{2} + \sqrt{-b x^{4} + a} \right |}\right )}{2 \, \sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/sqrt(-b*x^4 + a),x, algorithm="giac")
[Out]