Optimal. Leaf size=76 \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]
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Rubi [A] time = 0.075931, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a - b*x^n]*Sqrt[a + b*x^n],x]
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Rubi in Sympy [A] time = 19.4373, size = 63, normalized size = 0.83 \[ \frac{x \sqrt{a - b x^{n}} \sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{2 n} \\ \frac{n + \frac{1}{2}}{n} \end{matrix}\middle |{\frac{b^{2} x^{2 n}}{a^{2}}} \right )}}{\sqrt{1 - \frac{b^{2} x^{2 n}}{a^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a-b*x**n)**(1/2)*(a+b*x**n)**(1/2),x)
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Mathematica [A] time = 0.147357, size = 116, normalized size = 1.53 \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \left (a^2 n \sqrt{1-\frac{b^2 x^{2 n}}{a^2}} \, _2F_1\left (\frac{1}{2},\frac{1}{2 n};1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )+a^2-b^2 x^{2 n}\right )}{(n+1) \left (a^2-b^2 x^{2 n}\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a - b*x^n]*Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.068, size = 0, normalized size = 0. \[ \int \sqrt{a-b{x}^{n}}\sqrt{a+b{x}^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a-b*x^n)^(1/2)*(a+b*x^n)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a} \sqrt{-b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*sqrt(-b*x^n + a),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*sqrt(-b*x^n + a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a - b x^{n}} \sqrt{a + b x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a-b*x**n)**(1/2)*(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a} \sqrt{-b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*sqrt(-b*x^n + a),x, algorithm="giac")
[Out]