3.183 \(\int \sqrt{a-b x^n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=76 \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]

[Out]

(x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-1/2, 1/(2*n), (2 + n^(-1))
/2, (b^2*x^(2*n))/a^2])/Sqrt[1 - (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]

[Out]

(x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*Hypergeometric2F1[-1/2, 1/(2*n), (2 + n^(-1))
/2, (b^2*x^(2*n))/a^2])/Sqrt[1 - (b^2*x^(2*n))/a^2]

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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)

[Out]

x*sqrt(a - b*x**n)*sqrt(a + b*x**n)*hyper((-1/2, 1/(2*n)), ((n + 1/2)/n,), b**2*
x**(2*n)/a**2)/sqrt(1 - b**2*x**(2*n)/a**2)

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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]

(x*Sqrt[a - b*x^n]*Sqrt[a + b*x^n]*(a^2 - b^2*x^(2*n) + a^2*n*Sqrt[1 - (b^2*x^(2
*n))/a^2]*Hypergeometric2F1[1/2, 1/(2*n), 1 + 1/(2*n), (b^2*x^(2*n))/a^2]))/((1
+ n)*(a^2 - b^2*x^(2*n)))

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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)

[Out]

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")

[Out]

integrate(sqrt(b*x^n + a)*sqrt(-b*x^n + a), x)

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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]

Exception raised: TypeError

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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]

Integral(sqrt(a - b*x**n)*sqrt(a + b*x**n), x)

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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]

integrate(sqrt(b*x^n + a)*sqrt(-b*x^n + a), x)