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 = 0.0272434, 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, Rules used = {253, 246, 245} \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{2 n};\frac{1}{2} \left (2+\frac{1}{n}\right );\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]
Antiderivative was successfully verified.
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Rule 253
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \sqrt{a-b x^n} \sqrt{a+b x^n} \, dx &=\frac{\left (\sqrt{a-b x^n} \sqrt{a+b x^n}\right ) \int \sqrt{a^2-b^2 x^{2 n}} \, dx}{\sqrt{a^2-b^2 x^{2 n}}}\\ &=\frac{\left (\sqrt{a-b x^n} \sqrt{a+b x^n}\right ) \int \sqrt{1-\frac{b^2 x^{2 n}}{a^2}} \, dx}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}}\\ &=\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}}}\\ \end{align*}
Mathematica [A] time = 0.0164294, size = 76, normalized size = 1. \[ \frac{x \sqrt{a-b x^n} \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{1}{2 n};1+\frac{1}{2 n};\frac{b^2 x^{2 n}}{a^2}\right )}{\sqrt{1-\frac{b^2 x^{2 n}}{a^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.376, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a-b{x}^{n}}\sqrt{a+b{x}^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{n} + a} \sqrt{-b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a - b x^{n}} \sqrt{a + b x^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b x^{n} + a} \sqrt{-b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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