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 = 0.03, antiderivative size = 76, normalized size of antiderivative = 1.00, 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 245
Rule 246
Rule 253
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*}
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Mathematica [A] time = 0.02, size = 76, normalized size = 1.00 \[ \frac {x \sqrt {a-b x^n} \sqrt {a+b x^n} \, _2F_1\left (-\frac {1}{2},\frac {1}{2 n};1+\frac {1}{2 n};\frac {b^2 x^{2 n}}{a^2}\right )}{\sqrt {1-\frac {b^2 x^{2 n}}{a^2}}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} \sqrt {-b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.59, size = 0, normalized size = 0.00 \[ \int \sqrt {-b \,x^{n}+a}\, \sqrt {b \,x^{n}+a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} \sqrt {-b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \sqrt {a+b\,x^n}\,\sqrt {a-b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a - b x^{n}} \sqrt {a + b x^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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