Optimal. Leaf size=65 \[ \frac {x^{m+1} \sqrt {a+b x^{m-2}} \, _2F_1\left (1,-\frac {3 m}{2 (2-m)};\frac {1-2 m}{2-m};-\frac {b x^{m-2}}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 80, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {365, 364} \[ \frac {x^{m+1} \sqrt {\frac {b x^{m-2}}{a}+1} \, _2F_1\left (\frac {1}{2},-\frac {m+1}{2-m};\frac {1-2 m}{2-m};-\frac {b x^{m-2}}{a}\right )}{(m+1) \sqrt {a+b x^{m-2}}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {x^m}{\sqrt {a+b x^{-2+m}}} \, dx &=\frac {\sqrt {1+\frac {b x^{-2+m}}{a}} \int \frac {x^m}{\sqrt {1+\frac {b x^{-2+m}}{a}}} \, dx}{\sqrt {a+b x^{-2+m}}}\\ &=\frac {x^{1+m} \sqrt {1+\frac {b x^{-2+m}}{a}} \, _2F_1\left (\frac {1}{2},-\frac {1+m}{2-m};\frac {1-2 m}{2-m};-\frac {b x^{-2+m}}{a}\right )}{(1+m) \sqrt {a+b x^{-2+m}}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 110, normalized size = 1.69 \[ \frac {2 x \left (6 a x^2 \sqrt {\frac {a x^{2-m}}{b}+1} \, _2F_1\left (\frac {1}{2},\frac {m-8}{2 (m-2)};\frac {3 (m-4)}{2 (m-2)};-\frac {a x^{2-m}}{b}\right )+(m-8) \left (a x^2+b x^m\right )\right )}{b (m-8) (m+4) \sqrt {a+b x^{m-2}}} \]
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 \[ \int \frac {x^{m}}{\sqrt {b x^{m - 2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\sqrt {b \,x^{m -2}+a}}\, dx \]
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 \[ \int \frac {x^{m}}{\sqrt {b x^{m - 2} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{\sqrt {a+b\,x^{m-2}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 7.84, size = 94, normalized size = 1.45 \[ \frac {x x^{m} \Gamma \left (\frac {m}{m - 2} + \frac {1}{m - 2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{m - 2} + \frac {1}{m - 2} \\ \frac {m}{m - 2} + 1 + \frac {1}{m - 2} \end {matrix}\middle | {\frac {b x^{m} e^{i \pi }}{a x^{2}}} \right )}}{\sqrt {a} m \Gamma \left (\frac {m}{m - 2} + 1 + \frac {1}{m - 2}\right ) - 2 \sqrt {a} \Gamma \left (\frac {m}{m - 2} + 1 + \frac {1}{m - 2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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