Optimal. Leaf size=51 \[ -\frac{x^{m-1} \sqrt{a+b x^2} \, _2F_1\left (1,\frac{m}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right )}{a (1-m)} \]
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Rubi [A] time = 0.0199039, antiderivative size = 66, normalized size of antiderivative = 1.29, 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} \[ -\frac{x^{m-1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m+1}{2};-\frac{b x^2}{a}\right )}{(1-m) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^{-2+m}}{\sqrt{a+b x^2}} \, dx &=\frac{\sqrt{1+\frac{b x^2}{a}} \int \frac{x^{-2+m}}{\sqrt{1+\frac{b x^2}{a}}} \, dx}{\sqrt{a+b x^2}}\\ &=-\frac{x^{-1+m} \sqrt{1+\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{1}{2} (-1+m);\frac{1+m}{2};-\frac{b x^2}{a}\right )}{(1-m) \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0179246, size = 65, normalized size = 1.27 \[ \frac{x^{m-1} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m-1}{2};\frac{m-1}{2}+1;-\frac{b x^2}{a}\right )}{(m-1) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-2+m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}\, 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 \frac{x^{m - 2}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{m - 2}}{\sqrt{b x^{2} + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 26.4725, size = 53, normalized size = 1.04 \begin{align*} \frac{x^{m} \Gamma \left (\frac{m}{2} - \frac{1}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} - \frac{1}{2} \\ \frac{m}{2} + \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} x \Gamma \left (\frac{m}{2} + \frac{1}{2}\right )} \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 \frac{x^{m - 2}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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