Optimal. Leaf size=50 \[ \frac{x^{m+3} \sqrt{a+b x^2} \, _2F_1\left (1,\frac{m+4}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{a (m+3)} \]
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Rubi [A] time = 0.0195211, antiderivative size = 63, normalized size of antiderivative = 1.26, 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} \[ \frac{x^{m+3} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};-\frac{b x^2}{a}\right )}{(m+3) \sqrt{a+b x^2}} \]
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^{3+m} \sqrt{1+\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{3+m}{2};\frac{5+m}{2};-\frac{b x^2}{a}\right )}{(3+m) \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [A] time = 0.0167157, size = 65, normalized size = 1.3 \[ \frac{x^{m+3} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+3}{2};\frac{m+3}{2}+1;-\frac{b x^2}{a}\right )}{(m+3) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, 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 = 3.89411, size = 54, normalized size = 1.08 \begin{align*} \frac{x^{3} x^{m} \Gamma \left (\frac{m}{2} + \frac{3}{2}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + \frac{3}{2} \\ \frac{m}{2} + \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{m}{2} + \frac{5}{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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