Optimal. Leaf size=17 \[ x^{m+2} \sqrt{a+b x^2} \]
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Rubi [C] time = 0.0659783, antiderivative size = 127, normalized size of antiderivative = 7.47, number of steps used = 5, number of rules used = 2, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.047, Rules used = {365, 364} \[ \frac{a x^{m+2} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )}{\sqrt{a+b x^2}}+\frac{b (m+3) x^{m+4} \sqrt{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};-\frac{b x^2}{a}\right )}{(m+4) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \left (\frac{a (2+m) x^{1+m}}{\sqrt{a+b x^2}}+\frac{b (3+m) x^{3+m}}{\sqrt{a+b x^2}}\right ) \, dx &=(a (2+m)) \int \frac{x^{1+m}}{\sqrt{a+b x^2}} \, dx+(b (3+m)) \int \frac{x^{3+m}}{\sqrt{a+b x^2}} \, dx\\ &=\frac{\left (a (2+m) \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{x^{1+m}}{\sqrt{1+\frac{b x^2}{a}}} \, dx}{\sqrt{a+b x^2}}+\frac{\left (b (3+m) \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{x^{3+m}}{\sqrt{1+\frac{b x^2}{a}}} \, dx}{\sqrt{a+b x^2}}\\ &=\frac{a x^{2+m} \sqrt{1+\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{2+m}{2};\frac{4+m}{2};-\frac{b x^2}{a}\right )}{\sqrt{a+b x^2}}+\frac{b (3+m) x^{4+m} \sqrt{1+\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{2},\frac{4+m}{2};\frac{6+m}{2};-\frac{b x^2}{a}\right )}{(4+m) \sqrt{a+b x^2}}\\ \end{align*}
Mathematica [C] time = 0.0511731, size = 104, normalized size = 6.12 \[ \frac{x^{m+2} \sqrt{\frac{b x^2}{a}+1} \left (b (m+3) x^2 \, _2F_1\left (\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};-\frac{b x^2}{a}\right )+a (m+4) \, _2F_1\left (\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};-\frac{b x^2}{a}\right )\right )}{(m+4) \sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{a \left ( 2+m \right ){x}^{1+m}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{b \left ( 3+m \right ){x}^{3+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 [A] time = 1.56135, size = 22, normalized size = 1.29 \begin{align*} \sqrt{b x^{2} + a} x^{2} x^{m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57856, size = 39, normalized size = 2.29 \begin{align*} \frac{\sqrt{b x^{2} + a} x^{m + 3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 9.51591, size = 105, normalized size = 6.18 \begin{align*} \frac{\sqrt{a} x^{2} x^{m} \left (m + 2\right ) \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} + \frac{b x^{4} x^{m} \left (m + 3\right ) \Gamma \left (\frac{m}{2} + 2\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 2 \\ \frac{m}{2} + 3 \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt{a} \Gamma \left (\frac{m}{2} + 3\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{b{\left (m + 3\right )} x^{m + 3}}{\sqrt{b x^{2} + a}} + \frac{a{\left (m + 2\right )} x^{m + 1}}{\sqrt{b x^{2} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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