Optimal. Leaf size=66 \[ \frac{x^{m+2 n+1} \sqrt{a+b x^n} \, _2F_1\left (1,\frac{1}{2} \left (\frac{2 (m+1)}{n}+5\right );\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{a (m+2 n+1)} \]
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Rubi [A] time = 0.0262065, antiderivative size = 75, normalized size of antiderivative = 1.14, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {365, 364} \[ \frac{x^{m+2 n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+3 n+1}{n};-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{x^{m+2 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{x^{m+2 n}}{\sqrt{1+\frac{b x^n}{a}}} \, dx}{\sqrt{a+b x^n}}\\ &=\frac{x^{1+m+2 n} \sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{1}{2},\frac{1+m+2 n}{n};\frac{1+m+3 n}{n};-\frac{b x^n}{a}\right )}{(1+m+2 n) \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0231643, size = 77, normalized size = 1.17 \[ \frac{x^{m+2 n+1} \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{m+2 n+1}{n};\frac{m+2 n+1}{n}+1;-\frac{b x^n}{a}\right )}{(m+2 n+1) \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m+2\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, 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 \, n}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 3.81007, size = 65, normalized size = 0.98 \begin{align*} \frac{x x^{m} x^{2 n} \Gamma \left (\frac{m}{n} + 2 + \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{m}{n} + 2 + \frac{1}{n} \\ \frac{m}{n} + 3 + \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{\sqrt{a} n \Gamma \left (\frac{m}{n} + 3 + \frac{1}{n}\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 \, n}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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