Optimal. Leaf size=48 \[ \frac{x^2 \left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{3}{2}+\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 a} \]
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Rubi [A] time = 0.0144223, antiderivative size = 57, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {365, 364} \[ \frac{x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{2}{n};\frac{n+2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int x \sqrt{a+b x^n} \, dx &=\frac{\sqrt{a+b x^n} \int x \sqrt{1+\frac{b x^n}{a}} \, dx}{\sqrt{1+\frac{b x^n}{a}}}\\ &=\frac{x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{2}{n};\frac{2+n}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{1+\frac{b x^n}{a}}}\\ \end{align*}
Mathematica [A] time = 0.0185102, size = 57, normalized size = 1.19 \[ \frac{x^2 \sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},\frac{2}{n};1+\frac{2}{n};-\frac{b x^n}{a}\right )}{2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.058, size = 0, normalized size = 0. \begin{align*} \int x\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 \sqrt{b x^{n} + a} x\,{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 = 1.06585, size = 42, normalized size = 0.88 \begin{align*} \frac{\sqrt{a} x^{2} \Gamma \left (\frac{2}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{2}{n} \\ 1 + \frac{2}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac{2}{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 \sqrt{b x^{n} + a} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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