Optimal. Leaf size=45 \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
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Rubi [A] time = 0.0087533, antiderivative size = 45, normalized size of antiderivative = 1., 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 = {66, 64} \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 66
Rule 64
Rubi steps
\begin{align*} \int \sqrt{x} (a+b x)^n \, dx &=\left ((a+b x)^n \left (1+\frac{b x}{a}\right )^{-n}\right ) \int \sqrt{x} \left (1+\frac{b x}{a}\right )^n \, dx\\ &=\frac{2}{3} x^{3/2} (a+b x)^n \left (1+\frac{b x}{a}\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0074904, size = 45, normalized size = 1. \[ \frac{2}{3} x^{3/2} (a+b x)^n \left (\frac{b x}{a}+1\right )^{-n} \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};-\frac{b x}{a}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \left ( bx+a \right ) ^{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{\left (b x + a\right )}^{n} \sqrt{x}\,{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 ({\left (b x + a\right )}^{n} \sqrt{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 13.8212, size = 27, normalized size = 0.6 \begin{align*} \frac{2 a^{n} x^{\frac{3}{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - n \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{3} \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{\left (b x + a\right )}^{n} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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