Optimal. Leaf size=48 \[ \frac{2 x^m (a+b x)^{3/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{b x}{a}+1\right )}{3 b} \]
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Rubi [A] time = 0.0109918, antiderivative size = 48, 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 = {67, 65} \[ \frac{2 x^m (a+b x)^{3/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{b x}{a}+1\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 67
Rule 65
Rubi steps
\begin{align*} \int x^m \sqrt{a+b x} \, dx &=\left (x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \left (-\frac{b x}{a}\right )^m \sqrt{a+b x} \, dx\\ &=\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} (a+b x)^{3/2} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};1+\frac{b x}{a}\right )}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0298459, size = 48, normalized size = 1. \[ \frac{2 x^m (a+b x)^{3/2} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{3}{2},-m;\frac{5}{2};\frac{b x}{a}+1\right )}{3 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\sqrt{bx+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 \sqrt{b x + a} x^{m}\,{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 (\sqrt{b x + a} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.44426, size = 37, normalized size = 0.77 \begin{align*} \frac{\sqrt{a} x x^{m} \Gamma \left (m + 1\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, m + 1 \\ m + 2 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\Gamma \left (m + 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 \sqrt{b x + a} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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