Optimal. Leaf size=46 \[ \frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b x}{a}+1\right )}{b} \]
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Rubi [A] time = 0.010804, antiderivative size = 46, 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} \[ \frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b x}{a}+1\right )}{b} \]
Antiderivative was successfully verified.
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Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{x^m}{\sqrt{a+b x}} \, dx &=\left (x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^m}{\sqrt{a+b x}} \, dx\\ &=\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};1+\frac{b x}{a}\right )}{b}\\ \end{align*}
Mathematica [A] time = 0.0081201, size = 46, normalized size = 1. \[ \frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},-m;\frac{3}{2};\frac{b x}{a}+1\right )}{b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{{x}^{m}{\frac{1}{\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 \frac{x^{m}}{\sqrt{b x + a}}\,{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 (\frac{x^{m}}{\sqrt{b x + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 1.25598, size = 36, normalized size = 0.78 \begin{align*} \frac{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 )}}{\sqrt{a} \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 \frac{x^{m}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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