Optimal. Leaf size=48 \[ -\frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
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Rubi [A] time = 0.0116183, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {67, 65} \[ -\frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
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Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{x^{-1+m}}{\sqrt{a+b x}} \, dx &=-\frac{\left (b x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^{-1+m}}{\sqrt{a+b x}} \, dx}{a}\\ &=-\frac{2 x^m \left (-\frac{b x}{a}\right )^{-m} \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};1+\frac{b x}{a}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0291351, size = 48, normalized size = 1. \[ -\frac{2 x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},1-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1+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 - 1}}{\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 - 1}}{\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 = 5.56003, size = 31, normalized size = 0.65 \begin{align*} \frac{x^{m} \Gamma \left (m\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m \\ m + 1 \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} \Gamma \left (m + 1\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 - 1}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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