Optimal. Leaf size=49 \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]
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Rubi [A] time = 0.011609, antiderivative size = 49, 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} \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 67
Rule 65
Rubi steps
\begin{align*} \int \frac{x^{-2+m}}{\sqrt{a+b x}} \, dx &=\frac{\left (b^2 x^m \left (-\frac{b x}{a}\right )^{-m}\right ) \int \frac{\left (-\frac{b x}{a}\right )^{-2+m}}{\sqrt{a+b x}} \, dx}{a^2}\\ &=\frac{2 b x^m \left (-\frac{b x}{a}\right )^{-m} \sqrt{a+b x} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};1+\frac{b x}{a}\right )}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0307771, size = 49, normalized size = 1. \[ \frac{2 b x^m \sqrt{a+b x} \left (-\frac{b x}{a}\right )^{-m} \, _2F_1\left (\frac{1}{2},2-m;\frac{3}{2};\frac{b x}{a}+1\right )}{a^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-2+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 - 2}}{\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 - 2}}{\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 = 34.9807, size = 32, normalized size = 0.65 \begin{align*} \frac{x^{m} \Gamma \left (m - 1\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, m - 1 \\ m \end{matrix}\middle |{\frac{b x e^{i \pi }}{a}} \right )}}{\sqrt{a} x \Gamma \left (m\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 - 2}}{\sqrt{b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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