Optimal. Leaf size=37 \[ \frac{x^{m+1} \, _2F_1\left (1,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.0148449, antiderivative size = 37, 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 = {341, 64} \[ \frac{x^{m+1} \, _2F_1\left (1,2 (m+1);2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a (m+1)} \]
Antiderivative was successfully verified.
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Rule 341
Rule 64
Rubi steps
\begin{align*} \int \frac{x^m}{a+b \sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^{-1+2 (1+m)}}{a+b x} \, dx,x,\sqrt{x}\right )\\ &=\frac{x^{1+m} \, _2F_1\left (1,2 (1+m);3+2 m;-\frac{b \sqrt{x}}{a}\right )}{a (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0114614, size = 36, normalized size = 0.97 \[ \frac{x^{m+1} \, _2F_1\left (1,2 m+2;2 m+3;-\frac{b \sqrt{x}}{a}\right )}{a m+a} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.019, size = 0, normalized size = 0. \begin{align*} \int{{x}^{m} \left ( a+b\sqrt{x} \right ) ^{-1}}\, 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}}{b \sqrt{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{b \sqrt{x} x^{m} - a x^{m}}{b^{2} x - a^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.852701, size = 82, normalized size = 2.22 \begin{align*} \frac{4 m x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a \Gamma \left (2 m + 3\right )} + \frac{4 x x^{m} \Phi \left (\frac{b \sqrt{x} e^{i \pi }}{a}, 1, 2 m + 2\right ) \Gamma \left (2 m + 2\right )}{a \Gamma \left (2 m + 3\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}}{b \sqrt{x} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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