Optimal. Leaf size=40 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.0096072, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {364} \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+n+1}{n};-\frac{b x^n}{a}\right )}{a (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin{align*} \int \frac{x^m}{a+b x^n} \, dx &=\frac{x^{1+m} \, _2F_1\left (1,\frac{1+m}{n};\frac{1+m+n}{n};-\frac{b x^n}{a}\right )}{a (1+m)}\\ \end{align*}
Mathematica [A] time = 0.0074785, size = 41, normalized size = 1.02 \[ \frac{x^{m+1} \, _2F_1\left (1,\frac{m+1}{n};\frac{m+1}{n}+1;-\frac{b x^n}{a}\right )}{a (m+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{a+b{x}^{n}}}\, 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 x^{n} + 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}}{b x^{n} + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.996627, size = 95, normalized size = 2.38 \begin{align*} \frac{m x x^{m} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{m}{n} + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + \frac{1}{n}\right )}{a n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\right )} + \frac{x x^{m} \Phi \left (\frac{b x^{n} e^{i \pi }}{a}, 1, \frac{m}{n} + \frac{1}{n}\right ) \Gamma \left (\frac{m}{n} + \frac{1}{n}\right )}{a n^{2} \Gamma \left (\frac{m}{n} + 1 + \frac{1}{n}\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 x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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