Optimal. Leaf size=64 \[ \frac{x^m \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d m} \]
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Rubi [A] time = 0.0643364, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {514, 497, 135, 133} \[ \frac{x^m \left (a+\frac{b}{x}\right )^n \left (\frac{b}{a x}+1\right )^{-n} F_1\left (-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d m} \]
Antiderivative was successfully verified.
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Rule 514
Rule 497
Rule 135
Rule 133
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{c+d x} \, dx &=\int \frac{\left (a+\frac{b}{x}\right )^n x^{-1+m}}{d+\frac{c}{x}} \, dx\\ &=-\left (\left (\left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m} (a+b x)^n}{d+c x} \, dx,x,\frac{1}{x}\right )\right )\\ &=-\left (\left (\left (a+\frac{b}{x}\right )^n \left (1+\frac{b}{a x}\right )^{-n} \left (\frac{1}{x}\right )^m x^m\right ) \operatorname{Subst}\left (\int \frac{x^{-1-m} \left (1+\frac{b x}{a}\right )^n}{d+c x} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{\left (a+\frac{b}{x}\right )^n \left (1+\frac{b}{a x}\right )^{-n} x^m F_1\left (-m;-n,1;1-m;-\frac{b}{a x},-\frac{c}{d x}\right )}{d m}\\ \end{align*}
Mathematica [F] time = 0.0699305, size = 0, normalized size = 0. \[ \int \frac{\left (a+\frac{b}{x}\right )^n x^m}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.525, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{dx+c} \left ( a+{\frac{b}{x}} \right ) ^{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{{\left (a + \frac{b}{x}\right )}^{n} x^{m}}{d x + c}\,{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} \left (\frac{a x + b}{x}\right )^{n}}{d x + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m} \left (a + \frac{b}{x}\right )^{n}}{c + d x}\, dx \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{{\left (a + \frac{b}{x}\right )}^{n} x^{m}}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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