Optimal. Leaf size=25 \[ \frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m} \]
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Rubi [A] time = 0.0130205, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {647, 64} \[ \frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m} \]
Antiderivative was successfully verified.
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Rule 647
Rule 64
Rubi steps
\begin{align*} \int \frac{(d x)^m}{b x+c x^2} \, dx &=d \int \frac{(d x)^{-1+m}}{b+c x} \, dx\\ &=\frac{(d x)^m \, _2F_1\left (1,m;1+m;-\frac{c x}{b}\right )}{b m}\\ \end{align*}
Mathematica [A] time = 0.006013, size = 25, normalized size = 1. \[ \frac{(d x)^m \, _2F_1\left (1,m;m+1;-\frac{c x}{b}\right )}{b m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.374, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dx \right ) ^{m}}{c{x}^{2}+bx}}\, 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 (d x\right )^{m}}{c x^{2} + b x}\,{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{\left (d x\right )^{m}}{c x^{2} + b x}, 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{\left (d x\right )^{m}}{x \left (b + c x\right )}\, 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 (d x\right )^{m}}{c x^{2} + b x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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