Optimal. Leaf size=45 \[ -\frac{(c x)^m \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m) x} \]
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Rubi [A] time = 0.0322719, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {1142, 1584, 364} \[ -\frac{(c x)^m \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (1-m) x} \]
Antiderivative was successfully verified.
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Rule 1142
Rule 1584
Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^m}{b x^2+c x^4} \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \frac{x^m}{b x^2+c x^4} \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-2+m}}{b+c x^2} \, dx,x,x\right )\\ &=-\frac{(c x)^m \, _2F_1\left (1,\frac{1}{2} (-1+m);\frac{1+m}{2};-\frac{c x^2}{b}\right )}{b (1-m) x}\\ \end{align*}
Mathematica [A] time = 0.0111761, size = 42, normalized size = 0.93 \[ \frac{(c x)^m \, _2F_1\left (1,\frac{m-1}{2};\frac{m+1}{2};-\frac{c x^2}{b}\right )}{b (m-1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{m}}{c{x}^{4}+b{x}^{2}}}\, 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 (c x\right )^{m}}{c x^{4} + b x^{2}}\,{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 (c x\right )^{m}}{c x^{4} + b x^{2}}, 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 (c x\right )^{m}}{x^{2} \left (b + c x^{2}\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 (c x\right )^{m}}{c x^{4} + b x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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