Optimal. Leaf size=45 \[ -\frac{(c x)^m \, _2F_1\left (3,\frac{m-5}{2};\frac{m-3}{2};-\frac{c x^2}{b}\right )}{b^3 (5-m) x^5} \]
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Rubi [A] time = 0.0314853, 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} \[ -\frac{(c x)^m \, _2F_1\left (3,\frac{m-5}{2};\frac{m-3}{2};-\frac{c x^2}{b}\right )}{b^3 (5-m) x^5} \]
Antiderivative was successfully verified.
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Rule 1142
Rule 1584
Rule 364
Rubi steps
\begin{align*} \int \frac{(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \frac{x^m}{\left (b x^2+c x^4\right )^3} \, dx,x,x\right )\\ &=\left (x^{-m} (c x)^m\right ) \operatorname{Subst}\left (\int \frac{x^{-6+m}}{\left (b+c x^2\right )^3} \, dx,x,x\right )\\ &=-\frac{(c x)^m \, _2F_1\left (3,\frac{1}{2} (-5+m);\frac{1}{2} (-3+m);-\frac{c x^2}{b}\right )}{b^3 (5-m) x^5}\\ \end{align*}
Mathematica [A] time = 0.0127925, size = 44, normalized size = 0.98 \[ \frac{(c x)^m \, _2F_1\left (3,\frac{m-5}{2};\frac{m-5}{2}+1;-\frac{c x^2}{b}\right )}{b^3 (m-5) x^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.345, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( cx \right ) ^{m}}{ \left ( c{x}^{4}+b{x}^{2} \right ) ^{3}}}\, 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}}{{\left (c x^{4} + b x^{2}\right )}^{3}}\,{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^{3} x^{12} + 3 \, b c^{2} x^{10} + 3 \, b^{2} c x^{8} + b^{3} x^{6}}, 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^{6} \left (b + c x^{2}\right )^{3}}\, 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}}{{\left (c x^{4} + b x^{2}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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