Integrand size = 19, antiderivative size = 45 \[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {(c x)^m \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} (-5+m),\frac {1}{2} (-3+m),-\frac {c x^2}{b}\right )}{b^3 (5-m) x^5} \]
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Time = 0.02 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1156, 1598, 371} \[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=-\frac {(c x)^m \operatorname {Hypergeometric2F1}\left (3,\frac {m-5}{2},\frac {m-3}{2},-\frac {c x^2}{b}\right )}{b^3 (5-m) x^5} \]
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Rule 371
Rule 1156
Rule 1598
Rubi steps \begin{align*} \text {integral}& = \left (x^{-m} (c x)^m\right ) \text {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 ) \text {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*}
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\frac {(c x)^m \operatorname {Hypergeometric2F1}\left (3,\frac {1}{2} (-5+m),1+\frac {1}{2} (-5+m),-\frac {c x^2}{b}\right )}{b^3 (-5+m) x^5} \]
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\[\int \frac {\left (c x \right )^{m}}{\left (c \,x^{4}+b \,x^{2}\right )^{3}}d x\]
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\[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (c x^{4} + b x^{2}\right )}^{3}} \,d x } \]
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\[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\int \frac {\left (c x\right )^{m}}{x^{6} \left (b + c x^{2}\right )^{3}}\, dx \]
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\[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (c x^{4} + b x^{2}\right )}^{3}} \,d x } \]
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\[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\int { \frac {\left (c x\right )^{m}}{{\left (c x^{4} + b x^{2}\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(c x)^m}{\left (b x^2+c x^4\right )^3} \, dx=\int \frac {{\left (c\,x\right )}^m}{{\left (c\,x^4+b\,x^2\right )}^3} \,d x \]
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