Integrand size = 26, antiderivative size = 44 \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{1+m} \operatorname {Hypergeometric2F1}\left (6,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{a^6 d (1+m)} \]
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Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {28, 371} \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {(d x)^{m+1} \operatorname {Hypergeometric2F1}\left (6,\frac {m+1}{2},\frac {m+3}{2},-\frac {b x^2}{a}\right )}{a^6 d (m+1)} \]
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Rule 28
Rule 371
Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {(d x)^m}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = \frac {(d x)^{1+m} \, _2F_1\left (6,\frac {1+m}{2};\frac {3+m}{2};-\frac {b x^2}{a}\right )}{a^6 d (1+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x (d x)^m \operatorname {Hypergeometric2F1}\left (6,\frac {1+m}{2},1+\frac {1+m}{2},-\frac {b x^2}{a}\right )}{a^6 (1+m)} \]
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\[\int \frac {\left (d x \right )^{m}}{\left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{3}}d x\]
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\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {\left (d x\right )^{m}}{\left (a + b x^{2}\right )^{6}}\, dx \]
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\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]
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\[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int { \frac {\left (d x\right )^{m}}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\int \frac {{\left (d\,x\right )}^m}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^3} \,d x \]
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