Integrand size = 20, antiderivative size = 173 \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d (1+m)}-\frac {2 c (d x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m)} \]
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Time = 0.17 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1145, 371} \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right )}-\frac {2 c (d x)^{m+1} \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{2},\frac {m+3}{2},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d (m+1) \sqrt {b^2-4 a c} \left (\sqrt {b^2-4 a c}+b\right )} \]
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Rule 371
Rule 1145
Rubi steps \begin{align*} \text {integral}& = \frac {c \int \frac {(d x)^m}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {c \int \frac {(d x)^m}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}} \\ & = \frac {2 c (d x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b-\sqrt {b^2-4 a c}\right ) d (1+m)}-\frac {2 c (d x)^{1+m} \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (b+\sqrt {b^2-4 a c}\right ) d (1+m)} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 5 in optimal.
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.47 \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\frac {(d x)^m \text {RootSum}\left [a+b \text {$\#$1}^2+c \text {$\#$1}^4\&,\frac {\operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {\text {$\#$1}}{x-\text {$\#$1}}\right ) \left (\frac {x}{x-\text {$\#$1}}\right )^{-m}}{b \text {$\#$1}+2 c \text {$\#$1}^3}\&\right ]}{2 m} \]
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\[\int \frac {\left (d x \right )^{m}}{c \,x^{4}+b \,x^{2}+a}d x\]
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\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \]
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\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int \frac {\left (d x\right )^{m}}{a + b x^{2} + c x^{4}}\, dx \]
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\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \]
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\[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int { \frac {\left (d x\right )^{m}}{c x^{4} + b x^{2} + a} \,d x } \]
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Timed out. \[ \int \frac {(d x)^m}{a+b x^2+c x^4} \, dx=\int \frac {{\left (d\,x\right )}^m}{c\,x^4+b\,x^2+a} \,d x \]
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