Integrand size = 24, antiderivative size = 147 \[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {2 (d x)^{5/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 d \sqrt {a+b x^2+c x^4}} \]
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Time = 0.08 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1155, 524} \[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {2 (d x)^{5/2} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 d \sqrt {a+b x^2+c x^4}} \]
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Rule 524
Rule 1155
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}\right ) \int \frac {(d x)^{3/2}}{\sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \, dx}{\sqrt {a+b x^2+c x^4}} \\ & = \frac {2 (d x)^{5/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} F_1\left (\frac {5}{4};\frac {1}{2},\frac {1}{2};\frac {9}{4};-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 d \sqrt {a+b x^2+c x^4}} \\ \end{align*}
Time = 11.09 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.18 \[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {2 x (d x)^{3/2} \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )}{5 \sqrt {a+b x^2+c x^4}} \]
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\[\int \frac {\left (d x \right )^{\frac {3}{2}}}{\sqrt {c \,x^{4}+b \,x^{2}+a}}d x\]
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\[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
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\[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\sqrt {a + b x^{2} + c x^{4}}}\, dx \]
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\[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
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\[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{\sqrt {c x^{4} + b x^{2} + a}} \,d x } \]
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Timed out. \[ \int \frac {(d x)^{3/2}}{\sqrt {a+b x^2+c x^4}} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}}{\sqrt {c\,x^4+b\,x^2+a}} \,d x \]
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