Integrand size = 24, antiderivative size = 146 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=-\frac {2 a \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},-\frac {3}{2},\frac {3}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d \sqrt {d x} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}}} \]
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Time = 0.09 (sec) , antiderivative size = 146, 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} \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=-\frac {2 a \sqrt {a+b x^2+c x^4} \operatorname {AppellF1}\left (-\frac {1}{4},-\frac {3}{2},-\frac {3}{2},\frac {3}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{d \sqrt {d x} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1}} \]
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Rule 524
Rule 1155
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a+b x^2+c x^4}\right ) \int \frac {\left (1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )^{3/2}}{(d x)^{3/2}} \, dx}{\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}}}} \\ & = -\frac {2 a \sqrt {a+b x^2+c x^4} F_1\left (-\frac {1}{4};-\frac {3}{2},-\frac {3}{2};\frac {3}{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 )}{d \sqrt {d x} \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}}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(384\) vs. \(2(146)=292\).
Time = 11.46 (sec) , antiderivative size = 384, normalized size of antiderivative = 2.63 \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\frac {x \left (14 \left (-77 a^2-64 a b x^2+13 b^2 x^4-70 a c x^4+20 b c x^6+7 c^2 x^8\right )+896 a b x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},\frac {1}{2},\frac {7}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+24 \left (b^2+28 a c\right ) x^4 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},\frac {1}{2},\frac {11}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{539 (d x)^{3/2} \sqrt {a+b x^2+c x^4}} \]
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\[\int \frac {\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}{\left (d x \right )^{\frac {3}{2}}}d x\]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\int \frac {\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}{\left (d x\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\int { \frac {{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}}{\left (d x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2+c x^4\right )^{3/2}}{(d x)^{3/2}} \, dx=\int \frac {{\left (c\,x^4+b\,x^2+a\right )}^{3/2}}{{\left (d\,x\right )}^{3/2}} \,d x \]
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