Integrand size = 24, antiderivative size = 150 \[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 (d x)^{5/2} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},\frac {3}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 a d \sqrt {a+b x^2+c x^4}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 150, 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} \[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {2 (d x)^{5/2} \sqrt {\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^2}{\sqrt {b^2-4 a c}+b}+1} \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},\frac {3}{2},\frac {9}{4},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{5 a d \sqrt {a+b x^2+c x^4}} \]
[In]
[Out]
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}}{\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}} \, dx}{a \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 {3}{2},\frac {3}{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 a d \sqrt {a+b x^2+c x^4}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(348\) vs. \(2(150)=300\).
Time = 11.26 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32 \[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {d \sqrt {d x} \left (-5 \left (b+2 c x^2\right )+5 b \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},\frac {1}{2},\frac {5}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )+2 c x^2 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^2}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},\frac {1}{2},\frac {9}{4},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}},\frac {2 c x^2}{-b+\sqrt {b^2-4 a c}}\right )\right )}{5 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}} \]
[In]
[Out]
\[\int \frac {\left (d x \right )^{\frac {3}{2}}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{\frac {3}{2}}}d x\]
[In]
[Out]
\[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (d x\right )^{\frac {3}{2}}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {\left (d x\right )^{\frac {3}{2}}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(d x)^{3/2}}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {{\left (d\,x\right )}^{3/2}}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \]
[In]
[Out]