Integrand size = 23, antiderivative size = 262 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{c^2 \sqrt {c+\frac {d}{x^2}} x}-\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c^2}+\frac {2 \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {a+\frac {b}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}} \]
[Out]
Time = 0.21 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {382, 480, 597, 545, 429, 506, 422} \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\frac {2 \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{c^2 x \sqrt {c+\frac {d}{x^2}}}+\frac {2 x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}}{c^2}-\frac {x \sqrt {a+\frac {b}{x^2}}}{c \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {a+\frac {b}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]
[In]
[Out]
Rule 382
Rule 422
Rule 429
Rule 480
Rule 506
Rule 545
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {\text {Subst}\left (\int \frac {-2 a-b x^2}{x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c^2}-\frac {\text {Subst}\left (\int \frac {a b c+2 a b d x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{a c^2} \\ & = -\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c^2}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{c}-\frac {(2 b d) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = -\frac {2 d \sqrt {a+\frac {b}{x^2}}}{c^2 \sqrt {c+\frac {d}{x^2}} x}-\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c^2}-\frac {b \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}+\frac {(2 d) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {2 d \sqrt {a+\frac {b}{x^2}}}{c^2 \sqrt {c+\frac {d}{x^2}} x}-\frac {\sqrt {a+\frac {b}{x^2}} x}{c \sqrt {c+\frac {d}{x^2}}}+\frac {2 \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x}{c^2}+\frac {2 \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{c^{3/2} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}-\frac {b \sqrt {a+\frac {b}{x^2}} F\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 3.41 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \left (\sqrt {\frac {a}{b}} c x \left (b+a x^2\right )+2 i a d \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+i (b c-2 a d) \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right ),\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} c^2 \sqrt {c+\frac {d}{x^2}} \left (b+a x^2\right )} \]
[In]
[Out]
Time = 3.47 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (\sqrt {-\frac {c}{d}}\, a \,x^{3}+b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, F\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )-2 b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, E\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {c}{d}}\, b x \right ) \left (c \,x^{2}+d \right )}{x^{2} \left (a \,x^{2}+b \right ) \sqrt {-\frac {c}{d}}\, c \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}}}\) | \(185\) |
[In]
[Out]
none
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=-\frac {2 \, {\left (b c x^{2} + b d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a + 2 \, b\right )} c x^{2} + {\left (a + 2 \, b\right )} d\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (\frac {\sqrt {-\frac {b}{a}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (a c x^{3} + 2 \, a d x\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{a c^{3} x^{2} + a c^{2} d} \]
[In]
[Out]
\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{2}}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int { \frac {\sqrt {a + \frac {b}{x^{2}}}}{{\left (c + \frac {d}{x^{2}}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx=\int \frac {\sqrt {a+\frac {b}{x^2}}}{{\left (c+\frac {d}{x^2}\right )}^{3/2}} \,d x \]
[In]
[Out]