Integrand size = 23, antiderivative size = 84 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
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Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {422} \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \]
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Rule 422
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c+d x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.90 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {x \left (c+d x^2\right )+\frac {i c \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \left (E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{\sqrt {\frac {b}{a}}}}{a \sqrt {a+b x^2} \sqrt {c+d x^2}} \]
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Time = 2.37 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.15
method | result | size |
default | \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) c -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) c +\sqrt {-\frac {b}{a}}\, c x \right )}{\left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) a \sqrt {-\frac {b}{a}}}\) | \(181\) |
elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) x}{b a \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d}{b}-\frac {a d -b c}{b a}-\frac {c}{a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\) | \(314\) |
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none
Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {b x^{2} + a} \sqrt {d x^{2} + c} a x - {\left (b x^{2} + a\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) + {\left (b x^{2} + a\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c})}{a^{2} b x^{2} + a^{3}} \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x^{2}}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \]
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