Integrand size = 28, antiderivative size = 110 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=-\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \]
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Time = 0.02 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1166, 396, 223, 209} \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}}-\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}} \]
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Rule 209
Rule 223
Rule 396
Rule 1166
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {a+b x^2}{\sqrt {a-b x^2}} \, dx}{\sqrt {a^2-b^2 x^4}} \\ & = -\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {\left (3 a \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a-b x^2}} \, dx}{2 \sqrt {a^2-b^2 x^4}} \\ & = -\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {\left (3 a \sqrt {a-b x^2} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {a^2-b^2 x^4}} \\ & = -\frac {x \left (a-b x^2\right ) \sqrt {a+b x^2}}{2 \sqrt {a^2-b^2 x^4}}+\frac {3 a \sqrt {a-b x^2} \sqrt {a+b x^2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a-b x^2}}\right )}{2 \sqrt {b} \sqrt {a^2-b^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.78 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=-\frac {x \sqrt {a^2-b^2 x^4}}{2 \sqrt {a+b x^2}}+\frac {3 i a \log \left (-2 i \sqrt {b} x+\frac {2 \sqrt {a^2-b^2 x^4}}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b}} \]
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Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68
method | result | size |
default | \(\frac {\sqrt {-b^{2} x^{4}+a^{2}}\, \left (-x \sqrt {b}\, \sqrt {-b \,x^{2}+a}+3 \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) a \right )}{2 \sqrt {b \,x^{2}+a}\, \sqrt {-b \,x^{2}+a}\, \sqrt {b}}\) | \(75\) |
risch | \(-\frac {x \sqrt {-b \,x^{2}+a}\, \sqrt {\frac {-b^{2} x^{4}+a^{2}}{b \,x^{2}+a}}\, \sqrt {b \,x^{2}+a}}{2 \sqrt {-b^{2} x^{4}+a^{2}}}+\frac {3 a \arctan \left (\frac {\sqrt {b}\, x}{\sqrt {-b \,x^{2}+a}}\right ) \sqrt {\frac {-b^{2} x^{4}+a^{2}}{b \,x^{2}+a}}\, \sqrt {b \,x^{2}+a}}{2 \sqrt {b}\, \sqrt {-b^{2} x^{4}+a^{2}}}\) | \(131\) |
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none
Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.03 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\left [-\frac {2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x + 3 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {-b} \log \left (-\frac {2 \, b^{2} x^{4} + a b x^{2} - 2 \, \sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {-b} x - a^{2}}{b x^{2} + a}\right )}{4 \, {\left (b^{2} x^{2} + a b\right )}}, -\frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} b x + 3 \, {\left (a b x^{2} + a^{2}\right )} \sqrt {b} \arctan \left (\frac {\sqrt {-b^{2} x^{4} + a^{2}} \sqrt {b x^{2} + a} \sqrt {b}}{b^{2} x^{3} + a b x}\right )}{2 \, {\left (b^{2} x^{2} + a b\right )}}\right ] \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}}}{\sqrt {- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}} \,d x } \]
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\[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{\sqrt {-b^{2} x^{4} + a^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\sqrt {a^2-b^2 x^4}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}}{\sqrt {a^2-b^2\,x^4}} \,d x \]
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