Integrand size = 37, antiderivative size = 69 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\frac {\sqrt {b^2+a^2 x^4}}{x}+\sqrt {2} \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \]
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Time = 0.65 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {6857, 283, 311, 226, 1210, 1223, 1225, 1713, 211} \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\sqrt {2} \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right )+\frac {\sqrt {a^2 x^4+b^2}}{x} \]
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Rule 211
Rule 226
Rule 283
Rule 311
Rule 1210
Rule 1223
Rule 1225
Rule 1713
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {\sqrt {b^2+a^2 x^4}}{x^2}+\frac {2 a \sqrt {b^2+a^2 x^4}}{b+a x^2}\right ) \, dx \\ & = (2 a) \int \frac {\sqrt {b^2+a^2 x^4}}{b+a x^2} \, dx-\int \frac {\sqrt {b^2+a^2 x^4}}{x^2} \, dx \\ & = \frac {\sqrt {b^2+a^2 x^4}}{x}-\frac {2 \int \frac {a^2 b-a^3 x^2}{\sqrt {b^2+a^2 x^4}} \, dx}{a}-\left (2 a^2\right ) \int \frac {x^2}{\sqrt {b^2+a^2 x^4}} \, dx+\left (4 a b^2\right ) \int \frac {1}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx \\ & = \frac {\sqrt {b^2+a^2 x^4}}{x}+\frac {2 a x \sqrt {b^2+a^2 x^4}}{b+a x^2}-\frac {2 \sqrt {a} \sqrt {b} \left (b+a x^2\right ) \sqrt {\frac {b^2+a^2 x^4}{\left (b+a x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )|\frac {1}{2}\right )}{\sqrt {b^2+a^2 x^4}}+(2 a b) \int \frac {b-a x^2}{\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}} \, dx+(2 a b) \int \frac {1-\frac {a x^2}{b}}{\sqrt {b^2+a^2 x^4}} \, dx \\ & = \frac {\sqrt {b^2+a^2 x^4}}{x}+\left (2 a b^2\right ) \text {Subst}\left (\int \frac {1}{b+2 a b^2 x^2} \, dx,x,\frac {x}{\sqrt {b^2+a^2 x^4}}\right ) \\ & = \frac {\sqrt {b^2+a^2 x^4}}{x}+\sqrt {2} \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\frac {\sqrt {b^2+a^2 x^4}}{x}+\sqrt {2} \sqrt {a} \sqrt {b} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \]
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Time = 2.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {2}}{x}-\frac {2 a b \arctan \left (\frac {\sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{\sqrt {a b}}\right ) \sqrt {2}}{2}\) | \(63\) |
risch | \(\frac {\sqrt {x^{4} a^{2}+b^{2}}}{x}+\frac {b a \sqrt {2}\, \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a b x +\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}\right ) a}{a \,x^{2}+b}\right )\right )}{\sqrt {-a b}}\) | \(75\) |
default | \(\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}+2 \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a b x +\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}\right ) a}{a \,x^{2}+b}\right )\right ) x b a \right )}{2 \sqrt {-a b}\, x}\) | \(88\) |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}+2 \left (\ln \left (2\right )+\ln \left (\frac {\left (-2 a b x +\sqrt {2}\, \sqrt {-a b}\, \sqrt {x^{4} a^{2}+b^{2}}\right ) a}{a \,x^{2}+b}\right )\right ) x b a \right )}{2 \sqrt {-a b}\, x}\) | \(88\) |
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Time = 0.73 (sec) , antiderivative size = 162, normalized size of antiderivative = 2.35 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\left [\frac {\sqrt {2} \sqrt {-a b} x \log \left (\frac {a^{2} x^{4} - 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {-a b} x + b^{2}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{4} + b^{2}}}{2 \, x}, -\frac {\sqrt {2} \sqrt {a b} x \arctan \left (\frac {\sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {a b}}{2 \, a b x}\right ) - \sqrt {a^{2} x^{4} + b^{2}}}{x}\right ] \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\int \frac {\left (a x^{2} - b\right ) \sqrt {a^{2} x^{4} + b^{2}}}{x^{2} \left (a x^{2} + b\right )}\, dx \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]
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\[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (b+a x^2\right )} \, dx=\int -\frac {\sqrt {a^2\,x^4+b^2}\,\left (b-a\,x^2\right )}{x^2\,\left (a\,x^2+b\right )} \,d x \]
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