Integrand size = 20, antiderivative size = 43 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c}{a x}-\frac {(b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {464, 211} \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{a^{3/2} \sqrt {b}}-\frac {c}{a x} \]
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Rule 211
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {c}{a x}-\frac {(b c-a d) \int \frac {1}{a+b x^2} \, dx}{a} \\ & = -\frac {c}{a x}-\frac {(b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=-\frac {c}{a x}+\frac {(-b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}} \]
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Time = 2.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\left (a d -b c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a \sqrt {a b}}-\frac {c}{a x}\) | \(37\) |
risch | \(-\frac {c}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{3} \textit {\_Z}^{2} b +a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{3} b +2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) x +\left (-a^{3} d +a^{2} b c \right ) \textit {\_R} \right )\right )}{2}\) | \(99\) |
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Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.44 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=\left [\frac {\sqrt {-a b} {\left (b c - a d\right )} x \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, a b c}{2 \, a^{2} b x}, -\frac {\sqrt {a b} {\left (b c - a d\right )} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + a b c}{a^{2} b x}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (36) = 72\).
Time = 0.17 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.91 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=- \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b}} \left (a d - b c\right ) \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} - \frac {c}{a x} \]
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=-\frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=-\frac {{\left (b c - a d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {c}{a x} \]
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Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \frac {c+d x^2}{x^2 \left (a+b x^2\right )} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{3/2}\,\sqrt {b}}-\frac {c}{a\,x} \]
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