Integrand size = 23, antiderivative size = 33 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {c x}{b}-\frac {\sqrt {a} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {21, 327, 211} \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {c x}{b}-\frac {\sqrt {a} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
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Rule 21
Rule 211
Rule 327
Rubi steps \begin{align*} \text {integral}& = c \int \frac {x^2}{a+b x^2} \, dx \\ & = \frac {c x}{b}-\frac {(a c) \int \frac {1}{a+b x^2} \, dx}{b} \\ & = \frac {c x}{b}-\frac {\sqrt {a} c \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=c \left (\frac {x}{b}-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}\right ) \]
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Time = 2.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88
method | result | size |
default | \(c \left (\frac {x}{b}-\frac {a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b \sqrt {a b}}\right )\) | \(29\) |
risch | \(\frac {c x}{b}+\frac {\sqrt {-a b}\, c \ln \left (-\sqrt {-a b}\, x -a \right )}{2 b^{2}}-\frac {\sqrt {-a b}\, c \ln \left (\sqrt {-a b}\, x -a \right )}{2 b^{2}}\) | \(59\) |
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none
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=\left [\frac {c \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 2 \, c x}{2 \, b}, -\frac {c \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - c x}{b}\right ] \]
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Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.76 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=c \left (\frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (- b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {a}{b^{3}}} \log {\left (b \sqrt {- \frac {a}{b^{3}}} + x \right )}}{2} + \frac {x}{b}\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {c x}{b} \]
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Time = 0.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=-\frac {a c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {c x}{b} \]
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Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76 \[ \int \frac {x^2 \left (a c+b c x^2\right )}{\left (a+b x^2\right )^2} \, dx=\frac {c\,x}{b}-\frac {\sqrt {a}\,c\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{b^{3/2}} \]
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