Integrand size = 19, antiderivative size = 63 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^3}{3 b}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {398, 211} \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^2}{\sqrt {a} b^{5/2}}+\frac {d x (2 b c-a d)}{b^2}+\frac {d^2 x^3}{3 b} \]
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Rule 211
Rule 398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d (2 b c-a d)}{b^2}+\frac {d^2 x^2}{b}+\frac {b^2 c^2-2 a b c d+a^2 d^2}{b^2 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^3}{3 b}+\frac {(b c-a d)^2 \int \frac {1}{a+b x^2} \, dx}{b^2} \\ & = \frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^3}{3 b}+\frac {(b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.94 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {d x \left (6 b c-3 a d+b d x^2\right )}{3 b^2}+\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} b^{5/2}} \]
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Time = 2.71 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {d \left (-\frac {1}{3} b d \,x^{3}+a d x -2 b c x \right )}{b^{2}}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{2} \sqrt {a b}}\) | \(64\) |
risch | \(\frac {d^{2} x^{3}}{3 b}-\frac {d^{2} a x}{b^{2}}+\frac {2 d c x}{b}-\frac {\ln \left (b x +\sqrt {-a b}\right ) a^{2} d^{2}}{2 b^{2} \sqrt {-a b}}+\frac {\ln \left (b x +\sqrt {-a b}\right ) a c d}{b \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) c^{2}}{2 \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) a^{2} d^{2}}{2 b^{2} \sqrt {-a b}}-\frac {\ln \left (-b x +\sqrt {-a b}\right ) a c d}{b \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) c^{2}}{2 \sqrt {-a b}}\) | \(183\) |
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Time = 0.24 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.87 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\left [\frac {2 \, a b^{2} d^{2} x^{3} - 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{6 \, a b^{3}}, \frac {a b^{2} d^{2} x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (2 \, a b^{2} c d - a^{2} b d^{2}\right )} x}{3 \, a b^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (56) = 112\).
Time = 0.25 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.73 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=x \left (- \frac {a d^{2}}{b^{2}} + \frac {2 c d}{b}\right ) - \frac {\sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2} \log {\left (- \frac {a b^{2} \sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2} \log {\left (\frac {a b^{2} \sqrt {- \frac {1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x^{3}}{3 b} \]
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Time = 0.30 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b d^{2} x^{3} + 3 \, {\left (2 \, b c d - a d^{2}\right )} x}{3 \, b^{2}} \]
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Time = 0.27 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} d^{2} x^{3} + 6 \, b^{2} c d x - 3 \, a b d^{2} x}{3 \, b^{3}} \]
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Time = 5.14 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.43 \[ \int \frac {\left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {d^2\,x^3}{3\,b}-x\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{\sqrt {a}\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )\,{\left (a\,d-b\,c\right )}^2}{\sqrt {a}\,b^{5/2}} \]
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