Integrand size = 22, antiderivative size = 84 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^5}{5 b}-\frac {\sqrt {a} (b c-a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {472, 211} \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)^2}{b^{7/2}}+\frac {x (b c-a d)^2}{b^3}+\frac {d x^3 (2 b c-a d)}{3 b^2}+\frac {d^2 x^5}{5 b} \]
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Rule 211
Rule 472
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^2}{b^3}+\frac {d (2 b c-a d) x^2}{b^2}+\frac {d^2 x^4}{b}+\frac {-a b^2 c^2+2 a^2 b c d-a^3 d^2}{b^3 \left (a+b x^2\right )}\right ) \, dx \\ & = \frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^5}{5 b}-\frac {\left (a (b c-a d)^2\right ) \int \frac {1}{a+b x^2} \, dx}{b^3} \\ & = \frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^5}{5 b}-\frac {\sqrt {a} (b c-a d)^2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\frac {(b c-a d)^2 x}{b^3}+\frac {d (2 b c-a d) x^3}{3 b^2}+\frac {d^2 x^5}{5 b}-\frac {\sqrt {a} (-b c+a d)^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{7/2}} \]
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Time = 2.65 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.21
method | result | size |
default | \(\frac {\frac {1}{5} b^{2} d^{2} x^{5}-\frac {1}{3} x^{3} a b \,d^{2}+\frac {2}{3} x^{3} b^{2} c d +a^{2} d^{2} x -2 a b c d x +b^{2} c^{2} x}{b^{3}}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{b^{3} \sqrt {a b}}\) | \(102\) |
risch | \(\frac {d^{2} x^{5}}{5 b}-\frac {x^{3} a \,d^{2}}{3 b^{2}}+\frac {2 x^{3} c d}{3 b}+\frac {a^{2} d^{2} x}{b^{3}}-\frac {2 a c d x}{b^{2}}+\frac {c^{2} x}{b}+\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -a \right ) a^{2} d^{2}}{2 b^{4}}-\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -a \right ) a c d}{b^{3}}+\frac {\sqrt {-a b}\, \ln \left (-\sqrt {-a b}\, x -a \right ) c^{2}}{2 b^{2}}-\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -a \right ) a^{2} d^{2}}{2 b^{4}}+\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -a \right ) a c d}{b^{3}}-\frac {\sqrt {-a b}\, \ln \left (\sqrt {-a b}\, x -a \right ) c^{2}}{2 b^{2}}\) | \(233\) |
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Time = 0.25 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.74 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=\left [\frac {6 \, b^{2} d^{2} x^{5} + 10 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 30 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{30 \, b^{3}}, \frac {3 \, b^{2} d^{2} x^{5} + 5 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{3} - 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, b^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (73) = 146\).
Time = 0.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.31 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=x^{3} \left (- \frac {a d^{2}}{3 b^{2}} + \frac {2 c d}{3 b}\right ) + x \left (\frac {a^{2} d^{2}}{b^{3}} - \frac {2 a c d}{b^{2}} + \frac {c^{2}}{b}\right ) + \frac {\sqrt {- \frac {a}{b^{7}}} \left (a d - b c\right )^{2} \log {\left (- \frac {b^{3} \sqrt {- \frac {a}{b^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} - \frac {\sqrt {- \frac {a}{b^{7}}} \left (a d - b c\right )^{2} \log {\left (\frac {b^{3} \sqrt {- \frac {a}{b^{7}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac {d^{2} x^{5}}{5 b} \]
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Time = 0.29 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.25 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{2} d^{2} x^{5} + 5 \, {\left (2 \, b^{2} c d - a b d^{2}\right )} x^{3} + 15 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{15 \, b^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.35 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=-\frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{3}} + \frac {3 \, b^{4} d^{2} x^{5} + 10 \, b^{4} c d x^{3} - 5 \, a b^{3} d^{2} x^{3} + 15 \, b^{4} c^{2} x - 30 \, a b^{3} c d x + 15 \, a^{2} b^{2} d^{2} x}{15 \, b^{5}} \]
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Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.52 \[ \int \frac {x^2 \left (c+d x^2\right )^2}{a+b x^2} \, dx=x\,\left (\frac {c^2}{b}+\frac {a\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )}{b}\right )-x^3\,\left (\frac {a\,d^2}{3\,b^2}-\frac {2\,c\,d}{3\,b}\right )+\frac {d^2\,x^5}{5\,b}-\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {b}\,x\,{\left (a\,d-b\,c\right )}^2}{a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^{7/2}} \]
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