Integrand size = 24, antiderivative size = 166 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {\left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right ) x}{e^4}-\frac {2 c (c d-b e) x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (c d^2-b d e+a e^2\right ) \left (7 c d^2-e (3 b d+a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{9/2}} \]
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Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1171, 1824, 211} \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (a e^2-b d e+c d^2\right ) \left (7 c d^2-e (a e+3 b d)\right )}{2 d^{3/2} e^{9/2}}+\frac {x \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )}{e^4}+\frac {x \left (a e^2-b d e+c d^2\right )^2}{2 d e^4 \left (d+e x^2\right )}-\frac {2 c x^3 (c d-b e)}{3 e^3}+\frac {c^2 x^5}{5 e^2} \]
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Rule 211
Rule 1171
Rule 1824
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\int \frac {\frac {c^2 d^4-2 c d^2 e (b d-a e)+e^2 \left (b^2 d^2-2 a b d e-a^2 e^2\right )}{e^4}-\frac {2 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac {2 c d (c d-2 b e) x^4}{e^2}-\frac {2 c^2 d x^6}{e}}{d+e x^2} \, dx}{2 d} \\ & = \frac {\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\int \left (-\frac {2 d \left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right )}{e^4}+\frac {4 c d (c d-b e) x^2}{e^3}-\frac {2 c^2 d x^4}{e^2}+\frac {7 c^2 d^4-10 b c d^3 e+3 b^2 d^2 e^2+6 a c d^2 e^2-2 a b d e^3-a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{2 d} \\ & = \frac {\left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right ) x}{e^4}-\frac {2 c (c d-b e) x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (7 c^2 d^4-10 b c d^3 e+3 b^2 d^2 e^2+6 a c d^2 e^2-2 a b d e^3-a^2 e^4\right ) \int \frac {1}{d+e x^2} \, dx}{2 d e^4} \\ & = \frac {\left (3 c^2 d^2+b^2 e^2-2 c e (2 b d-a e)\right ) x}{e^4}-\frac {2 c (c d-b e) x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (7 c d^2-3 b d e-a e^2\right ) \left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{9/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {\left (3 c^2 d^2+b^2 e^2+2 c e (-2 b d+a e)\right ) x}{e^4}+\frac {2 c (-c d+b e) x^3}{3 e^3}+\frac {c^2 x^5}{5 e^2}+\frac {\left (c d^2+e (-b d+a e)\right )^2 x}{2 d e^4 \left (d+e x^2\right )}-\frac {\left (7 c^2 d^4+2 c d^2 e (-5 b d+3 a e)-e^2 \left (-3 b^2 d^2+2 a b d e+a^2 e^2\right )\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} e^{9/2}} \]
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Time = 0.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.28
method | result | size |
default | \(\frac {\frac {1}{5} e^{2} x^{5} c^{2}+\frac {2}{3} b c \,e^{2} x^{3}-\frac {2}{3} c^{2} d e \,x^{3}+2 e^{2} a c x +b^{2} e^{2} x -4 b c d e x +3 c^{2} d^{2} x}{e^{4}}+\frac {\frac {\left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a^{2} e^{4}+2 a b d \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}+10 b c \,d^{3} e -7 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \sqrt {e d}}}{e^{4}}\) | \(213\) |
risch | \(\frac {c^{2} x^{5}}{5 e^{2}}+\frac {2 b c \,x^{3}}{3 e^{2}}-\frac {2 c^{2} d \,x^{3}}{3 e^{3}}+\frac {2 c a x}{e^{2}}+\frac {b^{2} x}{e^{2}}-\frac {4 b c d x}{e^{3}}+\frac {3 c^{2} d^{2} x}{e^{4}}+\frac {\left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right ) x}{2 d \,e^{4} \left (e \,x^{2}+d \right )}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a^{2}}{4 \sqrt {-e d}\, d}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a b}{2 e \sqrt {-e d}}+\frac {3 d \ln \left (e x +\sqrt {-e d}\right ) a c}{2 e^{2} \sqrt {-e d}}+\frac {3 d \ln \left (e x +\sqrt {-e d}\right ) b^{2}}{4 e^{2} \sqrt {-e d}}-\frac {5 d^{2} \ln \left (e x +\sqrt {-e d}\right ) b c}{2 e^{3} \sqrt {-e d}}+\frac {7 d^{3} \ln \left (e x +\sqrt {-e d}\right ) c^{2}}{4 e^{4} \sqrt {-e d}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a^{2}}{4 \sqrt {-e d}\, d}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a b}{2 e \sqrt {-e d}}-\frac {3 d \ln \left (-e x +\sqrt {-e d}\right ) a c}{2 e^{2} \sqrt {-e d}}-\frac {3 d \ln \left (-e x +\sqrt {-e d}\right ) b^{2}}{4 e^{2} \sqrt {-e d}}+\frac {5 d^{2} \ln \left (-e x +\sqrt {-e d}\right ) b c}{2 e^{3} \sqrt {-e d}}-\frac {7 d^{3} \ln \left (-e x +\sqrt {-e d}\right ) c^{2}}{4 e^{4} \sqrt {-e d}}\) | \(457\) |
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Time = 0.25 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.61 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\left [\frac {12 \, c^{2} d^{2} e^{4} x^{7} - 4 \, {\left (7 \, c^{2} d^{3} e^{3} - 10 \, b c d^{2} e^{4}\right )} x^{5} + 20 \, {\left (7 \, c^{2} d^{4} e^{2} - 10 \, b c d^{3} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{3} + 15 \, {\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + {\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) + 30 \, {\left (7 \, c^{2} d^{5} e - 10 \, b c d^{4} e^{2} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x}{60 \, {\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}, \frac {6 \, c^{2} d^{2} e^{4} x^{7} - 2 \, {\left (7 \, c^{2} d^{3} e^{3} - 10 \, b c d^{2} e^{4}\right )} x^{5} + 10 \, {\left (7 \, c^{2} d^{4} e^{2} - 10 \, b c d^{3} e^{3} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{3} - 15 \, {\left (7 \, c^{2} d^{5} - 10 \, b c d^{4} e - 2 \, a b d^{2} e^{3} - a^{2} d e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{2} + {\left (7 \, c^{2} d^{4} e - 10 \, b c d^{3} e^{2} - 2 \, a b d e^{4} - a^{2} e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) + 15 \, {\left (7 \, c^{2} d^{5} e - 10 \, b c d^{4} e^{2} - 2 \, a b d^{2} e^{4} + a^{2} d e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x}{30 \, {\left (d^{2} e^{6} x^{2} + d^{3} e^{5}\right )}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (156) = 312\).
Time = 1.17 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.92 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\frac {c^{2} x^{5}}{5 e^{2}} + x^{3} \cdot \left (\frac {2 b c}{3 e^{2}} - \frac {2 c^{2} d}{3 e^{3}}\right ) + x \left (\frac {2 a c}{e^{2}} + \frac {b^{2}}{e^{2}} - \frac {4 b c d}{e^{3}} + \frac {3 c^{2} d^{2}}{e^{4}}\right ) + \frac {x \left (a^{2} e^{4} - 2 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} - 2 b c d^{3} e + c^{2} d^{4}\right )}{2 d^{2} e^{4} + 2 d e^{5} x^{2}} - \frac {\sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log {\left (- \frac {d^{2} e^{4} \sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right ) \log {\left (\frac {d^{2} e^{4} \sqrt {- \frac {1}{d^{3} e^{9}}} \left (a e^{2} - b d e + c d^{2}\right ) \left (a e^{2} + 3 b d e - 7 c d^{2}\right )}{a^{2} e^{4} + 2 a b d e^{3} - 6 a c d^{2} e^{2} - 3 b^{2} d^{2} e^{2} + 10 b c d^{3} e - 7 c^{2} d^{4}} + x \right )}}{4} \]
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Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=-\frac {{\left (7 \, c^{2} d^{4} - 10 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} - 2 \, a b d e^{3} - a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 \, \sqrt {d e} d e^{4}} + \frac {c^{2} d^{4} x - 2 \, b c d^{3} e x + b^{2} d^{2} e^{2} x + 2 \, a c d^{2} e^{2} x - 2 \, a b d e^{3} x + a^{2} e^{4} x}{2 \, {\left (e x^{2} + d\right )} d e^{4}} + \frac {3 \, c^{2} e^{8} x^{5} - 10 \, c^{2} d e^{7} x^{3} + 10 \, b c e^{8} x^{3} + 45 \, c^{2} d^{2} e^{6} x - 60 \, b c d e^{7} x + 15 \, b^{2} e^{8} x + 30 \, a c e^{8} x}{15 \, e^{10}} \]
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Time = 7.68 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.77 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^2} \, dx=x\,\left (\frac {b^2+2\,a\,c}{e^2}+\frac {2\,d\,\left (\frac {2\,c^2\,d}{e^3}-\frac {2\,b\,c}{e^2}\right )}{e}-\frac {c^2\,d^2}{e^4}\right )-x^3\,\left (\frac {2\,c^2\,d}{3\,e^3}-\frac {2\,b\,c}{3\,e^2}\right )+\frac {c^2\,x^5}{5\,e^2}+\frac {x\,\left (a^2\,e^4-2\,a\,b\,d\,e^3+2\,a\,c\,d^2\,e^2+b^2\,d^2\,e^2-2\,b\,c\,d^3\,e+c^2\,d^4\right )}{2\,d\,\left (e^5\,x^2+d\,e^4\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\left (-7\,c\,d^2+3\,b\,d\,e+a\,e^2\right )}{\sqrt {d}\,\left (a^2\,e^4+2\,a\,b\,d\,e^3-6\,a\,c\,d^2\,e^2-3\,b^2\,d^2\,e^2+10\,b\,c\,d^3\,e-7\,c^2\,d^4\right )}\right )\,\left (c\,d^2-b\,d\,e+a\,e^2\right )\,\left (-7\,c\,d^2+3\,b\,d\,e+a\,e^2\right )}{2\,d^{3/2}\,e^{9/2}} \]
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