Integrand size = 24, antiderivative size = 201 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=-\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4-6 c d^2 e (5 b d-a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \]
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Time = 0.27 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1171, 1828, 1167, 211} \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e^2 \left (3 a^2 e^2+2 a b d e+3 b^2 d^2\right )-6 c d^2 e (5 b d-a e)+35 c^2 d^4\right )}{8 d^{5/2} e^{9/2}}-\frac {x \left (-3 a e^2-5 b d e+13 c d^2\right ) \left (a e^2-b d e+c d^2\right )}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {x \left (a e^2-b d e+c d^2\right )^2}{4 d e^4 \left (d+e x^2\right )^2}-\frac {c x (3 c d-2 b e)}{e^4}+\frac {c^2 x^3}{3 e^3} \]
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Rule 211
Rule 1167
Rule 1171
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\int \frac {\frac {\left (c d^2-b d e-a e^2\right ) \left (c d^2-b d e+3 a e^2\right )}{e^4}-\frac {4 d \left (c^2 d^2+b^2 e^2-2 c e (b d-a e)\right ) x^2}{e^3}+\frac {4 c d (c d-2 b e) x^4}{e^2}-\frac {4 c^2 d x^6}{e}}{\left (d+e x^2\right )^2} \, dx}{4 d} \\ & = \frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\int \frac {\frac {11 c^2 d^4-2 c d^2 e (7 b d-3 a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )}{e^4}-\frac {16 c d^2 (c d-b e) x^2}{e^3}+\frac {8 c^2 d^2 x^4}{e^2}}{d+e x^2} \, dx}{8 d^2} \\ & = \frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\int \left (-\frac {8 c d^2 (3 c d-2 b e)}{e^4}+\frac {8 c^2 d^2 x^2}{e^3}+\frac {35 c^2 d^4-30 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+3 a^2 e^4}{e^4 \left (d+e x^2\right )}\right ) \, dx}{8 d^2} \\ & = -\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4-6 c d^2 e (5 b d-a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \int \frac {1}{d+e x^2} \, dx}{8 d^2 e^4} \\ & = -\frac {c (3 c d-2 b e) x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2-b d e+a e^2\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c d^2-5 b d e-3 a e^2\right ) \left (c d^2-b d e+a e^2\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4-6 c d^2 e (5 b d-a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {c (-3 c d+2 b e) x}{e^4}+\frac {c^2 x^3}{3 e^3}+\frac {\left (c d^2+e (-b d+a e)\right )^2 x}{4 d e^4 \left (d+e x^2\right )^2}-\frac {\left (13 c^2 d^4-2 c d^2 e (9 b d-5 a e)+e^2 \left (5 b^2 d^2-2 a b d e-3 a^2 e^2\right )\right ) x}{8 d^2 e^4 \left (d+e x^2\right )}+\frac {\left (35 c^2 d^4+6 c d^2 e (-5 b d+a e)+e^2 \left (3 b^2 d^2+2 a b d e+3 a^2 e^2\right )\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} e^{9/2}} \]
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Time = 0.29 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {c \left (\frac {1}{3} c \,x^{3} e +2 b e x -3 c d x \right )}{e^{4}}+\frac {\frac {\frac {e \left (3 a^{2} e^{4}+2 a b d \,e^{3}-10 a c \,d^{2} e^{2}-5 b^{2} d^{2} e^{2}+18 b c \,d^{3} e -13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-2 a b d \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}+14 b c \,d^{3} e -11 c^{2} d^{4}\right ) x}{8 d}}{\left (e \,x^{2}+d \right )^{2}}+\frac {\left (3 a^{2} e^{4}+2 a b d \,e^{3}+6 a c \,d^{2} e^{2}+3 b^{2} d^{2} e^{2}-30 b c \,d^{3} e +35 c^{2} d^{4}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{8 d^{2} \sqrt {e d}}}{e^{4}}\) | \(237\) |
risch | \(\frac {c^{2} x^{3}}{3 e^{3}}+\frac {2 c b x}{e^{3}}-\frac {3 c^{2} d x}{e^{4}}+\frac {\frac {e \left (3 a^{2} e^{4}+2 a b d \,e^{3}-10 a c \,d^{2} e^{2}-5 b^{2} d^{2} e^{2}+18 b c \,d^{3} e -13 c^{2} d^{4}\right ) x^{3}}{8 d^{2}}+\frac {\left (5 a^{2} e^{4}-2 a b d \,e^{3}-6 a c \,d^{2} e^{2}-3 b^{2} d^{2} e^{2}+14 b c \,d^{3} e -11 c^{2} d^{4}\right ) x}{8 d}}{e^{4} \left (e \,x^{2}+d \right )^{2}}-\frac {3 \ln \left (e x +\sqrt {-e d}\right ) a^{2}}{16 \sqrt {-e d}\, d^{2}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) a b}{8 e \sqrt {-e d}\, d}-\frac {3 \ln \left (e x +\sqrt {-e d}\right ) a c}{8 e^{2} \sqrt {-e d}}-\frac {3 \ln \left (e x +\sqrt {-e d}\right ) b^{2}}{16 e^{2} \sqrt {-e d}}+\frac {15 d \ln \left (e x +\sqrt {-e d}\right ) b c}{8 e^{3} \sqrt {-e d}}-\frac {35 d^{2} \ln \left (e x +\sqrt {-e d}\right ) c^{2}}{16 e^{4} \sqrt {-e d}}+\frac {3 \ln \left (-e x +\sqrt {-e d}\right ) a^{2}}{16 \sqrt {-e d}\, d^{2}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) a b}{8 e \sqrt {-e d}\, d}+\frac {3 \ln \left (-e x +\sqrt {-e d}\right ) a c}{8 e^{2} \sqrt {-e d}}+\frac {3 \ln \left (-e x +\sqrt {-e d}\right ) b^{2}}{16 e^{2} \sqrt {-e d}}-\frac {15 d \ln \left (-e x +\sqrt {-e d}\right ) b c}{8 e^{3} \sqrt {-e d}}+\frac {35 d^{2} \ln \left (-e x +\sqrt {-e d}\right ) c^{2}}{16 e^{4} \sqrt {-e d}}\) | \(482\) |
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Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (185) = 370\).
Time = 0.26 (sec) , antiderivative size = 794, normalized size of antiderivative = 3.95 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\left [\frac {16 \, c^{2} d^{3} e^{4} x^{7} - 16 \, {\left (7 \, c^{2} d^{4} e^{3} - 6 \, b c d^{3} e^{4}\right )} x^{5} - 2 \, {\left (175 \, c^{2} d^{5} e^{2} - 150 \, b c d^{4} e^{3} - 6 \, a b d^{2} e^{5} - 9 \, a^{2} d e^{6} + 15 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{3} - 3 \, {\left (35 \, c^{2} d^{6} - 30 \, b c d^{5} e + 2 \, a b d^{3} e^{3} + 3 \, a^{2} d^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + {\left (35 \, c^{2} d^{4} e^{2} - 30 \, b c d^{3} e^{3} + 2 \, a b d e^{5} + 3 \, a^{2} e^{6} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e - 30 \, b c d^{4} e^{2} + 2 \, a b d^{2} e^{4} + 3 \, a^{2} d e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} 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 ) - 6 \, {\left (35 \, c^{2} d^{6} e - 30 \, b c d^{5} e^{2} + 2 \, a b d^{3} e^{4} - 5 \, a^{2} d^{2} e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x}{48 \, {\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}, \frac {8 \, c^{2} d^{3} e^{4} x^{7} - 8 \, {\left (7 \, c^{2} d^{4} e^{3} - 6 \, b c d^{3} e^{4}\right )} x^{5} - {\left (175 \, c^{2} d^{5} e^{2} - 150 \, b c d^{4} e^{3} - 6 \, a b d^{2} e^{5} - 9 \, a^{2} d e^{6} + 15 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (35 \, c^{2} d^{6} - 30 \, b c d^{5} e + 2 \, a b d^{3} e^{3} + 3 \, a^{2} d^{2} e^{4} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{2} + {\left (35 \, c^{2} d^{4} e^{2} - 30 \, b c d^{3} e^{3} + 2 \, a b d e^{5} + 3 \, a^{2} e^{6} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{4}\right )} x^{4} + 2 \, {\left (35 \, c^{2} d^{5} e - 30 \, b c d^{4} e^{2} + 2 \, a b d^{2} e^{4} + 3 \, a^{2} d e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{3} e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 3 \, {\left (35 \, c^{2} d^{6} e - 30 \, b c d^{5} e^{2} + 2 \, a b d^{3} e^{4} - 5 \, a^{2} d^{2} e^{5} + 3 \, {\left (b^{2} + 2 \, a c\right )} d^{4} e^{3}\right )} x}{24 \, {\left (d^{3} e^{7} x^{4} + 2 \, d^{4} e^{6} x^{2} + d^{5} e^{5}\right )}}\right ] \]
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Time = 9.72 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.98 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {c^{2} x^{3}}{3 e^{3}} + x \left (\frac {2 b c}{e^{3}} - \frac {3 c^{2} d}{e^{4}}\right ) - \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log {\left (- d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{d^{5} e^{9}}} \cdot \left (3 a^{2} e^{4} + 2 a b d e^{3} + 6 a c d^{2} e^{2} + 3 b^{2} d^{2} e^{2} - 30 b c d^{3} e + 35 c^{2} d^{4}\right ) \log {\left (d^{3} e^{4} \sqrt {- \frac {1}{d^{5} e^{9}}} + x \right )}}{16} + \frac {x^{3} \cdot \left (3 a^{2} e^{5} + 2 a b d e^{4} - 10 a c d^{2} e^{3} - 5 b^{2} d^{2} e^{3} + 18 b c d^{3} e^{2} - 13 c^{2} d^{4} e\right ) + x \left (5 a^{2} d e^{4} - 2 a b d^{2} e^{3} - 6 a c d^{3} e^{2} - 3 b^{2} d^{3} e^{2} + 14 b c d^{4} e - 11 c^{2} d^{5}\right )}{8 d^{4} e^{4} + 16 d^{3} e^{5} x^{2} + 8 d^{2} e^{6} x^{4}} \]
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Exception generated. \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.32 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {{\left (35 \, c^{2} d^{4} - 30 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + 6 \, a c d^{2} e^{2} + 2 \, a b d e^{3} + 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{8 \, \sqrt {d e} d^{2} e^{4}} - \frac {13 \, c^{2} d^{4} e x^{3} - 18 \, b c d^{3} e^{2} x^{3} + 5 \, b^{2} d^{2} e^{3} x^{3} + 10 \, a c d^{2} e^{3} x^{3} - 2 \, a b d e^{4} x^{3} - 3 \, a^{2} e^{5} x^{3} + 11 \, c^{2} d^{5} x - 14 \, b c d^{4} e x + 3 \, b^{2} d^{3} e^{2} x + 6 \, a c d^{3} e^{2} x + 2 \, a b d^{2} e^{3} x - 5 \, a^{2} d e^{4} x}{8 \, {\left (e x^{2} + d\right )}^{2} d^{2} e^{4}} + \frac {c^{2} e^{6} x^{3} - 9 \, c^{2} d e^{5} x + 6 \, b c e^{6} x}{3 \, e^{9}} \]
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Time = 0.07 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.28 \[ \int \frac {\left (a+b x^2+c x^4\right )^2}{\left (d+e x^2\right )^3} \, dx=\frac {c^2\,x^3}{3\,e^3}-x\,\left (\frac {3\,c^2\,d}{e^4}-\frac {2\,b\,c}{e^3}\right )-\frac {\frac {x\,\left (-5\,a^2\,e^4+2\,a\,b\,d\,e^3+6\,a\,c\,d^2\,e^2+3\,b^2\,d^2\,e^2-14\,b\,c\,d^3\,e+11\,c^2\,d^4\right )}{8\,d}-\frac {x^3\,\left (3\,a^2\,e^5+2\,a\,b\,d\,e^4-10\,a\,c\,d^2\,e^3-5\,b^2\,d^2\,e^3+18\,b\,c\,d^3\,e^2-13\,c^2\,d^4\,e\right )}{8\,d^2}}{d^2\,e^4+2\,d\,e^5\,x^2+e^6\,x^4}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (3\,a^2\,e^4+2\,a\,b\,d\,e^3+6\,a\,c\,d^2\,e^2+3\,b^2\,d^2\,e^2-30\,b\,c\,d^3\,e+35\,c^2\,d^4\right )}{8\,d^{5/2}\,e^{9/2}} \]
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