Integrand size = 22, antiderivative size = 150 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\frac {\left (a+\frac {d (c d-b e)}{e^2}\right ) x}{6 d \left (d+e x^2\right )^3}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (b d+5 a e)\right ) x}{16 d^3 e^2 \left (d+e x^2\right )}+\frac {\left (c d^2+e (b d+5 a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{5/2}} \]
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Time = 0.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1171, 393, 205, 211} \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\frac {\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (e (5 a e+b d)+c d^2\right )}{16 d^{7/2} e^{5/2}}-\frac {x \left (7 c d^2-e (5 a e+b d)\right )}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac {x \left (a e^2-b d e+c d^2\right )}{6 d e^2 \left (d+e x^2\right )^3}+\frac {x \left (e (5 a e+b d)+c d^2\right )}{16 d^3 e^2 \left (d+e x^2\right )} \]
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Rule 205
Rule 211
Rule 393
Rule 1171
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^2\right )^3}-\frac {\int \frac {-5 a+\frac {d (c d-b e)}{e^2}-\frac {6 c d x^2}{e}}{\left (d+e x^2\right )^3} \, dx}{6 d} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^2\right )^3}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (b d+5 a e)\right ) \int \frac {1}{\left (d+e x^2\right )^2} \, dx}{8 d^2 e^2} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^2\right )^3}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (b d+5 a e)\right ) x}{16 d^3 e^2 \left (d+e x^2\right )}+\frac {\left (c d^2+e (b d+5 a e)\right ) \int \frac {1}{d+e x^2} \, dx}{16 d^3 e^2} \\ & = \frac {\left (c d^2-b d e+a e^2\right ) x}{6 d e^2 \left (d+e x^2\right )^3}-\frac {\left (7 c d^2-e (b d+5 a e)\right ) x}{24 d^2 e^2 \left (d+e x^2\right )^2}+\frac {\left (c d^2+e (b d+5 a e)\right ) x}{16 d^3 e^2 \left (d+e x^2\right )}+\frac {\left (c d^2+e (b d+5 a e)\right ) \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{5/2}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\frac {x \left (c d^2 \left (-3 d^2-8 d e x^2+3 e^2 x^4\right )+e \left (b d \left (-3 d^2+8 d e x^2+3 e^2 x^4\right )+a e \left (33 d^2+40 d e x^2+15 e^2 x^4\right )\right )\right )}{48 d^3 e^2 \left (d+e x^2\right )^3}+\frac {\left (c d^2+e (b d+5 a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2} e^{5/2}} \]
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Time = 0.57 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\frac {\left (5 a \,e^{2}+b d e +c \,d^{2}\right ) x^{5}}{16 d^{3}}+\frac {\left (5 a \,e^{2}+b d e -c \,d^{2}\right ) x^{3}}{6 d^{2} e}+\frac {\left (11 a \,e^{2}-b d e -c \,d^{2}\right ) x}{16 d \,e^{2}}}{\left (e \,x^{2}+d \right )^{3}}+\frac {\left (5 a \,e^{2}+b d e +c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{16 d^{3} e^{2} \sqrt {e d}}\) | \(130\) |
risch | \(\frac {\frac {\left (5 a \,e^{2}+b d e +c \,d^{2}\right ) x^{5}}{16 d^{3}}+\frac {\left (5 a \,e^{2}+b d e -c \,d^{2}\right ) x^{3}}{6 d^{2} e}+\frac {\left (11 a \,e^{2}-b d e -c \,d^{2}\right ) x}{16 d \,e^{2}}}{\left (e \,x^{2}+d \right )^{3}}-\frac {5 \ln \left (e x +\sqrt {-e d}\right ) a}{32 \sqrt {-e d}\, d^{3}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) b}{32 \sqrt {-e d}\, e \,d^{2}}-\frac {\ln \left (e x +\sqrt {-e d}\right ) c}{32 \sqrt {-e d}\, e^{2} d}+\frac {5 \ln \left (-e x +\sqrt {-e d}\right ) a}{32 \sqrt {-e d}\, d^{3}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) b}{32 \sqrt {-e d}\, e \,d^{2}}+\frac {\ln \left (-e x +\sqrt {-e d}\right ) c}{32 \sqrt {-e d}\, e^{2} d}\) | \(245\) |
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Time = 0.25 (sec) , antiderivative size = 530, normalized size of antiderivative = 3.53 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\left [\frac {6 \, {\left (c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{5} - 16 \, {\left (c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{3} - 3 \, {\left ({\left (c d^{2} e^{3} + b d e^{4} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + b d^{4} e + 5 \, a d^{3} e^{2} + 3 \, {\left (c d^{3} e^{2} + b d^{2} e^{3} + 5 \, a d e^{4}\right )} x^{4} + 3 \, {\left (c d^{4} e + b d^{3} e^{2} + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \sqrt {-d e} \log \left (\frac {e x^{2} - 2 \, \sqrt {-d e} x - d}{e x^{2} + d}\right ) - 6 \, {\left (c d^{5} e + b d^{4} e^{2} - 11 \, a d^{3} e^{3}\right )} x}{96 \, {\left (d^{4} e^{6} x^{6} + 3 \, d^{5} e^{5} x^{4} + 3 \, d^{6} e^{4} x^{2} + d^{7} e^{3}\right )}}, \frac {3 \, {\left (c d^{3} e^{3} + b d^{2} e^{4} + 5 \, a d e^{5}\right )} x^{5} - 8 \, {\left (c d^{4} e^{2} - b d^{3} e^{3} - 5 \, a d^{2} e^{4}\right )} x^{3} + 3 \, {\left ({\left (c d^{2} e^{3} + b d e^{4} + 5 \, a e^{5}\right )} x^{6} + c d^{5} + b d^{4} e + 5 \, a d^{3} e^{2} + 3 \, {\left (c d^{3} e^{2} + b d^{2} e^{3} + 5 \, a d e^{4}\right )} x^{4} + 3 \, {\left (c d^{4} e + b d^{3} e^{2} + 5 \, a d^{2} e^{3}\right )} x^{2}\right )} \sqrt {d e} \arctan \left (\frac {\sqrt {d e} x}{d}\right ) - 3 \, {\left (c d^{5} e + b d^{4} e^{2} - 11 \, a d^{3} e^{3}\right )} x}{48 \, {\left (d^{4} e^{6} x^{6} + 3 \, d^{5} e^{5} x^{4} + 3 \, d^{6} e^{4} x^{2} + d^{7} e^{3}\right )}}\right ] \]
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Time = 1.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.61 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=- \frac {\sqrt {- \frac {1}{d^{7} e^{5}}} \cdot \left (5 a e^{2} + b d e + c d^{2}\right ) \log {\left (- d^{4} e^{2} \sqrt {- \frac {1}{d^{7} e^{5}}} + x \right )}}{32} + \frac {\sqrt {- \frac {1}{d^{7} e^{5}}} \cdot \left (5 a e^{2} + b d e + c d^{2}\right ) \log {\left (d^{4} e^{2} \sqrt {- \frac {1}{d^{7} e^{5}}} + x \right )}}{32} + \frac {x^{5} \cdot \left (15 a e^{4} + 3 b d e^{3} + 3 c d^{2} e^{2}\right ) + x^{3} \cdot \left (40 a d e^{3} + 8 b d^{2} e^{2} - 8 c d^{3} e\right ) + x \left (33 a d^{2} e^{2} - 3 b d^{3} e - 3 c d^{4}\right )}{48 d^{6} e^{2} + 144 d^{5} e^{3} x^{2} + 144 d^{4} e^{4} x^{4} + 48 d^{3} e^{5} x^{6}} \]
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Exception generated. \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.97 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\frac {{\left (c d^{2} + b d e + 5 \, a e^{2}\right )} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{16 \, \sqrt {d e} d^{3} e^{2}} + \frac {3 \, c d^{2} e^{2} x^{5} + 3 \, b d e^{3} x^{5} + 15 \, a e^{4} x^{5} - 8 \, c d^{3} e x^{3} + 8 \, b d^{2} e^{2} x^{3} + 40 \, a d e^{3} x^{3} - 3 \, c d^{4} x - 3 \, b d^{3} e x + 33 \, a d^{2} e^{2} x}{48 \, {\left (e x^{2} + d\right )}^{3} d^{3} e^{2}} \]
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Time = 7.58 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^2+c x^4}{\left (d+e x^2\right )^4} \, dx=\frac {\frac {x^5\,\left (c\,d^2+b\,d\,e+5\,a\,e^2\right )}{16\,d^3}-\frac {x\,\left (c\,d^2+b\,d\,e-11\,a\,e^2\right )}{16\,d\,e^2}+\frac {x^3\,\left (-c\,d^2+b\,d\,e+5\,a\,e^2\right )}{6\,d^2\,e}}{d^3+3\,d^2\,e\,x^2+3\,d\,e^2\,x^4+e^3\,x^6}+\frac {\mathrm {atan}\left (\frac {\sqrt {e}\,x}{\sqrt {d}}\right )\,\left (c\,d^2+b\,d\,e+5\,a\,e^2\right )}{16\,d^{7/2}\,e^{5/2}} \]
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