Integrand size = 30, antiderivative size = 104 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=-\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}} \]
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Time = 0.07 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {1816, 211} \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=\frac {b c-a d}{3 a^2 x^3}-\frac {a^2 e-a b d+b^2 c}{a^3 x}-\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{7/2} \sqrt {b}}-\frac {c}{5 a x^5} \]
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Rule 211
Rule 1816
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c}{a x^6}+\frac {-b c+a d}{a^2 x^4}+\frac {b^2 c-a b d+a^2 e}{a^3 x^2}+\frac {-b^3 c+a b^2 d-a^2 b e+a^3 f}{a^3 \left (a+b x^2\right )}\right ) \, dx \\ & = -\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{a^3} \\ & = -\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}-\frac {b^2 c-a b d+a^2 e}{a^3 x}-\frac {\left (b^3 c-a b^2 d+a^2 b e-a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=-\frac {c}{5 a x^5}+\frac {b c-a d}{3 a^2 x^3}+\frac {-b^2 c+a b d-a^2 e}{a^3 x}+\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{7/2} \sqrt {b}} \]
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Time = 3.42 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {c}{5 a \,x^{5}}-\frac {a d -b c}{3 a^{2} x^{3}}-\frac {a^{2} e -a b d +b^{2} c}{a^{3} x}+\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{3} \sqrt {a b}}\) | \(94\) |
risch | \(\frac {-\frac {\left (a^{2} e -a b d +b^{2} c \right ) x^{4}}{a^{3}}-\frac {\left (a d -b c \right ) x^{2}}{3 a^{2}}-\frac {c}{5 a}}{x^{5}}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) f}{2 \sqrt {-a b}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b e}{2 \sqrt {-a b}\, a}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b^{2} d}{2 \sqrt {-a b}\, a^{2}}+\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) b^{3} c}{2 \sqrt {-a b}\, a^{3}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) f}{2 \sqrt {-a b}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b e}{2 \sqrt {-a b}\, a}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b^{2} d}{2 \sqrt {-a b}\, a^{2}}-\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) b^{3} c}{2 \sqrt {-a b}\, a^{3}}\) | \(261\) |
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Time = 0.27 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.37 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=\left [\frac {15 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt {-a b} x^{5} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 6 \, a^{3} b c - 30 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e\right )} x^{4} + 10 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}}{30 \, a^{4} b x^{5}}, -\frac {15 \, {\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \sqrt {a b} x^{5} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, a^{3} b c + 15 \, {\left (a b^{3} c - a^{2} b^{2} d + a^{3} b e\right )} x^{4} - 5 \, {\left (a^{2} b^{2} c - a^{3} b d\right )} x^{2}}{15 \, a^{4} b x^{5}}\right ] \]
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Time = 2.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.61 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=- \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (- a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{7} b}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log {\left (a^{4} \sqrt {- \frac {1}{a^{7} b}} + x \right )}}{2} + \frac {- 3 a^{2} c + x^{4} \left (- 15 a^{2} e + 15 a b d - 15 b^{2} c\right ) + x^{2} \left (- 5 a^{2} d + 5 a b c\right )}{15 a^{3} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, {\left (b^{2} c - a b d + a^{2} e\right )} x^{4} + 3 \, a^{2} c - 5 \, {\left (a b c - a^{2} d\right )} x^{2}}{15 \, a^{3} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{3}} - \frac {15 \, b^{2} c x^{4} - 15 \, a b d x^{4} + 15 \, a^{2} e x^{4} - 5 \, a b c x^{2} + 5 \, a^{2} d x^{2} + 3 \, a^{2} c}{15 \, a^{3} x^{5}} \]
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Time = 5.99 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^6 \left (a+b x^2\right )} \, dx=-\frac {\frac {c}{5\,a}+\frac {x^2\,\left (a\,d-b\,c\right )}{3\,a^2}+\frac {x^4\,\left (e\,a^2-d\,a\,b+c\,b^2\right )}{a^3}}{x^5}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{a^{7/2}\,\sqrt {b}} \]
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