Integrand size = 30, antiderivative size = 121 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (5 b^3 c-3 a b^2 d+a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} b^{3/2}} \]
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Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1819, 1275, 211} \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {2 b c-a d}{a^3 x}+\frac {x \left (\frac {b^2 c}{a^2}-\frac {b d}{a}-\frac {a f}{b}+e\right )}{2 a \left (a+b x^2\right )}-\frac {c}{3 a^2 x^3}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (a^3 f+a^2 b e-3 a b^2 d+5 b^3 c\right )}{2 a^{7/2} b^{3/2}} \]
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Rule 211
Rule 1275
Rule 1819
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{2 a \left (a+b x^2\right )}-\frac {\int \frac {-2 c+2 \left (\frac {b c}{a}-d\right ) x^2+\left (-\frac {b^2 c}{a^2}+\frac {b d}{a}-e-\frac {a f}{b}\right ) x^4}{x^4 \left (a+b x^2\right )} \, dx}{2 a} \\ & = \frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 c}{a x^4}-\frac {2 (-2 b c+a d)}{a^2 x^2}+\frac {-5 b^3 c+3 a b^2 d-a^2 b e-a^3 f}{a^2 b \left (a+b x^2\right )}\right ) \, dx}{2 a} \\ & = -\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (5 b^3 c-3 a b^2 d+a^2 b e+a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{2 a^3 b} \\ & = -\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}+\frac {\left (\frac {b^2 c}{a^2}-\frac {b d}{a}+e-\frac {a f}{b}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (5 b^3 c-3 a b^2 d+a^2 b e+a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} b^{3/2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=-\frac {c}{3 a^2 x^3}+\frac {2 b c-a d}{a^3 x}-\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a^3 b \left (a+b x^2\right )}+\frac {\left (5 b^3 c-3 a b^2 d+a^2 b e+a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{7/2} b^{3/2}} \]
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Time = 3.46 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {c}{3 a^{2} x^{3}}-\frac {a d -2 b c}{a^{3} x}+\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{2 b \left (b \,x^{2}+a \right )}+\frac {\left (f \,a^{3}+a^{2} b e -3 a \,b^{2} d +5 b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 b \sqrt {a b}}}{a^{3}}\) | \(116\) |
risch | \(\frac {-\frac {\left (f \,a^{3}-a^{2} b e +3 a \,b^{2} d -5 b^{3} c \right ) x^{4}}{2 a^{3} b}-\frac {\left (3 a d -5 b c \right ) x^{2}}{3 a^{2}}-\frac {c}{3 a}}{x^{3} \left (b \,x^{2}+a \right )}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) f}{4 \sqrt {-a b}\, b}-\frac {\ln \left (-\sqrt {-a b}\, x +a \right ) e}{4 \sqrt {-a b}\, a}+\frac {3 b \ln \left (-\sqrt {-a b}\, x +a \right ) d}{4 \sqrt {-a b}\, a^{2}}-\frac {5 b^{2} \ln \left (-\sqrt {-a b}\, x +a \right ) c}{4 \sqrt {-a b}\, a^{3}}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) f}{4 \sqrt {-a b}\, b}+\frac {\ln \left (-\sqrt {-a b}\, x -a \right ) e}{4 \sqrt {-a b}\, a}-\frac {3 b \ln \left (-\sqrt {-a b}\, x -a \right ) d}{4 \sqrt {-a b}\, a^{2}}+\frac {5 b^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) c}{4 \sqrt {-a b}\, a^{3}}\) | \(284\) |
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Time = 0.28 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.12 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=\left [-\frac {4 \, a^{3} b^{2} c - 6 \, {\left (5 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} - 4 \, {\left (5 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d\right )} x^{2} + 3 \, {\left ({\left (5 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e + a^{3} b f\right )} x^{5} + {\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d + a^{3} b e + a^{4} f\right )} x^{3}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{12 \, {\left (a^{4} b^{3} x^{5} + a^{5} b^{2} x^{3}\right )}}, -\frac {2 \, a^{3} b^{2} c - 3 \, {\left (5 \, a b^{4} c - 3 \, a^{2} b^{3} d + a^{3} b^{2} e - a^{4} b f\right )} x^{4} - 2 \, {\left (5 \, a^{2} b^{3} c - 3 \, a^{3} b^{2} d\right )} x^{2} - 3 \, {\left ({\left (5 \, b^{4} c - 3 \, a b^{3} d + a^{2} b^{2} e + a^{3} b f\right )} x^{5} + {\left (5 \, a b^{3} c - 3 \, a^{2} b^{2} d + a^{3} b e + a^{4} f\right )} x^{3}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{6 \, {\left (a^{4} b^{3} x^{5} + a^{5} b^{2} x^{3}\right )}}\right ] \]
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Time = 5.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.75 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=- \frac {\sqrt {- \frac {1}{a^{7} b^{3}}} \left (a^{3} f + a^{2} b e - 3 a b^{2} d + 5 b^{3} c\right ) \log {\left (- a^{4} b \sqrt {- \frac {1}{a^{7} b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{7} b^{3}}} \left (a^{3} f + a^{2} b e - 3 a b^{2} d + 5 b^{3} c\right ) \log {\left (a^{4} b \sqrt {- \frac {1}{a^{7} b^{3}}} + x \right )}}{4} + \frac {- 2 a^{2} b c + x^{4} \left (- 3 a^{3} f + 3 a^{2} b e - 9 a b^{2} d + 15 b^{3} c\right ) + x^{2} \left (- 6 a^{2} b d + 10 a b^{2} c\right )}{6 a^{4} b x^{3} + 6 a^{3} b^{2} x^{5}} \]
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Time = 0.28 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {3 \, {\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e - a^{3} f\right )} x^{4} - 2 \, a^{2} b c + 2 \, {\left (5 \, a b^{2} c - 3 \, a^{2} b d\right )} x^{2}}{6 \, {\left (a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )}} + \frac {{\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e + a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3} b} \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {{\left (5 \, b^{3} c - 3 \, a b^{2} d + a^{2} b e + a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{3} b} + \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{2 \, {\left (b x^{2} + a\right )} a^{3} b} + \frac {6 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{3} x^{3}} \]
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Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x^2+e x^4+f x^6}{x^4 \left (a+b x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (f\,a^3+e\,a^2\,b-3\,d\,a\,b^2+5\,c\,b^3\right )}{2\,a^{7/2}\,b^{3/2}}-\frac {\frac {c}{3\,a}+\frac {x^2\,\left (3\,a\,d-5\,b\,c\right )}{3\,a^2}-\frac {x^4\,\left (-f\,a^3+e\,a^2\,b-3\,d\,a\,b^2+5\,c\,b^3\right )}{2\,a^3\,b}}{b\,x^5+a\,x^3} \]
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