Integrand size = 27, antiderivative size = 118 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\frac {(b e-2 a f) x}{b^3}+\frac {f x^3}{3 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (b^3 c+a b^2 d-3 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \]
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Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1828, 1167, 211} \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\frac {x \left (c-\frac {a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}+\frac {\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (5 a^3 f-3 a^2 b e+a b^2 d+b^3 c\right )}{2 a^{3/2} b^{7/2}}+\frac {x (b e-2 a f)}{b^3}+\frac {f x^3}{3 b^2} \]
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Rule 211
Rule 1167
Rule 1828
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{2 a \left (a+b x^2\right )}-\frac {\int \frac {-\frac {b^3 c+a b^2 d-a^2 b e+a^3 f}{b^3}-\frac {2 a (b e-a f) x^2}{b^2}-\frac {2 a f x^4}{b}}{a+b x^2} \, dx}{2 a} \\ & = \frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{2 a \left (a+b x^2\right )}-\frac {\int \left (-\frac {2 a (b e-2 a f)}{b^3}-\frac {2 a f x^2}{b^2}+\frac {-b^3 c-a b^2 d+3 a^2 b e-5 a^3 f}{b^3 \left (a+b x^2\right )}\right ) \, dx}{2 a} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^3}{3 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (b^3 c+a b^2 d-3 a^2 b e+5 a^3 f\right ) \int \frac {1}{a+b x^2} \, dx}{2 a b^3} \\ & = \frac {(b e-2 a f) x}{b^3}+\frac {f x^3}{3 b^2}+\frac {\left (c-\frac {a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{2 a \left (a+b x^2\right )}+\frac {\left (b^3 c+a b^2 d-3 a^2 b e+5 a^3 f\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\frac {(b e-2 a f) x}{b^3}+\frac {f x^3}{3 b^2}-\frac {\left (-b^3 c+a b^2 d-a^2 b e+a^3 f\right ) x}{2 a b^3 \left (a+b x^2\right )}+\frac {\left (b^3 c+a b^2 d-3 a^2 b e+5 a^3 f\right ) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{7/2}} \]
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Time = 3.46 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {-\frac {1}{3} f \,x^{3} b +2 a f x -b e x}{b^{3}}+\frac {-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{2 a \left (b \,x^{2}+a \right )}+\frac {\left (5 f \,a^{3}-3 a^{2} b e +a \,b^{2} d +b^{3} c \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 a \sqrt {a b}}}{b^{3}}\) | \(114\) |
risch | \(\frac {f \,x^{3}}{3 b^{2}}-\frac {2 a f x}{b^{3}}+\frac {e x}{b^{2}}-\frac {\left (f \,a^{3}-a^{2} b e +a \,b^{2} d -b^{3} c \right ) x}{2 a \,b^{3} \left (b \,x^{2}+a \right )}-\frac {5 a^{2} \ln \left (b x +\sqrt {-a b}\right ) f}{4 b^{3} \sqrt {-a b}}+\frac {3 a \ln \left (b x +\sqrt {-a b}\right ) e}{4 b^{2} \sqrt {-a b}}-\frac {\ln \left (b x +\sqrt {-a b}\right ) d}{4 b \sqrt {-a b}}-\frac {c \ln \left (b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}+\frac {5 a^{2} \ln \left (-b x +\sqrt {-a b}\right ) f}{4 b^{3} \sqrt {-a b}}-\frac {3 a \ln \left (-b x +\sqrt {-a b}\right ) e}{4 b^{2} \sqrt {-a b}}+\frac {\ln \left (-b x +\sqrt {-a b}\right ) d}{4 b \sqrt {-a b}}+\frac {c \ln \left (-b x +\sqrt {-a b}\right )}{4 \sqrt {-a b}\, a}\) | \(264\) |
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Time = 0.31 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.08 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, a^{2} b^{3} f x^{5} + 4 \, {\left (3 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{3} - 3 \, {\left (a b^{3} c + a^{2} b^{2} d - 3 \, a^{3} b e + 5 \, a^{4} f + {\left (b^{4} c + a b^{3} d - 3 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 6 \, {\left (a b^{4} c - a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x}{12 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}, \frac {2 \, a^{2} b^{3} f x^{5} + 2 \, {\left (3 \, a^{2} b^{3} e - 5 \, a^{3} b^{2} f\right )} x^{3} + 3 \, {\left (a b^{3} c + a^{2} b^{2} d - 3 \, a^{3} b e + 5 \, a^{4} f + {\left (b^{4} c + a b^{3} d - 3 \, a^{2} b^{2} e + 5 \, a^{3} b f\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 3 \, {\left (a b^{4} c - a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 5 \, a^{4} b f\right )} x}{6 \, {\left (a^{2} b^{5} x^{2} + a^{3} b^{4}\right )}}\right ] \]
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Time = 0.80 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.70 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=x \left (- \frac {2 a f}{b^{3}} + \frac {e}{b^{2}}\right ) + \frac {x \left (- a^{3} f + a^{2} b e - a b^{2} d + b^{3} c\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \cdot \left (5 a^{3} f - 3 a^{2} b e + a b^{2} d + b^{3} c\right ) \log {\left (- a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{7}}} \cdot \left (5 a^{3} f - 3 a^{2} b e + a b^{2} d + b^{3} c\right ) \log {\left (a^{2} b^{3} \sqrt {- \frac {1}{a^{3} b^{7}}} + x \right )}}{4} + \frac {f x^{3}}{3 b^{2}} \]
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Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac {b f x^{3} + 3 \, {\left (b e - 2 \, a f\right )} x}{3 \, b^{3}} + \frac {{\left (b^{3} c + a b^{2} d - 3 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} \]
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Time = 0.39 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.04 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=\frac {{\left (b^{3} c + a b^{2} d - 3 \, a^{2} b e + 5 \, a^{3} f\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b^{3}} + \frac {b^{3} c x - a b^{2} d x + a^{2} b e x - a^{3} f x}{2 \, {\left (b x^{2} + a\right )} a b^{3}} + \frac {b^{4} f x^{3} + 3 \, b^{4} e x - 6 \, a b^{3} f x}{3 \, b^{6}} \]
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Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^2+e x^4+f x^6}{\left (a+b x^2\right )^2} \, dx=x\,\left (\frac {e}{b^2}-\frac {2\,a\,f}{b^3}\right )+\frac {f\,x^3}{3\,b^2}+\frac {x\,\left (-f\,a^3+e\,a^2\,b-d\,a\,b^2+c\,b^3\right )}{2\,a\,\left (b^4\,x^2+a\,b^3\right )}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (5\,f\,a^3-3\,e\,a^2\,b+d\,a\,b^2+c\,b^3\right )}{2\,a^{3/2}\,b^{7/2}} \]
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