Integrand size = 20, antiderivative size = 59 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {464, 331, 211} \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b c-a d)}{a^{5/2}}+\frac {b c-a d}{a^2 x}-\frac {c}{3 a x^3} \]
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Rule 211
Rule 331
Rule 464
Rubi steps \begin{align*} \text {integral}& = -\frac {c}{3 a x^3}-\frac {(3 b c-3 a d) \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{3 a} \\ & = -\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {(b (b c-a d)) \int \frac {1}{a+b x^2} \, dx}{a^2} \\ & = -\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}+\frac {\sqrt {b} (b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {c}{3 a x^3}+\frac {b c-a d}{a^2 x}-\frac {\sqrt {b} (-b c+a d) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{5/2}} \]
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Time = 2.64 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {c}{3 a \,x^{3}}-\frac {a d -b c}{a^{2} x}-\frac {\left (a d -b c \right ) b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{a^{2} \sqrt {a b}}\) | \(55\) |
risch | \(\frac {-\frac {\left (a d -b c \right ) x^{2}}{a^{2}}-\frac {c}{3 a}}{x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{2}+a^{2} b \,d^{2}-2 a \,b^{2} c d +c^{2} b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{5}+2 a^{2} b \,d^{2}-4 a \,b^{2} c d +2 c^{2} b^{3}\right ) x +\left (a^{4} d -a^{3} b c \right ) \textit {\_R} \right )\right )}{2}\) | \(121\) |
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Time = 0.25 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.31 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=\left [-\frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) - 6 \, {\left (b c - a d\right )} x^{2} + 2 \, a c}{6 \, a^{2} x^{3}}, \frac {3 \, {\left (b c - a d\right )} x^{3} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (49) = 98\).
Time = 0.23 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.19 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} - \frac {\sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right ) \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}} \left (a d - b c\right )}{a b d - b^{2} c} + x \right )}}{2} + \frac {- a c + x^{2} \left (- 3 a d + 3 b c\right )}{3 a^{2} x^{3}} \]
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Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, {\left (b c - a d\right )} x^{2} - a c}{3 \, a^{2} x^{3}} \]
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Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{2} c - a b d\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}} + \frac {3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]
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Time = 5.42 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x^2}{x^4 \left (a+b x^2\right )} \, dx=-\frac {\frac {c}{3\,a}+\frac {x^2\,\left (a\,d-b\,c\right )}{a^2}}{x^3}-\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (a\,d-b\,c\right )}{a^{5/2}} \]
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