Integrand size = 40, antiderivative size = 19 \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {\arctan \left (\frac {c x^3}{a+b x^2}\right )}{c} \]
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Time = 0.07 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2119, 211} \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {\arctan \left (\frac {c x^3}{a+b x^2}\right )}{c} \]
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Rule 211
Rule 2119
Rubi steps \begin{align*} \text {integral}& = \left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{a^2+9 a^2 c^2 x^2} \, dx,x,\frac {x^3}{3 a+3 b x^2}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {c x^3}{a+b x^2}\right )}{c} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 4.58 \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {1}{2} \text {RootSum}\left [a^2+2 a b \text {$\#$1}^2+b^2 \text {$\#$1}^4+c^2 \text {$\#$1}^6\&,\frac {3 a \log (x-\text {$\#$1}) \text {$\#$1}+b \log (x-\text {$\#$1}) \text {$\#$1}^3}{2 a b+2 b^2 \text {$\#$1}^2+3 c^2 \text {$\#$1}^4}\&\right ] \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.58
method | result | size |
parallelrisch | \(\frac {i \ln \left (i b \,x^{2}+c \,x^{3}+i a \right )-i \ln \left (-i b \,x^{2}+c \,x^{3}-i a \right )}{2 c}\) | \(49\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c^{2} \textit {\_Z}^{6}+b^{2} \textit {\_Z}^{4}+2 a \,\textit {\_Z}^{2} b +a^{2}\right )}{\sum }\frac {\left (\textit {\_R}^{4} b +3 \textit {\_R}^{2} a \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{5} c^{2}+2 \textit {\_R}^{3} b^{2}+2 a b \textit {\_R}}\right )}{2}\) | \(75\) |
risch | \(-\frac {\arctan \left (\frac {c \,x^{5} b}{a^{2}}-\frac {c \,x^{3}}{a}+\frac {b^{3} x^{3}}{a^{2} c}+\frac {b^{2} x}{a c}\right )}{c}-\frac {\arctan \left (-\frac {c \,x^{3}}{a}+\frac {c x}{b}-\frac {b^{2} x}{a c}\right )}{c}+\frac {\arctan \left (\frac {c x}{b}\right )}{c}\) | \(96\) |
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Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 4.37 \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {\arctan \left (\frac {c x}{b}\right ) - \arctan \left (\frac {b c^{2} x^{5} + a b^{2} x + {\left (b^{3} - a c^{2}\right )} x^{3}}{a^{2} c}\right ) + \arctan \left (\frac {b c^{2} x^{3} + {\left (b^{3} - a c^{2}\right )} x}{a b c}\right )}{c} \]
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Result contains complex when optimal does not.
Time = 0.61 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.32 \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {- \frac {i \log {\left (- \frac {i a}{c} - \frac {i b x^{2}}{c} + x^{3} \right )}}{2} + \frac {i \log {\left (\frac {i a}{c} + \frac {i b x^{2}}{c} + x^{3} \right )}}{2}}{c} \]
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\[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )} x^{2}}{c^{2} x^{6} + b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )} x^{2}}{c^{2} x^{6} + b^{2} x^{4} + 2 \, a b x^{2} + a^{2}} \,d x } \]
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Time = 9.55 (sec) , antiderivative size = 252, normalized size of antiderivative = 13.26 \[ \int \frac {x^2 \left (3 a+b x^2\right )}{a^2+2 a b x^2+b^2 x^4+c^2 x^6} \, dx=\frac {\mathrm {atan}\left (\frac {27\,a\,c^5\,x^3}{27\,a^2\,c^4-4\,a\,b^3\,c^2}-\frac {27\,b\,c^5\,x^5}{27\,a^2\,c^4-4\,a\,b^3\,c^2}-\frac {31\,b^3\,c^3\,x^3}{27\,a^2\,c^4-4\,a\,b^3\,c^2}+\frac {4\,b^6\,c\,x^3}{27\,a^3\,c^4-4\,a^2\,b^3\,c^2}+\frac {4\,b^5\,c\,x}{27\,a^2\,c^4-4\,a\,b^3\,c^2}+\frac {4\,b^4\,c^3\,x^5}{27\,a^3\,c^4-4\,a^2\,b^3\,c^2}-\frac {27\,a\,b^2\,c^3\,x}{27\,a^2\,c^4-4\,a\,b^3\,c^2}\right )+\mathrm {atan}\left (\frac {c\,x^3}{a}-\frac {c\,x}{b}+\frac {b^2\,x}{a\,c}\right )+\mathrm {atan}\left (\frac {c\,x}{b}\right )}{c} \]
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