Integrand size = 42, antiderivative size = 40 \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\frac {\arctan \left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2119, 211} \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\frac {\arctan \left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rule 211
Rule 2119
Rubi steps \begin{align*} \text {integral}& = \left (3 e^2\right ) \text {Subst}\left (\int \frac {1}{e^2+36 d e^2 f x^2} \, dx,x,\frac {x^3}{3 e+6 f x^2}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.12 \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\frac {\text {RootSum}\left [e^2+4 e f \text {$\#$1}^2+4 f^2 \text {$\#$1}^4+4 d f \text {$\#$1}^6\&,\frac {3 e \log (x-\text {$\#$1}) \text {$\#$1}+2 f \log (x-\text {$\#$1}) \text {$\#$1}^3}{e+2 f \text {$\#$1}^2+3 d \text {$\#$1}^4}\&\right ]}{8 f} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.85
method | result | size |
default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )}{\sum }\frac {\left (2 \textit {\_R}^{4} f +3 e \,\textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{3 d \,\textit {\_R}^{5}+2 \textit {\_R}^{3} f +e \textit {\_R}}}{8 f}\) | \(74\) |
risch | \(-\frac {\ln \left (-2 d^{2} f^{2} x^{3}+2 \left (-d f \right )^{\frac {3}{2}} f \,x^{2}+\left (-d f \right )^{\frac {3}{2}} e \right )}{4 \sqrt {-d f}}+\frac {\ln \left (2 d^{2} f^{2} x^{3}+2 \left (-d f \right )^{\frac {3}{2}} f \,x^{2}+\left (-d f \right )^{\frac {3}{2}} e \right )}{4 \sqrt {-d f}}\) | \(84\) |
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Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (30) = 60\).
Time = 0.31 (sec) , antiderivative size = 208, normalized size of antiderivative = 5.20 \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, e f x^{2} - e^{2} - 4 \, {\left (2 \, f x^{5} + e x^{3}\right )} \sqrt {-d f}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}}\right )}{4 \, d f}, \frac {\sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{f}\right ) - \sqrt {d f} \arctan \left (\frac {2 \, {\left (2 \, d f x^{5} - {\left (d e - 2 \, f^{2}\right )} x^{3} + e f x\right )} \sqrt {d f}}{d e^{2}}\right ) + \sqrt {d f} \arctan \left (\frac {{\left (2 \, d f x^{3} - {\left (d e - 2 \, f^{2}\right )} x\right )} \sqrt {d f}}{d e f}\right )}{2 \, d f}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (36) = 72\).
Time = 0.65 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.25 \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=- \frac {\sqrt {- \frac {1}{d f}} \log {\left (- \frac {e \sqrt {- \frac {1}{d f}}}{2} - f x^{2} \sqrt {- \frac {1}{d f}} + x^{3} \right )}}{4} + \frac {\sqrt {- \frac {1}{d f}} \log {\left (\frac {e \sqrt {- \frac {1}{d f}}}{2} + f x^{2} \sqrt {- \frac {1}{d f}} + x^{3} \right )}}{4} \]
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\[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\int { \frac {{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}} \,d x } \]
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\[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\int { \frac {{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}} \,d x } \]
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Time = 17.47 (sec) , antiderivative size = 278, normalized size of antiderivative = 6.95 \[ \int \frac {x^2 \left (3 e+2 f x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^6} \, dx=\frac {\mathrm {atan}\left (\frac {2\,f^2\,x+2\,d\,f\,x^3-d\,e\,x}{\sqrt {d}\,e\,\sqrt {f}}\right )-\mathrm {atan}\left (\frac {1984\,d^{3/2}\,f^{9/2}\,x^3}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {1728\,d^{5/2}\,f^{7/2}\,x^5}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {512\,\sqrt {d}\,f^{13/2}\,x^3}{128\,d\,e^2\,f^4-432\,d^2\,e^3\,f^2}+\frac {512\,d^{3/2}\,f^{11/2}\,x^5}{128\,d\,e^2\,f^4-432\,d^2\,e^3\,f^2}-\frac {256\,\sqrt {d}\,f^{11/2}\,x}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {864\,d^{3/2}\,e\,f^{7/2}\,x}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}-\frac {864\,d^{5/2}\,e\,f^{5/2}\,x^3}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}\right )+\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {f}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]
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