Optimal. Leaf size=40 \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]
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Rubi [A] time = 0.13, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2094, 205} \[ \frac {\tan ^{-1}\left (\frac {2 \sqrt {d} \sqrt {f} x^3}{e+2 f x^2}\right )}{2 \sqrt {d} \sqrt {f}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 2094
Rubi steps
\begin {align*} \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 (3 e^2\right ) \operatorname {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*}
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Mathematica [C] time = 0.05, size = 85, normalized size = 2.12 \[ \frac {\text {RootSum}\left [4 \text {$\#$1}^6 d f+4 \text {$\#$1}^4 f^2+4 \text {$\#$1}^2 e f+e^2\& ,\frac {2 \text {$\#$1}^3 f \log (x-\text {$\#$1})+3 \text {$\#$1} e \log (x-\text {$\#$1})}{3 \text {$\#$1}^4 d+2 \text {$\#$1}^2 f+e}\& \right ]}{8 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 208, normalized size = 5.20 \[ \left [-\frac {\sqrt {-d f} \log \left (\frac {4 \, d f x^{6} - 4 \, f^{2} x^{4} - 4 \, e f x^{2} - e^{2} - 4 \, {\left (2 \, f x^{5} + e x^{3}\right )} \sqrt {-d f}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}}\right )}{4 \, d f}, \frac {\sqrt {d f} \arctan \left (\frac {\sqrt {d f} x}{f}\right ) - \sqrt {d f} \arctan \left (\frac {2 \, {\left (2 \, d f x^{5} - {\left (d e - 2 \, f^{2}\right )} x^{3} + e f x\right )} \sqrt {d f}}{d e^{2}}\right ) + \sqrt {d f} \arctan \left (\frac {{\left (2 \, d f x^{3} - {\left (d e - 2 \, f^{2}\right )} x\right )} \sqrt {d f}}{d e f}\right )}{2 \, d f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.15, size = 74, normalized size = 1.85 \[ \frac {\left (2 \RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )^{4} f +3 \RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )^{2} e \right ) \ln \left (-\RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )+x \right )}{8 f \left (3 d \RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )^{5}+2 f \RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )^{3}+e \RootOf \left (4 d f \,\textit {\_Z}^{6}+4 f^{2} \textit {\_Z}^{4}+4 e f \,\textit {\_Z}^{2}+e^{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (2 \, f x^{2} + 3 \, e\right )} x^{2}}{4 \, d f x^{6} + 4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.21, size = 278, normalized size = 6.95 \[ \frac {\mathrm {atan}\left (\frac {2\,f^2\,x+2\,d\,f\,x^3-d\,e\,x}{\sqrt {d}\,e\,\sqrt {f}}\right )-\mathrm {atan}\left (\frac {1984\,d^{3/2}\,f^{9/2}\,x^3}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {1728\,d^{5/2}\,f^{7/2}\,x^5}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {512\,\sqrt {d}\,f^{13/2}\,x^3}{128\,d\,e^2\,f^4-432\,d^2\,e^3\,f^2}+\frac {512\,d^{3/2}\,f^{11/2}\,x^5}{128\,d\,e^2\,f^4-432\,d^2\,e^3\,f^2}-\frac {256\,\sqrt {d}\,f^{11/2}\,x}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}+\frac {864\,d^{3/2}\,e\,f^{7/2}\,x}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}-\frac {864\,d^{5/2}\,e\,f^{5/2}\,x^3}{432\,d^2\,e^2\,f^2-128\,d\,e\,f^4}\right )+\mathrm {atan}\left (\frac {\sqrt {d}\,x}{\sqrt {f}}\right )}{2\,\sqrt {d}\,\sqrt {f}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.13, size = 90, normalized size = 2.25 \[ - \frac {\sqrt {- \frac {1}{d f}} \log {\left (- \frac {e \sqrt {- \frac {1}{d f}}}{2} - f x^{2} \sqrt {- \frac {1}{d f}} + x^{3} \right )}}{4} + \frac {\sqrt {- \frac {1}{d f}} \log {\left (\frac {e \sqrt {- \frac {1}{d f}}}{2} + f x^{2} \sqrt {- \frac {1}{d f}} + x^{3} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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