Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.127675, antiderivative size = 40, normalized size of antiderivative = 1., 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, 208} \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 2094
Rule 208
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{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^3}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [C] time = 0.0518767, size = 85, normalized size = 2.12 \[ \frac{\text{RootSum}\left [-4 \text{$\#$1}^6 d f+4 \text{$\#$1}^2 e f+4 \text{$\#$1}^4 f^2+e^2\& ,\frac{2 \text{$\#$1}^3 f \log (x-\text{$\#$1})+3 \text{$\#$1} e \log (x-\text{$\#$1})}{-3 \text{$\#$1}^4 d+2 \text{$\#$1}^2 f+e}\& \right ]}{8 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.105, size = 77, normalized size = 1.9 \begin{align*} -{\frac{1}{8\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,df{{\it \_Z}}^{6}-4\,{f}^{2}{{\it \_Z}}^{4}-4\,fe{{\it \_Z}}^{2}-{e}^{2} \right ) }{\frac{ \left ( 2\,{{\it \_R}}^{4}f+3\,{{\it \_R}}^{2}e \right ) \ln \left ( x-{\it \_R} \right ) }{3\,d{{\it \_R}}^{5}-2\,f{{\it \_R}}^{3}-e{\it \_R}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.23696, size = 467, normalized size = 11.68 \begin{align*} \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 ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.0913, size = 80, normalized size = 2. \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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