Optimal. Leaf size=40 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.0891171, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2094, 208} \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 2094
Rule 208
Rubi steps
\begin{align*} \int \frac{x \left (2 e-2 f x^3\right )}{e^2+4 e f x^3-4 d f x^4+4 f^2 x^6} \, dx &=-\left (\left (2 e^2\right ) \operatorname{Subst}\left (\int \frac{1}{e^2-16 d e^2 f x^2} \, dx,x,\frac{x^2}{-2 e-4 f x^3}\right )\right )\\ &=\frac{\tanh ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^2}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [C] time = 0.0481161, size = 86, normalized size = 2.15 \[ -\frac{\text{RootSum}\left [-4 \text{$\#$1}^4 d f+4 \text{$\#$1}^3 e f+4 \text{$\#$1}^6 f^2+e^2\& ,\frac{\text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{-4 \text{$\#$1}^2 d+6 \text{$\#$1}^4 f+3 \text{$\#$1} e}\& \right ]}{2 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.009, size = 74, normalized size = 1.9 \begin{align*} -{\frac{1}{2\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{f}^{2}{{\it \_Z}}^{6}-4\,df{{\it \_Z}}^{4}+4\,fe{{\it \_Z}}^{3}+{e}^{2} \right ) }{\frac{ \left ({{\it \_R}}^{4}f-{\it \_R}\,e \right ) \ln \left ( x-{\it \_R} \right ) }{6\,f{{\it \_R}}^{5}-4\,d{{\it \_R}}^{3}+3\,e{{\it \_R}}^{2}}}} \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*} -2 \, \int \frac{{\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.3109, size = 348, normalized size = 8.7 \begin{align*} \left [\frac{\sqrt{d f} \log \left (\frac{4 \, f^{2} x^{6} + 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2} + 4 \,{\left (2 \, f x^{5} + e x^{2}\right )} \sqrt{d f}}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\right )}{4 \, d f}, -\frac{\sqrt{-d f} \arctan \left (\frac{\sqrt{-d f} x}{d}\right ) - \sqrt{-d f} \arctan \left (\frac{{\left (2 \, f x^{4} - 2 \, d x^{2} + e x\right )} \sqrt{-d f}}{d e}\right )}{2 \, d f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16377, size = 66, normalized size = 1.65 \begin{align*} - \frac{\sqrt{\frac{1}{d f}} \log{\left (- d x^{2} \sqrt{\frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} + \frac{\sqrt{\frac{1}{d f}} \log{\left (d x^{2} \sqrt{\frac{1}{d f}} + \frac{e}{2 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{2 \,{\left (f x^{3} - e\right )} x}{4 \, f^{2} x^{6} - 4 \, d f x^{4} + 4 \, e f x^{3} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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