Optimal. Leaf size=38 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.0627856, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {2093, 205} \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 2093
Rule 205
Rubi steps
\begin{align*} \int \frac{e-4 f x^3}{e^2+4 d f x^2+4 e f x^3+4 f^2 x^6} \, dx &=\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 e+4 f x^3}\right )\\ &=\frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x}{e+2 f x^3}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [C] time = 0.0606442, size = 87, normalized size = 2.29 \[ -\frac{\text{RootSum}\left [4 \text{$\#$1}^2 d f+4 \text{$\#$1}^3 e f+4 \text{$\#$1}^6 f^2+e^2\& ,\frac{4 \text{$\#$1}^3 f \log (x-\text{$\#$1})-e \log (x-\text{$\#$1})}{3 \text{$\#$1}^2 e+6 \text{$\#$1}^5 f+2 \text{$\#$1} d}\& \right ]}{4 f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.008, size = 70, normalized size = 1.8 \begin{align*}{\frac{1}{4\,f}\sum _{{\it \_R}={\it RootOf} \left ( 4\,{f}^{2}{{\it \_Z}}^{6}+4\,fe{{\it \_Z}}^{3}+4\,df{{\it \_Z}}^{2}+{e}^{2} \right ) }{\frac{ \left ( -4\,{{\it \_R}}^{3}f+e \right ) \ln \left ( x-{\it \_R} \right ) }{6\,f{{\it \_R}}^{5}+3\,e{{\it \_R}}^{2}+2\,d{\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{4 \, f x^{3} - e}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d 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 = 1.47591, size = 347, normalized size = 9.13 \begin{align*} \left [-\frac{\sqrt{-d f} \log \left (\frac{4 \, f^{2} x^{6} + 4 \, e f x^{3} - 4 \, d f x^{2} + e^{2} + 4 \,{\left (2 \, f x^{4} + e x\right )} \sqrt{-d f}}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac{\sqrt{d f} \arctan \left (\frac{\sqrt{d f} x^{2}}{d}\right ) - \sqrt{d f} \arctan \left (\frac{{\left (2 \, f x^{5} + e x^{2} + 2 \, d 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 [B] time = 0.795029, size = 70, normalized size = 1.84 \begin{align*} \frac{\sqrt{- \frac{1}{d f}} \log{\left (- d x \sqrt{- \frac{1}{d f}} + \frac{e}{2 f} + x^{3} \right )}}{4} - \frac{\sqrt{- \frac{1}{d f}} \log{\left (d x \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{4 \, f x^{3} - e}{4 \, f^{2} x^{6} + 4 \, e f x^{3} + 4 \, d f x^{2} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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