Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]
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Rubi [A] time = 0.0589649, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6, 1352, 618, 204} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 6
Rule 1352
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{x^2}{e^2+4 e f x^3+4 d f x^6+4 f^2 x^6} \, dx &=\int \frac{x^2}{e^2+4 e f x^3+4 \left (d f+f^2\right ) x^6} \, dx\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{e^2+4 e f x+4 \left (d f+f^2\right ) x^2} \, dx,x,x^3\right )\\ &=-\left (\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-16 d e^2 f-x^2} \, dx,x,4 f \left (e+2 (d+f) x^3\right )\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (e+2 (d+f) x^3\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}}\\ \end{align*}
Mathematica [A] time = 0.0195887, size = 42, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{f} \left (2 x^3 (d+f)+e\right )}{\sqrt{d} e}\right )}{6 \sqrt{d} e \sqrt{f}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 42, normalized size = 1. \begin{align*}{\frac{1}{6\,e}\arctan \left ({\frac{2\, \left ( 4\,df+4\,{f}^{2} \right ){x}^{3}+4\,fe}{4\,e}{\frac{1}{\sqrt{df}}}} \right ){\frac{1}{\sqrt{df}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24589, size = 342, normalized size = 8.14 \begin{align*} \left [-\frac{\sqrt{-d f} \log \left (\frac{4 \,{\left (d^{2} f + 2 \, d f^{2} + f^{3}\right )} x^{6} + 4 \,{\left (d e f + e f^{2}\right )} x^{3} - d e^{2} + e^{2} f - 2 \,{\left (2 \,{\left (d e + e f\right )} x^{3} + e^{2}\right )} \sqrt{-d f}}{4 \,{\left (d f + f^{2}\right )} x^{6} + 4 \, e f x^{3} + e^{2}}\right )}{12 \, d e f}, \frac{\sqrt{d f} \arctan \left (\frac{{\left (2 \,{\left (d + f\right )} x^{3} + e\right )} \sqrt{d f}}{d e}\right )}{6 \, d e f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.693828, size = 78, normalized size = 1.86 \begin{align*} \frac{- \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{3} + \frac{- d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{12} + \frac{\sqrt{- \frac{1}{d f}} \log{\left (x^{3} + \frac{d e \sqrt{- \frac{1}{d f}} + e}{2 d + 2 f} \right )}}{12}}{e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.80116, size = 51, normalized size = 1.21 \begin{align*} \frac{\arctan \left (\frac{{\left (2 \, d f x^{3} + 2 \, f^{2} x^{3} + f e\right )} e^{\left (-1\right )}}{\sqrt{d f}}\right ) e^{\left (-1\right )}}{6 \, \sqrt{d f}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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