Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
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Rubi [A] time = 0.219073, antiderivative size = 61, normalized size of antiderivative = 1.45, number of steps used = 2, number of rules used = 2, integrand size = 51, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.039, Rules used = {2094, 205} \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} \left (1-m^2\right ) x^{m+1}}{(1-m) (m+1) \left (e+2 f x^2\right )}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Rule 2094
Rule 205
Rubi steps
\begin{align*} \int \frac{x^m \left (e (1+m)+2 f (-1+m) x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^{2+2 m}} \, dx &=-\left (\left (e^2 (1-m) (1+m)\right ) \operatorname{Subst}\left (\int \frac{1}{e^2+4 d e^2 f (-1+m)^2 (1+m)^2 x^2} \, dx,x,\frac{x^{1+m}}{e (-1+m) (1+m)+2 f (-1+m) (1+m) x^2}\right )\right )\\ &=\frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} \left (1-m^2\right ) x^{1+m}}{(1-m) (1+m) \left (e+2 f x^2\right )}\right )}{2 \sqrt{d} \sqrt{f}}\\ \end{align*}
Mathematica [A] time = 0.316644, size = 42, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.053, size = 78, normalized size = 1.9 \begin{align*} -{\frac{1}{4}\ln \left ({x}^{m}+{\frac{2\,f{x}^{2}+e}{2\,dfx}\sqrt{-df}} \right ){\frac{1}{\sqrt{-df}}}}+{\frac{1}{4}\ln \left ({x}^{m}-{\frac{2\,f{x}^{2}+e}{2\,dfx}\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (2 \, f{\left (m - 1\right )} x^{2} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{4} + 4 \, e f x^{2} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37995, size = 321, normalized size = 7.64 \begin{align*} \left [-\frac{\sqrt{-d f} \log \left (-\frac{4 \, f^{2} x^{4} - 4 \, d f x^{2} x^{2 \, m} + 4 \, e f x^{2} + 4 \,{\left (2 \, f x^{3} + e x\right )} \sqrt{-d f} x^{m} + e^{2}}{4 \, f^{2} x^{4} + 4 \, d f x^{2} x^{2 \, m} + 4 \, e f x^{2} + e^{2}}\right )}{4 \, d f}, -\frac{\sqrt{d f} \arctan \left (\frac{{\left (2 \, f x^{2} + e\right )} \sqrt{d f}}{2 \, d f x x^{m}}\right )}{2 \, d f}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (m - 1\right )} x^{2} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{4} + 4 \, e f x^{2} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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