Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
[Out]
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Rubi [A] time = 0.340527, 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 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^2}\right )}{2 \sqrt{d} \sqrt{f}} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^(2 + 2*m)),x]
[Out]
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Rubi in Sympy [A] time = 106.903, size = 37, normalized size = 0.88 \[ \frac{\operatorname{atan}{\left (\frac{2 \sqrt{d} \sqrt{f} x^{m + 1}}{e + 2 f x^{2}} \right )}}{2 \sqrt{d} \sqrt{f}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**m*(e*(1+m)+2*f*(-1+m)*x**2)/(e**2+4*e*f*x**2+4*f**2*x**4+4*d*f*x**(2+2*m)),x)
[Out]
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Mathematica [A] time = 0.0835232, 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.
[In] Integrate[(x^m*(e*(1 + m) + 2*f*(-1 + m)*x^2))/(e^2 + 4*e*f*x^2 + 4*f^2*x^4 + 4*d*f*x^(2 + 2*m)),x]
[Out]
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Maple [B] time = 0.09, size = 78, normalized size = 1.9 \[ -{\frac{1}{4}\ln \left ({x}^{m}+{\frac{2\,f{x}^{2}+e}{2\,dfx}\sqrt{-df}} \right ){\frac{1}{\sqrt{-df}}}}+{\frac{1}{4}\ln \left ({x}^{m}-{\frac{2\,f{x}^{2}+e}{2\,dfx}\sqrt{-df}} \right ){\frac{1}{\sqrt{-df}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^m*(e*(1+m)+2*f*(-1+m)*x^2)/(e^2+4*e*f*x^2+4*f^2*x^4+4*d*f*x^(2+2*m)),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, f{\left (m - 1\right )} x^{2} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{4} + 4 \, e f x^{2} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*f*(m - 1)*x^2 + e*(m + 1))*x^m/(4*f^2*x^4 + 4*e*f*x^2 + 4*d*f*x^(2*m + 2) + e^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.30025, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (\frac{4 \, \sqrt{-d f} d f x^{2} x^{2 \, m} + 4 \,{\left (2 \, d f^{2} x^{3} + d e f x\right )} x^{m} -{\left (4 \, f^{2} x^{4} + 4 \, e f x^{2} + e^{2}\right )} \sqrt{-d f}}{4 \, f^{2} x^{4} + 4 \, d f x^{2} x^{2 \, m} + 4 \, e f x^{2} + e^{2}}\right )}{4 \, \sqrt{-d f}}, -\frac{\arctan \left (\frac{2 \, f x^{2} + e}{2 \, \sqrt{d f} x x^{m}}\right )}{2 \, \sqrt{d f}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*f*(m - 1)*x^2 + e*(m + 1))*x^m/(4*f^2*x^4 + 4*e*f*x^2 + 4*d*f*x^(2*m + 2) + e^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**m*(e*(1+m)+2*f*(-1+m)*x**2)/(e**2+4*e*f*x**2+4*f**2*x**4+4*d*f*x**(2+2*m)),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, f{\left (m - 1\right )} x^{2} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, f^{2} x^{4} + 4 \, e f x^{2} + 4 \, d f x^{2 \, m + 2} + e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*f*(m - 1)*x^2 + e*(m + 1))*x^m/(4*f^2*x^4 + 4*e*f*x^2 + 4*d*f*x^(2*m + 2) + e^2),x, algorithm="giac")
[Out]