3.373 \(\int \frac{x^m \left (e (1+m)+2 f (-1+m) x^2\right )}{e^2+4 e f x^2+4 f^2 x^4+4 d f x^{2+2 m}} \, dx\)

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]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])

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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]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])

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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]

atan(2*sqrt(d)*sqrt(f)*x**(m + 1)/(e + 2*f*x**2))/(2*sqrt(d)*sqrt(f))

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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]

ArcTan[(2*Sqrt[d]*Sqrt[f]*x^(1 + m))/(e + 2*f*x^2)]/(2*Sqrt[d]*Sqrt[f])

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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]

-1/4/(-d*f)^(1/2)*ln(x^m+1/2*(2*f*x^2+e)*(-d*f)^(1/2)/d/f/x)+1/4/(-d*f)^(1/2)*ln
(x^m-1/2*(2*f*x^2+e)*(-d*f)^(1/2)/d/f/x)

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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]

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)

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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]

[1/4*log((4*sqrt(-d*f)*d*f*x^2*x^(2*m) + 4*(2*d*f^2*x^3 + d*e*f*x)*x^m - (4*f^2*
x^4 + 4*e*f*x^2 + e^2)*sqrt(-d*f))/(4*f^2*x^4 + 4*d*f*x^2*x^(2*m) + 4*e*f*x^2 +
e^2))/sqrt(-d*f), -1/2*arctan(1/2*(2*f*x^2 + e)/(sqrt(d*f)*x*x^m))/sqrt(d*f)]

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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]

Timed 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]

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)