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

Optimal. Leaf size=42 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^n}\right )}{2 \sqrt{d} \sqrt{f}} \]

[Out]

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

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Rubi [A]  time = 0.382072, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 56, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{\tan ^{-1}\left (\frac{2 \sqrt{d} \sqrt{f} x^{m+1}}{e+2 f x^n}\right )}{2 \sqrt{d} \sqrt{f}} \]

Antiderivative was successfully verified.

[In]  Int[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)),x]

[Out]

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

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Rubi in Sympy [A]  time = 74.6395, size = 63, normalized size = 1.5 \[ \frac{\operatorname{atan}{\left (\frac{2 \sqrt{d} \sqrt{f} x^{m + 1} \left (m + 1\right ) \left (m - n + 1\right )}{e \left (m + 1\right ) \left (m - n + 1\right ) + 2 f x^{n} \left (m + 1\right ) \left (m - n + 1\right )} \right )}}{2 \sqrt{d} \sqrt{f}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m*(e*(1+m)+2*f*(1+m-n)*x**n)/(e**2+4*d*f*x**(2+2*m)+4*e*f*x**n+4*f**2*x**(2*n)),x)

[Out]

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

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Mathematica [A]  time = 0.267333, size = 0, normalized size = 0. \[ \int \frac{x^m \left (e (1+m)+2 f (1+m-n) x^n\right )}{e^2+4 d f x^{2+2 m}+4 e f x^n+4 f^2 x^{2 n}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e*f*x^n + 4*f^2*x^(2*n)),x]

[Out]

Integrate[(x^m*(e*(1 + m) + 2*f*(1 + m - n)*x^n))/(e^2 + 4*d*f*x^(2 + 2*m) + 4*e
*f*x^n + 4*f^2*x^(2*n)), x]

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Maple [B]  time = 0.111, size = 84, normalized size = 2. \[ -{\frac{1}{4}\ln \left ({x}^{n}+{\frac{1}{2\,f} \left ( 2\,dfx{x}^{m}+e\sqrt{-df} \right ){\frac{1}{\sqrt{-df}}}} \right ){\frac{1}{\sqrt{-df}}}}+{\frac{1}{4}\ln \left ({x}^{n}+{\frac{1}{2\,f} \left ( -2\,dfx{x}^{m}+e\sqrt{-df} \right ){\frac{1}{\sqrt{-df}}}} \right ){\frac{1}{\sqrt{-df}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(e*(1+m)+2*f*(1+m-n)*x^n)/(e^2+4*d*f*x^(2+2*m)+4*e*f*x^n+4*f^2*x^(2*n)),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, f{\left (m - n + 1\right )} x^{n} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, d f x^{2 \, m + 2} + 4 \, f^{2} x^{2 \, n} + 4 \, e f x^{n} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) + 4*f^2*x^(2*n) + 4*e*f*x^n + e^2),x, algorithm="maxima")

[Out]

integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) + 4*f^2*x^(2*
n) + 4*e*f*x^n + e^2), x)

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Fricas [A]  time = 0.343854, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{4 \, \sqrt{-d f} d f x^{2} x^{2 \, m} + 4 \, d e f x x^{m} - 4 \, \sqrt{-d f} f^{2} x^{2 \, n} - \sqrt{-d f} e^{2} + 4 \,{\left (2 \, d f^{2} x x^{m} - \sqrt{-d f} e f\right )} x^{n}}{4 \, d f x^{2} x^{2 \, m} + 4 \, f^{2} x^{2 \, n} + 4 \, e f x^{n} + e^{2}}\right )}{4 \, \sqrt{-d f}}, -\frac{\arctan \left (\frac{2 \, \sqrt{d f} f x^{n} + \sqrt{d f} e}{2 \, d f x x^{m}}\right )}{2 \, \sqrt{d f}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) + 4*f^2*x^(2*n) + 4*e*f*x^n + e^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (2 \, f{\left (m - n + 1\right )} x^{n} + e{\left (m + 1\right )}\right )} x^{m}}{4 \, d f x^{2 \, m + 2} + 4 \, f^{2} x^{2 \, n} + 4 \, e f x^{n} + e^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) + 4*f^2*x^(2*n) + 4*e*f*x^n + e^2),x, algorithm="giac")

[Out]

integrate((2*f*(m - n + 1)*x^n + e*(m + 1))*x^m/(4*d*f*x^(2*m + 2) + 4*f^2*x^(2*
n) + 4*e*f*x^n + e^2), x)