3.2736 \(\int \frac{x^m}{a+b x^{2+2 m}} \, dx\)

Optimal. Leaf size=33 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]

[Out]

ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*(1 + m))

_______________________________________________________________________________________

Rubi [A]  time = 0.0504952, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a + b*x^(2 + 2*m)),x]

[Out]

ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*(1 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.25988, size = 26, normalized size = 0.79 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a+b*x**(2+2*m)),x)

[Out]

x**(m + 1)*hyper((1, 1/2), (3/2,), -b*x**(2*m + 2)/a)/(a*(m + 1))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0178915, size = 33, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} (m+1)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a + b*x^(2 + 2*m)),x]

[Out]

ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*(1 + m))

_______________________________________________________________________________________

Maple [B]  time = 0.069, size = 61, normalized size = 1.9 \[ -{\frac{1}{2+2\,m}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{2+2\,m}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^(2+2*m)),x)

[Out]

-1/2/(-a*b)^(1/2)/(1+m)*ln(x^m-a/x/(-a*b)^(1/2))+1/2/(-a*b)^(1/2)/(1+m)*ln(x^m+a
/x/(-a*b)^(1/2))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="maxima")

[Out]

integrate(x^m/(b*x^(2*m + 2) + a), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.24023, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (\frac{\sqrt{-a b} b x^{2} x^{2 \, m} + 2 \, a b x x^{m} - \sqrt{-a b} a}{b x^{2} x^{2 \, m} + a}\right )}{2 \, \sqrt{-a b}{\left (m + 1\right )}}, -\frac{\arctan \left (\frac{a}{\sqrt{a b} x x^{m}}\right )}{\sqrt{a b}{\left (m + 1\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="fricas")

[Out]

[1/2*log((sqrt(-a*b)*b*x^2*x^(2*m) + 2*a*b*x*x^m - sqrt(-a*b)*a)/(b*x^2*x^(2*m)
+ a))/(sqrt(-a*b)*(m + 1)), -arctan(a/(sqrt(a*b)*x*x^m))/(sqrt(a*b)*(m + 1))]

_______________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**m/(a+b*x**(2+2*m)),x)

[Out]

Exception raised: TypeError

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{b x^{2 \, m + 2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(b*x^(2*m + 2) + a),x, algorithm="giac")

[Out]

integrate(x^m/(b*x^(2*m + 2) + a), x)