3.2737 \(\int \frac{x^m}{\left (a+b x^{2+2 m}\right )^2} \, dx\)

Optimal. Leaf size=67 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} (m+1)}+\frac{x^{m+1}}{2 a (m+1) \left (a+b x^{2 (m+1)}\right )} \]

[Out]

x^(1 + m)/(2*a*(1 + m)*(a + b*x^(2*(1 + m)))) + ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[
a]]/(2*a^(3/2)*Sqrt[b]*(1 + m))

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Rubi [A]  time = 0.0739474, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{2 a^{3/2} \sqrt{b} (m+1)}+\frac{x^{m+1}}{2 a (m+1) \left (a+b x^{2 (m+1)}\right )} \]

Antiderivative was successfully verified.

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

[Out]

x^(1 + m)/(2*a*(1 + m)*(a + b*x^(2*(1 + m)))) + ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[
a]]/(2*a^(3/2)*Sqrt[b]*(1 + m))

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Rubi in Sympy [A]  time = 4.98206, size = 27, normalized size = 0.4 \[ \frac{x^{m + 1}{{}_{2}F_{1}\left (\begin{matrix} 2, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{b x^{2 m + 2}}{a}} \right )}}{a^{2} \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.115383, size = 59, normalized size = 0.88 \[ \frac{\frac{\tan ^{-1}\left (\frac{\sqrt{b} x^{m+1}}{\sqrt{a}}\right )}{a^{3/2} \sqrt{b}}+\frac{x^{m+1}}{a^2+a b x^{2 m+2}}}{2 m+2} \]

Antiderivative was successfully verified.

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

[Out]

(x^(1 + m)/(a^2 + a*b*x^(2 + 2*m)) + ArcTan[(Sqrt[b]*x^(1 + m))/Sqrt[a]]/(a^(3/2
)*Sqrt[b]))/(2 + 2*m)

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Maple [A]  time = 0.056, size = 95, normalized size = 1.4 \[{\frac{x{x}^{m}}{ \left ( 2+2\,m \right ) a \left ( a+b{x}^{2} \left ({x}^{m} \right ) ^{2} \right ) }}-{\frac{1}{ \left ( 4+4\,m \right ) a}\ln \left ({x}^{m}-{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}}+{\frac{1}{ \left ( 4+4\,m \right ) a}\ln \left ({x}^{m}+{\frac{a}{x}{\frac{1}{\sqrt{-ab}}}} \right ){\frac{1}{\sqrt{-ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a+b*x^(2+2*m))^2,x)

[Out]

1/2*x/(1+m)/a*x^m/(a+b*x^2*(x^m)^2)-1/4/(-a*b)^(1/2)/(1+m)/a*ln(x^m-a/x/(-a*b)^(
1/2))+1/4/(-a*b)^(1/2)/(1+m)/a*ln(x^m+a/x/(-a*b)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x x^{m}}{2 \,{\left (a b{\left (m + 1\right )} x^{2} x^{2 \, m} + a^{2}{\left (m + 1\right )}\right )}} + \int \frac{x^{m}}{2 \,{\left (a b x^{2} x^{2 \, m} + a^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*x*x^m/(a*b*(m + 1)*x^2*x^(2*m) + a^2*(m + 1)) + integrate(1/2*x^m/(a*b*x^2*x
^(2*m) + a^2), x)

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Fricas [A]  time = 0.237882, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{-a b} x x^{m} +{\left (b x^{2} x^{2 \, m} + a\right )} \log \left (\frac{\sqrt{-a b} b x^{2} x^{2 \, m} + 2 \, a b x x^{m} - \sqrt{-a b} a}{b x^{2} x^{2 \, m} + a}\right )}{4 \,{\left ({\left (a b m + a b\right )} \sqrt{-a b} x^{2} x^{2 \, m} +{\left (a^{2} m + a^{2}\right )} \sqrt{-a b}\right )}}, \frac{\sqrt{a b} x x^{m} -{\left (b x^{2} x^{2 \, m} + a\right )} \arctan \left (\frac{a}{\sqrt{a b} x x^{m}}\right )}{2 \,{\left ({\left (a b m + a b\right )} \sqrt{a b} x^{2} x^{2 \, m} +{\left (a^{2} m + a^{2}\right )} \sqrt{a b}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(2*sqrt(-a*b)*x*x^m + (b*x^2*x^(2*m) + a)*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)))/((a*b*m + a*b)*sqrt(-a*b)*x^2*x
^(2*m) + (a^2*m + a^2)*sqrt(-a*b)), 1/2*(sqrt(a*b)*x*x^m - (b*x^2*x^(2*m) + a)*a
rctan(a/(sqrt(a*b)*x*x^m)))/((a*b*m + a*b)*sqrt(a*b)*x^2*x^(2*m) + (a^2*m + a^2)
*sqrt(a*b))]

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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))**2,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^m/(b*x^(2*m + 2) + a)^2, x)