3.512 \(\int \frac{x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx\)

Optimal. Leaf size=48 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])

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Rubi [A]  time = 0.0714896, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right ) \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])

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Rubi in Sympy [A]  time = 8.92025, size = 42, normalized size = 0.88 \[ \frac{x^{2 n} \left (2 a + 2 b x^{n}\right )}{4 a n \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)/(a**2+2*a*b*x**n+b**2*x**(2*n))**(3/2),x)

[Out]

x**(2*n)*(2*a + 2*b*x**n)/(4*a*n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(3/2))

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Mathematica [A]  time = 0.0410631, size = 40, normalized size = 0.83 \[ -\frac{a+2 b x^n}{2 b^2 n \left (a+b x^n\right ) \sqrt{\left (a+b x^n\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(3/2),x]

[Out]

-(a + 2*b*x^n)/(2*b^2*n*(a + b*x^n)*Sqrt[(a + b*x^n)^2])

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Maple [A]  time = 0.038, size = 37, normalized size = 0.8 \[ -{\frac{2\,b{x}^{n}+a}{2\, \left ( a+b{x}^{n} \right ) ^{3}{b}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)/(a^2+2*a*b*x^n+b^2*x^(2*n))^(3/2),x)

[Out]

-1/2*((a+b*x^n)^2)^(1/2)/(a+b*x^n)^3*(2*b*x^n+a)/b^2/n

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Maxima [A]  time = 0.764608, size = 55, normalized size = 1.15 \[ -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

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Fricas [A]  time = 0.267415, size = 55, normalized size = 1.15 \[ -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="fricas")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(2*n - 1)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(3/2), x)