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