Optimal. Leaf size=24 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]
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Rubi [A] time = 0.0229047, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/(a + b*x^n)^3,x]
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Rubi in Sympy [A] time = 3.08278, size = 17, normalized size = 0.71 \[ \frac{x^{2 n}}{2 a n \left (a + b x^{n}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)/(a+b*x**n)**3,x)
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Mathematica [A] time = 0.020997, size = 27, normalized size = 1.12 \[ -\frac{a+2 b x^n}{2 b^2 n \left (a+b x^n\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/(a + b*x^n)^3,x]
[Out]
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Maple [A] time = 0.039, size = 36, normalized size = 1.5 \[{\frac{1}{ \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{bn}}-{\frac{a}{2\,{b}^{2}n}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)/(a+b*x^n)^3,x)
[Out]
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Maxima [A] time = 1.51647, size = 55, normalized size = 2.29 \[ -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.217433, size = 55, normalized size = 2.29 \[ -\frac{2 \, b x^{n} + a}{2 \,{\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a)^3,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+b*x**n)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/(b*x^n + a)^3,x, algorithm="giac")
[Out]