Optimal. Leaf size=88 \[ \frac{a}{6 b^2 n \left (a+b x^n\right )^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{5 b^2 n \left (a+b x^n\right )^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
[Out]
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Rubi [A] time = 0.122044, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094 \[ \frac{a}{6 b^2 n \left (a+b x^n\right )^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{5 b^2 n \left (a+b x^n\right )^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(7/2),x]
[Out]
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Rubi in Sympy [A] time = 15.2018, size = 73, normalized size = 0.83 \[ \frac{a \left (2 a + 2 b x^{n}\right )}{12 b^{2} n \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{7}{2}}} - \frac{1}{5 b^{2} n \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{5}{2}}} \]
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))**(7/2),x)
[Out]
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Mathematica [A] time = 0.0492915, size = 40, normalized size = 0.45 \[ -\frac{a+6 b x^n}{30 b^2 n \left (a+b x^n\right )^5 \sqrt{\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/(a^2 + 2*a*b*x^n + b^2*x^(2*n))^(7/2),x]
[Out]
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Maple [A] time = 0.048, size = 37, normalized size = 0.4 \[ -{\frac{6\,b{x}^{n}+a}{30\, \left ( a+b{x}^{n} \right ) ^{7}{b}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]
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))^(7/2),x)
[Out]
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Maxima [A] time = 0.767999, size = 131, normalized size = 1.49 \[ -\frac{6 \, b x^{n} + a}{30 \,{\left (b^{8} n x^{6 \, n} + 6 \, a b^{7} n x^{5 \, n} + 15 \, a^{2} b^{6} n x^{4 \, n} + 20 \, a^{3} b^{5} n x^{3 \, n} + 15 \, a^{4} b^{4} n x^{2 \, n} + 6 \, a^{5} b^{3} n x^{n} + a^{6} b^{2} n\right )}} \]
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)^(7/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.271406, size = 131, normalized size = 1.49 \[ -\frac{6 \, b x^{n} + a}{30 \,{\left (b^{8} n x^{6 \, n} + 6 \, a b^{7} n x^{5 \, n} + 15 \, a^{2} b^{6} n x^{4 \, n} + 20 \, a^{3} b^{5} n x^{3 \, n} + 15 \, a^{4} b^{4} n x^{2 \, n} + 6 \, a^{5} b^{3} n x^{n} + a^{6} b^{2} n\right )}} \]
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)^(7/2),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))**(7/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{7}{2}}}\,{d x} \]
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)^(7/2),x, algorithm="giac")
[Out]