3.514 \(\int \frac{x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{7/2}} \, dx\)

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]

a/(6*b^2*n*(a + b*x^n)^5*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) - 1/(5*b^2*n*(a +
b*x^n)^4*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])

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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]

a/(6*b^2*n*(a + b*x^n)^5*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]) - 1/(5*b^2*n*(a +
b*x^n)^4*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])

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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]

a*(2*a + 2*b*x**n)/(12*b**2*n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(7/2)) - 1/(5
*b**2*n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(5/2))

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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]

-(a + 6*b*x^n)/(30*b^2*n*(a + b*x^n)^5*Sqrt[(a + b*x^n)^2])

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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]

-1/30*((a+b*x^n)^2)^(1/2)/(a+b*x^n)^7*(6*b*x^n+a)/b^2/n

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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]

-1/30*(6*b*x^n + a)/(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)

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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]

-1/30*(6*b*x^n + a)/(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)

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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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{7}{2}}}\,{d x} \]

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]

integrate(x^(2*n - 1)/(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(7/2), x)