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

Optimal. Leaf size=43 \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]

[Out]

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

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Rubi [A]  time = 0.0291258, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{-\frac{n+1}{2 n}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.73376, size = 41, normalized size = 0.95 \[ \frac{x \left (2 a + 2 b x^{n}\right ) \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{- \frac{n + 1}{2 n}}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0954756, size = 32, normalized size = 0.74 \[ \frac{x \left (a+b x^n\right ) \left (\left (a+b x^n\right )^2\right )^{-\frac{n+1}{2 n}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.053, size = 51, normalized size = 1.2 \[{1 \left ( x+{\frac{bx{{\rm e}^{n\ln \left ( x \right ) }}}{a}} \right ) \left ({{\rm e}^{{\frac{ \left ( 1+n \right ) \ln \left ({a}^{2}+2\,ab{{\rm e}^{n\ln \left ( x \right ) }}+{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \right ) }{2\,n}}}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x+b/a*x*exp(n*ln(x)))/exp(1/2*(1+n)/n*ln(a^2+2*a*b*exp(n*ln(x))+b^2*exp(n*ln(x)
)^2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.28077, size = 61, normalized size = 1.42 \[ \frac{b x x^{n} + a x}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{n + 1}{2 \, n}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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