3.510 \(\int x^{-1+2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx\)

Optimal. Leaf size=99 \[ \frac{a x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a+b x^n\right )}+\frac{b^2 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )} \]

[Out]

(a*x^(2*n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(2*n*(a + b*x^n)) + (b^2*x^(3*n)
*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*n*(a*b + b^2*x^n))

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Rubi [A]  time = 0.0809695, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{a x^{2 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a+b x^n\right )}+\frac{b^2 x^{3 n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(a*x^(2*n)*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(2*n*(a + b*x^n)) + (b^2*x^(3*n)
*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(3*n*(a*b + b^2*x^n))

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Rubi in Sympy [A]  time = 9.1947, size = 80, normalized size = 0.81 \[ \frac{a b x^{2 n} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{3 n \left (2 a b + 2 b^{2} x^{n}\right )} + \frac{x^{2 n} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{3 n} \]

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))**(1/2),x)

[Out]

a*b*x**(2*n)*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/(3*n*(2*a*b + 2*b**2*x**n))
+ x**(2*n)*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/(3*n)

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Mathematica [A]  time = 0.0274296, size = 44, normalized size = 0.44 \[ \frac{x^{2 n} \sqrt{\left (a+b x^n\right )^2} \left (3 a+2 b x^n\right )}{6 n \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

(x^(2*n)*Sqrt[(a + b*x^n)^2]*(3*a + 2*b*x^n))/(6*n*(a + b*x^n))

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Maple [A]  time = 0.034, size = 64, normalized size = 0.7 \[{\frac{b \left ({x}^{n} \right ) ^{3}}{ \left ( 3\,a+3\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{a \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]

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))^(1/2),x)

[Out]

1/3*((a+b*x^n)^2)^(1/2)/(a+b*x^n)*b/n*(x^n)^3+1/2*((a+b*x^n)^2)^(1/2)/(a+b*x^n)*
a/n*(x^n)^2

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Maxima [A]  time = 0.760705, size = 30, normalized size = 0.3 \[ \frac{2 \, b x^{3 \, n} + 3 \, a x^{2 \, n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^(3*n) + 3*a*x^(2*n))/n

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Fricas [A]  time = 0.268874, size = 30, normalized size = 0.3 \[ \frac{2 \, b x^{3 \, n} + 3 \, a x^{2 \, n}}{6 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*x^(3*n) + 3*a*x^(2*n))/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))**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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