3.520 \(\int \frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{b^2 x^{n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(x*(a + b*x^n))) - (b^2*x^(-1 + n)*Sqr
t[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((1 - n)*(a*b + b^2*x^n))

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Rubi [A]  time = 0.0793567, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ -\frac{b^2 x^{n-1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{x \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

-((a*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/(x*(a + b*x^n))) - (b^2*x^(-1 + n)*Sqr
t[a^2 + 2*a*b*x^n + b^2*x^(2*n)])/((1 - n)*(a*b + b^2*x^n))

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Rubi in Sympy [A]  time = 9.31757, size = 76, normalized size = 0.81 \[ \frac{2 a b n \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{x \left (- n + 1\right ) \left (2 a b + 2 b^{2} x^{n}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{x \left (- n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0321772, size = 42, normalized size = 0.45 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (-a n+a+b x^n\right )}{(n-1) x \left (a+b x^n\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.029, size = 61, normalized size = 0.7 \[ -{\frac{a}{ \left ( a+b{x}^{n} \right ) x}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( -1+n \right ) x}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.753678, size = 30, normalized size = 0.32 \[ -\frac{a{\left (n - 1\right )} - b x^{n}}{{\left (n - 1\right )} x} \]

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,x, algorithm="maxima")

[Out]

-(a*(n - 1) - b*x^n)/((n - 1)*x)

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Fricas [A]  time = 0.277106, size = 31, normalized size = 0.33 \[ -\frac{a n - b x^{n} - a}{{\left (n - 1\right )} x} \]

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,x, algorithm="fricas")

[Out]

-(a*n - b*x^n - a)/((n - 1)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{n}\right )^{2}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((a + b*x**n)**2)/x**2, x)

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

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,x, algorithm="giac")

[Out]

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