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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]