Optimal. Leaf size=96 \[ -\frac{b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
[Out]
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Rubi [A] time = 0.078336, antiderivative size = 96, 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 \[ -\frac{b^2 x^{n-2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(2-n) \left (a b+b^2 x^n\right )}-\frac{a \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 x^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 9.37828, size = 80, normalized size = 0.83 \[ \frac{2 a b n \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{x^{2} \left (- n + 2\right ) \left (4 a b + 4 b^{2} x^{n}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{x^{2} \left (- n + 2\right )} \]
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**3,x)
[Out]
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Mathematica [A] time = 0.0348301, size = 47, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^n\right )^2} \left (2 b x^n-a (n-2)\right )}{2 (n-2) x^2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]/x^3,x]
[Out]
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Maple [A] time = 0.028, size = 61, normalized size = 0.6 \[ -{\frac{a}{ \left ( 2\,a+2\,b{x}^{n} \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( -2+n \right ){x}^{2}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]
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^3,x)
[Out]
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Maxima [A] time = 0.759294, size = 30, normalized size = 0.31 \[ -\frac{a{\left (n - 2\right )} - 2 \, b x^{n}}{2 \,{\left (n - 2\right )} x^{2}} \]
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^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272887, size = 31, normalized size = 0.32 \[ -\frac{a n - 2 \, b x^{n} - 2 \, a}{2 \,{\left (n - 2\right )} x^{2}} \]
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^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (a + b x^{n}\right )^{2}}}{x^{3}}\, dx \]
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**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{x^{3}}\,{d x} \]
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^3,x, algorithm="giac")
[Out]