Optimal. Leaf size=88 \[ \frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
[Out]
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Rubi [A] time = 0.0467569, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ \frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]
[Out]
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Rubi in Sympy [A] time = 3.28758, size = 76, normalized size = 0.86 \[ \frac{2 a b n x \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (n + 1\right ) \left (2 a b + 2 b^{2} x^{n}\right )} + \frac{x \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{n + 1} \]
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)
[Out]
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Mathematica [A] time = 0.0200341, size = 39, normalized size = 0.44 \[ \frac{x \sqrt{\left (a+b x^n\right )^2} \left (a n+a+b x^n\right )}{(n+1) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]
[Out]
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Maple [A] time = 0.02, size = 56, normalized size = 0.6 \[{\frac{ax}{a+b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{bx{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 1+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]
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)
[Out]
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Maxima [A] time = 0.749602, size = 26, normalized size = 0.3 \[ \frac{a{\left (n + 1\right )} x + b x x^{n}}{n + 1} \]
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, algorithm="maxima")
[Out]
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Fricas [A] time = 0.27853, size = 27, normalized size = 0.31 \[ \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \]
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, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}\, dx \]
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)
[Out]
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GIAC/XCAS [A] time = 0.27933, size = 34, normalized size = 0.39 \[{\left (a x + \frac{b x^{n + 1}}{n + 1}\right )}{\rm sign}\left (b x^{n} + a\right ) \]
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, algorithm="giac")
[Out]