Optimal. Leaf size=93 \[ \frac{b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
[Out]
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Rubi [A] time = 0.0662483, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{b^2 x^{n+2} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+2) \left (a b+b^2 x^n\right )}+\frac{a x^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]
[Out]
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Rubi in Sympy [A] time = 8.61075, size = 80, normalized size = 0.86 \[ \frac{2 a b n x^{2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{\left (n + 2\right ) \left (4 a b + 4 b^{2} x^{n}\right )} + \frac{x^{2} \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}}{n + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0266408, size = 46, normalized size = 0.49 \[ \frac{x^2 \sqrt{\left (a+b x^n\right )^2} \left (a (n+2)+2 b x^n\right )}{2 (n+2) \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)],x]
[Out]
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Maple [A] time = 0.022, size = 61, normalized size = 0.7 \[{\frac{a{x}^{2}}{2\,a+2\,b{x}^{n}}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{b{x}^{2}{x}^{n}}{ \left ( a+b{x}^{n} \right ) \left ( 2+n \right ) }\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x)
[Out]
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Maxima [A] time = 0.755021, size = 34, normalized size = 0.37 \[ \frac{2 \, b x^{2} x^{n} + a{\left (n + 2\right )} x^{2}}{2 \,{\left (n + 2\right )}} \]
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,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272414, size = 38, normalized size = 0.41 \[ \frac{2 \, b x^{2} x^{n} +{\left (a n + 2 \, a\right )} x^{2}}{2 \,{\left (n + 2\right )}} \]
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,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int x \sqrt{\left (a + b x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.278873, size = 74, normalized size = 0.8 \[ \frac{a n x^{2}{\rm sign}\left (b x^{n} + a\right ) + 2 \, b x^{2} e^{\left (n{\rm ln}\left (x\right )\right )}{\rm sign}\left (b x^{n} + a\right ) + 2 \, a x^{2}{\rm sign}\left (b x^{n} + a\right )}{2 \,{\left (n + 2\right )}} \]
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,x, algorithm="giac")
[Out]