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

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]

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

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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]

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

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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]

2*a*b*n*x**2*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/((n + 2)*(4*a*b + 4*b**2*x*
*n)) + x**2*sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n))/(n + 2)

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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]

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

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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]

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

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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]

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

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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]

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

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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]

Integral(x*sqrt((a + b*x**n)**2), x)

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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]

1/2*(a*n*x^2*sign(b*x^n + a) + 2*b*x^2*e^(n*ln(x))*sign(b*x^n + a) + 2*a*x^2*sig
n(b*x^n + a))/(n + 2)