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

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]

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

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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]

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

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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]

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

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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]

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

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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]

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

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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]

(a*(n + 1)*x + b*x*x^n)/(n + 1)

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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]

(b*x*x^n + (a*n + a)*x)/(n + 1)

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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]

Integral(sqrt(a**2 + 2*a*b*x**n + b**2*x**(2*n)), x)

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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]

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