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 = 0.0185866, 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, Rules used = {1343} \[ \frac{b^2 x^{n+1} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(n+1) \left (a b+b^2 x^n\right )}+\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n} \]

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))

Rule 1343

Int[((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^(2*n))^p/(b + 2*c*x
^n)^(2*p), Int[(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin{align*} \int \sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int \left (2 a b+2 b^2 x^n\right ) \, dx}{2 a b+2 b^2 x^n}\\ &=\frac{a x \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}+\frac{b^2 x^{1+n} \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{(1+n) \left (a b+b^2 x^n\right )}\\ \end{align*}

Mathematica [A]  time = 0.01403, 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 = 0.013, size = 56, normalized size = 0.6 \begin{align*}{\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}}} \end{align*}

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 = 0.960536, size = 26, normalized size = 0.3 \begin{align*} \frac{a{\left (n + 1\right )} x + b x x^{n}}{n + 1} \end{align*}

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, algorithm="maxima")

[Out]

(a*(n + 1)*x + b*x*x^n)/(n + 1)

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Fricas [A]  time = 1.59111, size = 45, normalized size = 0.51 \begin{align*} \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \end{align*}

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, 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. \begin{align*} \int \sqrt{a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}\, dx \end{align*}

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 [A]  time = 1.1015, size = 34, normalized size = 0.39 \begin{align*}{\left (a x + \frac{b x^{n + 1}}{n + 1}\right )} \mathrm{sgn}\left (b x^{n} + a\right ) \end{align*}

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, algorithm="giac")

[Out]

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