3.6.18 \(\int \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx\) [518]

Optimal. Leaf size=88 \[ \frac {a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{a+b x^n}+\frac {b^2 x^{1+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(1+n) \left (a b+b^2 x^n\right )} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 = {1357} \begin {gather*} \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} \end {gather*}

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 1357

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*}

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Mathematica [A]
time = 0.01, size = 39, normalized size = 0.44 \begin {gather*} \frac {x \sqrt {\left (a+b x^n\right )^2} \left (a+a n+b x^n\right )}{(1+n) \left (a+b x^n\right )} \end {gather*}

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 = 0.64

method	result	size
risch	\(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a x}{a +b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b x \,x^{n}}{\left (a +b \,x^{n}\right ) \left (1+n \right )}\)	\(56\)

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,method=_RETURNVERBOSE)

[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 = 0.30, size = 19, normalized size = 0.22 \begin {gather*} \frac {a {\left (n + 1\right )} x + b x x^{n}}{n + 1} \end {gather*}

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 = 0.37, size = 20, normalized size = 0.23 \begin {gather*} \frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a^{2} + 2 a b x^{n} + b^{2} x^{2 n}}\, dx \end {gather*}

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 = 5.67, size = 25, normalized size = 0.28 \begin {gather*} {\left (a x + \frac {b x^{n + 1}}{n + 1}\right )} \mathrm {sgn}\left (b x^{n} + a\right ) \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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