3.6.17 \(\int x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, dx\) [517]

Optimal. Leaf size=93 \[ \frac {a x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 \left (a+b x^n\right )}+\frac {b^2 x^{2+n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{(2+n) \left (a b+b^2 x^n\right )} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1369, 14} \begin {gather*} \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 )} \end {gather*}

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 1369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rubi steps

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

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Mathematica [A]
time = 0.02, size = 46, normalized size = 0.49 \begin {gather*} \frac {x^2 \sqrt {\left (a+b x^n\right )^2} \left (a (2+n)+2 b x^n\right )}{2 (2+n) \left (a+b x^n\right )} \end {gather*}

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 = 0.02, size = 61, normalized size = 0.66

method	result	size
risch	\(\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, a \,x^{2}}{2 a +2 b \,x^{n}}+\frac {\sqrt {\left (a +b \,x^{n}\right )^{2}}\, b \,x^{2} x^{n}}{\left (a +b \,x^{n}\right ) \left (2+n \right )}\)	\(61\)

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,method=_RETURNVERBOSE)

[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 = 0.30, size = 25, normalized size = 0.27 \begin {gather*} \frac {2 \, b x^{2} x^{n} + a {\left (n + 2\right )} x^{2}}{2 \, {\left (n + 2\right )}} \end {gather*}

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, 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 = 0.35, size = 28, normalized size = 0.30 \begin {gather*} \frac {2 \, b x^{2} x^{n} + {\left (a n + 2 \, a\right )} x^{2}}{2 \, {\left (n + 2\right )}} \end {gather*}

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

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 [A]
time = 3.89, size = 53, normalized size = 0.57 \begin {gather*} \frac {2 \, b x^{2} x^{n} \mathrm {sgn}\left (b x^{n} + a\right ) + a n x^{2} \mathrm {sgn}\left (b x^{n} + a\right ) + 2 \, a x^{2} \mathrm {sgn}\left (b x^{n} + a\right )}{2 \, {\left (n + 2\right )}} \end {gather*}

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, algorithm="giac")

[Out]

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

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

[Out]

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

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