3.491 \(\int (a+x^2) (x-\sqrt{a+x^2})^n \, dx\)

Optimal. Leaf size=116 \[ -\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{n-3}}{8 (3-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^{n-1}}{8 (1-n)}+\frac{3 a \left (x-\sqrt{a+x^2}\right )^{n+1}}{8 (n+1)}+\frac{\left (x-\sqrt{a+x^2}\right )^{n+3}}{8 (n+3)} \]

[Out]

-(a^3*(x - Sqrt[a + x^2])^(-3 + n))/(8*(3 - n)) - (3*a^2*(x - Sqrt[a + x^2])^(-1 + n))/(8*(1 - n)) + (3*a*(x -
Sqrt[a + x^2])^(1 + n))/(8*(1 + n)) + (x - Sqrt[a + x^2])^(3 + n)/(8*(3 + n))

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Rubi [A]  time = 0.0638217, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2122, 270} \[ -\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{n-3}}{8 (3-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^{n-1}}{8 (1-n)}+\frac{3 a \left (x-\sqrt{a+x^2}\right )^{n+1}}{8 (n+1)}+\frac{\left (x-\sqrt{a+x^2}\right )^{n+3}}{8 (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[(a + x^2)*(x - Sqrt[a + x^2])^n,x]

[Out]

-(a^3*(x - Sqrt[a + x^2])^(-3 + n))/(8*(3 - n)) - (3*a^2*(x - Sqrt[a + x^2])^(-1 + n))/(8*(1 - n)) + (3*a*(x -
Sqrt[a + x^2])^(1 + n))/(8*(1 + n)) + (x - Sqrt[a + x^2])^(3 + n)/(8*(3 + n))

Rule 2122

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1*(i/c)^m)/(2^(2*m + 1)*e*f^(2*m)), Subst[Int[(x^n*(d^2 + a*f^2 - 2*d*x + x^2)^(2*m + 1))/(-d + x)^(2*(m +
1)), x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /; FreeQ[{a, c, d, e, f, g, i, n}, x] && EqQ[e^2 - c*f^2, 0] && E
qQ[c*g - a*i, 0] && IntegerQ[2*m] && (IntegerQ[m] || GtQ[i/c, 0])

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \left (a+x^2\right ) \left (x-\sqrt{a+x^2}\right )^n \, dx &=\frac{1}{8} \operatorname{Subst}\left (\int x^{-4+n} \left (a+x^2\right )^3 \, dx,x,x-\sqrt{a+x^2}\right )\\ &=\frac{1}{8} \operatorname{Subst}\left (\int \left (a^3 x^{-4+n}+3 a^2 x^{-2+n}+3 a x^n+x^{2+n}\right ) \, dx,x,x-\sqrt{a+x^2}\right )\\ &=-\frac{a^3 \left (x-\sqrt{a+x^2}\right )^{-3+n}}{8 (3-n)}-\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^{-1+n}}{8 (1-n)}+\frac{3 a \left (x-\sqrt{a+x^2}\right )^{1+n}}{8 (1+n)}+\frac{\left (x-\sqrt{a+x^2}\right )^{3+n}}{8 (3+n)}\\ \end{align*}

Mathematica [A]  time = 0.118305, size = 100, normalized size = 0.86 \[ \frac{1}{8} \left (x-\sqrt{a+x^2}\right )^{n-3} \left (\frac{3 a^2 \left (x-\sqrt{a+x^2}\right )^2}{n-1}+\frac{a^3}{n-3}+\frac{\left (x-\sqrt{a+x^2}\right )^6}{n+3}+\frac{3 a \left (x-\sqrt{a+x^2}\right )^4}{n+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + x^2)*(x - Sqrt[a + x^2])^n,x]

[Out]

((x - Sqrt[a + x^2])^(-3 + n)*(a^3/(-3 + n) + (3*a^2*(x - Sqrt[a + x^2])^2)/(-1 + n) + (3*a*(x - Sqrt[a + x^2]
)^4)/(1 + n) + (x - Sqrt[a + x^2])^6/(3 + n)))/8

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}+a \right ) \left ( x-\sqrt{{x}^{2}+a} \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{2} + a\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+a)*(x-(x^2+a)^(1/2))^n,x, algorithm="maxima")

[Out]

integrate((x^2 + a)*(x - sqrt(x^2 + a))^n, x)

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Fricas [A]  time = 1.36644, size = 174, normalized size = 1.5 \begin{align*} -\frac{{\left (3 \,{\left (n^{2} - 1\right )} x^{3} + 3 \,{\left (a n^{2} - 3 \, a\right )} x +{\left (a n^{3} +{\left (n^{3} - n\right )} x^{2} - 7 \, a n\right )} \sqrt{x^{2} + a}\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}}{n^{4} - 10 \, n^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+a)*(x-(x^2+a)^(1/2))^n,x, algorithm="fricas")

[Out]

-(3*(n^2 - 1)*x^3 + 3*(a*n^2 - 3*a)*x + (a*n^3 + (n^3 - n)*x^2 - 7*a*n)*sqrt(x^2 + a))*(x - sqrt(x^2 + a))^n/(
n^4 - 10*n^2 + 9)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + x^{2}\right ) \left (x - \sqrt{a + x^{2}}\right )^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+a)*(x-(x**2+a)**(1/2))**n,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{2} + a\right )}{\left (x - \sqrt{x^{2} + a}\right )}^{n}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+a)*(x-(x^2+a)^(1/2))^n,x, algorithm="giac")

[Out]

integrate((x^2 + a)*(x - sqrt(x^2 + a))^n, x)