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

Optimal. Leaf size=108 \[ -\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-3}}{8 (3-n)}-\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^{n-1}}{8 (1-n)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+1}}{8 (n+1)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+3}}{8 (n+3)} \]

[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))

________________________________________________________________________________________

Rubi [A]  time = 0.0626827, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2122, 270} \[ -\frac{a^3 \left (\sqrt{a+x^2}+x\right )^{n-3}}{8 (3-n)}-\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^{n-1}}{8 (1-n)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+1}}{8 (n+1)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+3}}{8 (n+3)} \]

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 = 0.12373, size = 92, normalized size = 0.85 \[ \frac{1}{8} \left (\sqrt{a+x^2}+x\right )^{n-3} \left (\frac{3 a^2 \left (\sqrt{a+x^2}+x\right )^2}{n-1}+\frac{a^3}{n-3}+\frac{\left (\sqrt{a+x^2}+x\right )^6}{n+3}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^4}{n+1}\right ) \]

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

________________________________________________________________________________________

Maple [C]  time = 0.01, size = 167, normalized size = 1.6 \begin{align*}{\frac{{2}^{n}{x}^{3+n}}{3+n}{\mbox{$_3$F$_2$}(-{\frac{n}{2}},{\frac{1}{2}}-{\frac{n}{2}},-{\frac{3}{2}}-{\frac{n}{2}};\,1-n,-{\frac{1}{2}}-{\frac{n}{2}};\,-{\frac{a}{{x}^{2}}})}}+{\frac{n}{4\,\sqrt{\pi }}{a}^{{\frac{3}{2}}+{\frac{n}{2}}} \left ( 8\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n \left ( -2+2\,n \right ) } \left ({\frac{an}{{x}^{2}}}+n-1 \right ) \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}}+4\,{\frac{\sqrt{\pi }{x}^{1+n}{a}^{-1/2-n/2}}{ \left ( 1+n \right ) n}\sqrt{1+{\frac{a}{{x}^{2}}}} \left ( \sqrt{1+{\frac{a}{{x}^{2}}}}+1 \right ) ^{-1+n}} \right ) } \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]

2^n/(3+n)*x^(3+n)*hypergeom([-1/2*n,1/2-1/2*n,-3/2-1/2*n],[1-n,-1/2-1/2*n],-a/x^2)+1/4*a^(3/2+1/2*n)/Pi^(1/2)*
n*(8*Pi^(1/2)/(1+n)/n*x^(1+n)*a^(-1/2-1/2*n)*(a/x^2*n+n-1)/(-2+2*n)*((1+a/x^2)^(1/2)+1)^(-1+n)+4*Pi^(1/2)/(1+n
)/n*x^(1+n)*a^(-1/2-1/2*n)*(1+a/x^2)^(1/2)*((1+a/x^2)^(1/2)+1)^(-1+n))

________________________________________________________________________________________

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)

________________________________________________________________________________________

Fricas [A]  time = 1.35338, size = 174, normalized size = 1.61 \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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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]

Timed out

________________________________________________________________________________________

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)