3.495 \(\int (a+x^2)^{5/2} (x+\sqrt{a+x^2})^n \, dx\)

Optimal. Leaf size=187 \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]

[Out]

-(a^6*(x + Sqrt[a + x^2])^(-6 + n))/(64*(6 - n)) - (3*a^5*(x + Sqrt[a + x^2])^(-4 + n))/(32*(4 - n)) - (15*a^4
*(x + Sqrt[a + x^2])^(-2 + n))/(64*(2 - n)) + (5*a^3*(x + Sqrt[a + x^2])^n)/(16*n) + (15*a^2*(x + Sqrt[a + x^2
])^(2 + n))/(64*(2 + n)) + (3*a*(x + Sqrt[a + x^2])^(4 + n))/(32*(4 + n)) + (x + Sqrt[a + x^2])^(6 + n)/(64*(6
 + n))

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Rubi [A]  time = 0.117178, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2122, 270} \[ -\frac{a^6 \left (\sqrt{a+x^2}+x\right )^{n-6}}{64 (6-n)}-\frac{3 a^5 \left (\sqrt{a+x^2}+x\right )^{n-4}}{32 (4-n)}-\frac{15 a^4 \left (\sqrt{a+x^2}+x\right )^{n-2}}{64 (2-n)}+\frac{5 a^3 \left (\sqrt{a+x^2}+x\right )^n}{16 n}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^{n+2}}{64 (n+2)}+\frac{3 a \left (\sqrt{a+x^2}+x\right )^{n+4}}{32 (n+4)}+\frac{\left (\sqrt{a+x^2}+x\right )^{n+6}}{64 (n+6)} \]

Antiderivative was successfully verified.

[In]

Int[(a + x^2)^(5/2)*(x + Sqrt[a + x^2])^n,x]

[Out]

-(a^6*(x + Sqrt[a + x^2])^(-6 + n))/(64*(6 - n)) - (3*a^5*(x + Sqrt[a + x^2])^(-4 + n))/(32*(4 - n)) - (15*a^4
*(x + Sqrt[a + x^2])^(-2 + n))/(64*(2 - n)) + (5*a^3*(x + Sqrt[a + x^2])^n)/(16*n) + (15*a^2*(x + Sqrt[a + x^2
])^(2 + n))/(64*(2 + n)) + (3*a*(x + Sqrt[a + x^2])^(4 + n))/(32*(4 + n)) + (x + Sqrt[a + x^2])^(6 + n)/(64*(6
 + 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 )^{5/2} \left (x+\sqrt{a+x^2}\right )^n \, dx &=\frac{1}{64} \operatorname{Subst}\left (\int x^{-7+n} \left (a+x^2\right )^6 \, dx,x,x+\sqrt{a+x^2}\right )\\ &=\frac{1}{64} \operatorname{Subst}\left (\int \left (a^6 x^{-7+n}+6 a^5 x^{-5+n}+15 a^4 x^{-3+n}+20 a^3 x^{-1+n}+15 a^2 x^{1+n}+6 a x^{3+n}+x^{5+n}\right ) \, dx,x,x+\sqrt{a+x^2}\right )\\ &=-\frac{a^6 \left (x+\sqrt{a+x^2}\right )^{-6+n}}{64 (6-n)}-\frac{3 a^5 \left (x+\sqrt{a+x^2}\right )^{-4+n}}{32 (4-n)}-\frac{15 a^4 \left (x+\sqrt{a+x^2}\right )^{-2+n}}{64 (2-n)}+\frac{5 a^3 \left (x+\sqrt{a+x^2}\right )^n}{16 n}+\frac{15 a^2 \left (x+\sqrt{a+x^2}\right )^{2+n}}{64 (2+n)}+\frac{3 a \left (x+\sqrt{a+x^2}\right )^{4+n}}{32 (4+n)}+\frac{\left (x+\sqrt{a+x^2}\right )^{6+n}}{64 (6+n)}\\ \end{align*}

Mathematica [A]  time = 0.328109, size = 157, normalized size = 0.84 \[ \frac{1}{64} \left (\sqrt{a+x^2}+x\right )^n \left (\frac{a^6}{(n-6) \left (\sqrt{a+x^2}+x\right )^6}+\frac{6 a^5}{(n-4) \left (\sqrt{a+x^2}+x\right )^4}+\frac{15 a^4}{(n-2) \left (\sqrt{a+x^2}+x\right )^2}+\frac{15 a^2 \left (\sqrt{a+x^2}+x\right )^2}{n+2}+\frac{20 a^3}{n}+\frac{6 a \left (\sqrt{a+x^2}+x\right )^4}{n+4}+\frac{\left (\sqrt{a+x^2}+x\right )^6}{n+6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + x^2)^(5/2)*(x + Sqrt[a + x^2])^n,x]

[Out]

((x + Sqrt[a + x^2])^n*((20*a^3)/n + a^6/((-6 + n)*(x + Sqrt[a + x^2])^6) + (6*a^5)/((-4 + n)*(x + Sqrt[a + x^
2])^4) + (15*a^4)/((-2 + n)*(x + Sqrt[a + x^2])^2) + (15*a^2*(x + Sqrt[a + x^2])^2)/(2 + n) + (6*a*(x + Sqrt[a
 + x^2])^4)/(4 + n) + (x + Sqrt[a + x^2])^6/(6 + n)))/64

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Maple [F]  time = 0.015, size = 0, normalized size = 0. \begin{align*} \int \left ({x}^{2}+a \right ) ^{{\frac{5}{2}}} \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)^(5/2)*(x+(x^2+a)^(1/2))^n,x)

[Out]

int((x^2+a)^(5/2)*(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 )}^{\frac{5}{2}}{\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)^(5/2)*(x+(x^2+a)^(1/2))^n,x, algorithm="maxima")

[Out]

integrate((x^2 + a)^(5/2)*(x + sqrt(x^2 + a))^n, x)

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Fricas [A]  time = 1.34751, size = 459, normalized size = 2.45 \begin{align*} \frac{{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} +{\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \,{\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \,{\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} - 6 \,{\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \,{\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} +{\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt{x^{2} + a}\right )}{\left (x + \sqrt{x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+a)^(5/2)*(x+(x^2+a)^(1/2))^n,x, algorithm="fricas")

[Out]

(a^3*n^6 - 50*a^3*n^4 + (n^6 - 20*n^4 + 64*n^2)*x^6 + 544*a^3*n^2 + 3*(a*n^6 - 30*a*n^4 + 104*a*n^2)*x^4 - 720
*a^3 + 3*(a^2*n^6 - 40*a^2*n^4 + 264*a^2*n^2)*x^2 - 6*((n^5 - 20*n^3 + 64*n)*x^5 + 2*(a*n^5 - 30*a*n^3 + 104*a
*n)*x^3 + (a^2*n^5 - 40*a^2*n^3 + 264*a^2*n)*x)*sqrt(x^2 + a))*(x + sqrt(x^2 + a))^n/(n^7 - 56*n^5 + 784*n^3 -
 2304*n)

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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)**(5/2)*(x+(x**2+a)**(1/2))**n,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (x^{2} + a\right )}^{\frac{5}{2}}{\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)^(5/2)*(x+(x^2+a)^(1/2))^n,x, algorithm="giac")

[Out]

integrate((x^2 + a)^(5/2)*(x + sqrt(x^2 + a))^n, x)