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

Optimal. Leaf size=201 \[ \frac {a^6 \left (x-\sqrt {a+x^2}\right )^{n-6}}{64 (6-n)}+\frac {3 a^5 \left (x-\sqrt {a+x^2}\right )^{n-4}}{32 (4-n)}+\frac {15 a^4 \left (x-\sqrt {a+x^2}\right )^{n-2}}{64 (2-n)}-\frac {5 a^3 \left (x-\sqrt {a+x^2}\right )^n}{16 n}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^{n+2}}{64 (n+2)}-\frac {3 a \left (x-\sqrt {a+x^2}\right )^{n+4}}{32 (n+4)}-\frac {\left (x-\sqrt {a+x^2}\right )^{n+6}}{64 (n+6)} \]

[Out]

1/64*a^6*(x-(x^2+a)^(1/2))^(-6+n)/(6-n)+3/32*a^5*(x-(x^2+a)^(1/2))^(-4+n)/(4-n)+15/64*a^4*(x-(x^2+a)^(1/2))^(-
2+n)/(2-n)-5/16*a^3*(x-(x^2+a)^(1/2))^n/n-15/64*a^2*(x-(x^2+a)^(1/2))^(2+n)/(2+n)-3/32*a*(x-(x^2+a)^(1/2))^(4+
n)/(4+n)-1/64*(x-(x^2+a)^(1/2))^(6+n)/(6+n)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2122, 270} \[ \frac {a^6 \left (x-\sqrt {a+x^2}\right )^{n-6}}{64 (6-n)}+\frac {3 a^5 \left (x-\sqrt {a+x^2}\right )^{n-4}}{32 (4-n)}+\frac {15 a^4 \left (x-\sqrt {a+x^2}\right )^{n-2}}{64 (2-n)}-\frac {5 a^3 \left (x-\sqrt {a+x^2}\right )^n}{16 n}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^{n+2}}{64 (n+2)}-\frac {3 a \left (x-\sqrt {a+x^2}\right )^{n+4}}{32 (n+4)}-\frac {\left (x-\sqrt {a+x^2}\right )^{n+6}}{64 (n+6)} \]

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 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]

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])

Rubi steps

\begin {align*} \int \left (a+x^2\right )^{5/2} \left (x-\sqrt {a+x^2}\right )^n \, dx &=-\left (\frac {1}{64} \operatorname {Subst}\left (\int x^{-7+n} \left (a+x^2\right )^6 \, dx,x,x-\sqrt {a+x^2}\right )\right )\\ &=-\left (\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 )\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 = 0.49, size = 173, normalized size = 0.86 \[ \frac {1}{64} \left (x-\sqrt {a+x^2}\right )^n \left (-\frac {a^6}{(n-6) \left (x-\sqrt {a+x^2}\right )^6}-\frac {6 a^5}{(n-4) \left (x-\sqrt {a+x^2}\right )^4}-\frac {15 a^4}{(n-2) \left (x-\sqrt {a+x^2}\right )^2}-\frac {20 a^3}{n}-\frac {15 a^2 \left (x-\sqrt {a+x^2}\right )^2}{n+2}-\frac {6 a \left (x-\sqrt {a+x^2}\right )^4}{n+4}-\frac {\left (x-\sqrt {a+x^2}\right )^6}{n+6}\right ) \]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 204, normalized size = 1.01 \[ -\frac {{\left (a^{3} n^{6} - 50 \, a^{3} n^{4} + {\left (n^{6} - 20 \, n^{4} + 64 \, n^{2}\right )} x^{6} + 544 \, a^{3} n^{2} + 3 \, {\left (a n^{6} - 30 \, a n^{4} + 104 \, a n^{2}\right )} x^{4} - 720 \, a^{3} + 3 \, {\left (a^{2} n^{6} - 40 \, a^{2} n^{4} + 264 \, a^{2} n^{2}\right )} x^{2} + 6 \, {\left ({\left (n^{5} - 20 \, n^{3} + 64 \, n\right )} x^{5} + 2 \, {\left (a n^{5} - 30 \, a n^{3} + 104 \, a n\right )} x^{3} + {\left (a^{2} n^{5} - 40 \, a^{2} n^{3} + 264 \, a^{2} n\right )} x\right )} \sqrt {x^{2} + a}\right )} {\left (x - \sqrt {x^{2} + a}\right )}^{n}}{n^{7} - 56 \, n^{5} + 784 \, n^{3} - 2304 \, n} \]

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 - 72
0*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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]

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)

________________________________________________________________________________________

maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (x^{2}+a \right )^{\frac {5}{2}} \left (x -\sqrt {x^{2}+a}\right )^{n}\, dx \]

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)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x^{2} + a\right )}^{\frac {5}{2}} {\left (x - \sqrt {x^{2} + a}\right )}^{n}\,{d x} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (x-\sqrt {x^2+a}\right )}^n\,{\left (x^2+a\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + x^{2}\right )^{\frac {5}{2}} \left (x - \sqrt {a + x^{2}}\right )^{n}\, dx \]

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]

Integral((a + x**2)**(5/2)*(x - sqrt(a + x**2))**n, x)

________________________________________________________________________________________