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

Optimal. Leaf size=63 \[ -\frac {16 \left (x-\sqrt {a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac {n+4}{2};\frac {n+6}{2};-\frac {\left (x-\sqrt {x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]

[Out]

-16*hypergeom([4, 2+1/2*n],[3+1/2*n],-(x-(x^2+a)^(1/2))^2/a)*(x-(x^2+a)^(1/2))^(4+n)/a^4/(4+n)

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Rubi [A]  time = 0.07, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2122, 364} \[ -\frac {16 \left (x-\sqrt {a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac {n+4}{2};\frac {n+6}{2};-\frac {\left (x-\sqrt {x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(x - Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, (6 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a^4*
(4 + n))

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 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 \frac {\left (x-\sqrt {a+x^2}\right )^n}{\left (a+x^2\right )^{5/2}} \, dx &=-\left (16 \operatorname {Subst}\left (\int \frac {x^{3+n}}{\left (a+x^2\right )^4} \, dx,x,x-\sqrt {a+x^2}\right )\right )\\ &=-\frac {16 \left (x-\sqrt {a+x^2}\right )^{4+n} \, _2F_1\left (4,\frac {4+n}{2};\frac {6+n}{2};-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a^4 (4+n)}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 65, normalized size = 1.03 \[ -\frac {16 \left (x-\sqrt {a+x^2}\right )^{n+4} \, _2F_1\left (4,\frac {n+4}{2};\frac {n+4}{2}+1;-\frac {\left (x-\sqrt {x^2+a}\right )^2}{a}\right )}{a^4 (n+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*(x - Sqrt[a + x^2])^(4 + n)*Hypergeometric2F1[4, (4 + n)/2, 1 + (4 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(
a^4*(4 + n))

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + a} {\left (x - \sqrt {x^{2} + a}\right )}^{n}}{x^{6} + 3 \, a x^{4} + 3 \, a^{2} x^{2} + a^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(x^2 + a)*(x - sqrt(x^2 + a))^n/(x^6 + 3*a*x^4 + 3*a^2*x^2 + a^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x-\sqrt {x^2+a}\right )}^n}{{\left (x^2+a\right )}^{5/2}} \,d x \]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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