3.5.93 \(\int \frac {(x-\sqrt {a+x^2})^n}{a+x^2} \, dx\) [493]

Optimal. Leaf size=63 \[ \frac {2 \left (x-\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2147, 371} \begin {gather*} \frac {2 \left (x-\sqrt {a+x^2}\right )^{n+1} \, _2F_1\left (1,\frac {n+1}{2};\frac {n+3}{2};-\frac {\left (x-\sqrt {x^2+a}\right )^2}{a}\right )}{a (n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(x - Sqrt[a + x^2])^(1 + n)*Hypergeometric2F1[1, (1 + n)/2, (3 + n)/2, -((x - Sqrt[a + x^2])^2/a)])/(a*(1 +
n))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2147

Int[((g_) + (i_.)*(x_)^2)^(m_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dis
t[(1/(2^(2*m + 1)*e*f^(2*m)))*(i/c)^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}{a+x^2} \, dx &=2 \text {Subst}\left (\int \frac {x^n}{a+x^2} \, dx,x,x-\sqrt {a+x^2}\right )\\ &=\frac {2 \left (x-\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 65, normalized size = 1.03 \begin {gather*} \frac {2 \left (x-\sqrt {a+x^2}\right )^{1+n} \, _2F_1\left (1,\frac {1+n}{2};1+\frac {1+n}{2};-\frac {\left (x-\sqrt {a+x^2}\right )^2}{a}\right )}{a (1+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x -\sqrt {x^{2}+a}\right )^{n}}{x^{2}+a}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x-\sqrt {x^2+a}\right )}^n}{x^2+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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