3.2643 \(\int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx\)

Optimal. Leaf size=50 \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]

[Out]

-2/(a*n*x^(n/2)) + (2*Sqrt[b]*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(3/2)*n)

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Rubi [A]  time = 0.0226298, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {345, 193, 321, 205} \[ \frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}-\frac{2 x^{-n/2}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n/2)/(a + b*x^n),x]

[Out]

-2/(a*n*x^(n/2)) + (2*Sqrt[b]*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/(a^(3/2)*n)

Rule 345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a + b*x^Simplify[n/(m +
1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{-1-\frac{n}{2}}}{a+b x^n} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{a+\frac{b}{x^2}} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{b+a x^2} \, dx,x,x^{-n/2}\right )}{n}\\ &=-\frac{2 x^{-n/2}}{a n}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{b+a x^2} \, dx,x,x^{-n/2}\right )}{a n}\\ &=-\frac{2 x^{-n/2}}{a n}+\frac{2 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x^{-n/2}}{\sqrt{b}}\right )}{a^{3/2} n}\\ \end{align*}

Mathematica [C]  time = 0.0029385, size = 32, normalized size = 0.64 \[ -\frac{2 x^{-n/2} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x^n}{a}\right )}{a n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n/2)/(a + b*x^n),x]

[Out]

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

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Maple [A]  time = 0., size = 79, normalized size = 1.6 \begin{align*} -2\,{\frac{1}{an{x}^{n/2}}}+{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}-{\frac{1}{b}\sqrt{-ab}} \right ) }-{\frac{1}{{a}^{2}n}\sqrt{-ab}\ln \left ({x}^{{\frac{n}{2}}}+{\frac{1}{b}\sqrt{-ab}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/n/(x^(1/2*n))+(-a*b)^(1/2)/a^2/n*ln(x^(1/2*n)-1/b*(-a*b)^(1/2))-(-a*b)^(1/2)/a^2/n*ln(x^(1/2*n)+1/b*(-a*b
)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -b \int \frac{x^{\frac{1}{2} \, n}}{a b x x^{n} + a^{2} x}\,{d x} - \frac{2}{a n x^{\frac{1}{2} \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.10182, size = 282, normalized size = 5.64 \begin{align*} \left [-\frac{2 \, x x^{-\frac{1}{2} \, n - 1} - \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} x^{-n - 2} + 2 \, a x x^{-\frac{1}{2} \, n - 1} \sqrt{-\frac{b}{a}} - b}{a x^{2} x^{-n - 2} + b}\right )}{a n}, -\frac{2 \,{\left (x x^{-\frac{1}{2} \, n - 1} + \sqrt{\frac{b}{a}} \arctan \left (\frac{\sqrt{\frac{b}{a}}}{x x^{-\frac{1}{2} \, n - 1}}\right )\right )}}{a n}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-(2*x*x^(-1/2*n - 1) - sqrt(-b/a)*log((a*x^2*x^(-n - 2) + 2*a*x*x^(-1/2*n - 1)*sqrt(-b/a) - b)/(a*x^2*x^(-n -
2) + b)))/(a*n), -2*(x*x^(-1/2*n - 1) + sqrt(b/a)*arctan(sqrt(b/a)/(x*x^(-1/2*n - 1))))/(a*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**(-1-1/2*n)/(a+b*x**n),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{1}{2} \, n - 1}}{b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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