3.2663 \(\int \frac{x^{-1-n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=53 \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{x^{-n} \sqrt{a+b x^n}}{a n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0253995, antiderivative size = 53, 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 = {266, 51, 63, 208} \[ \frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{a^{3/2} n}-\frac{x^{-n} \sqrt{a+b x^n}}{a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)/Sqrt[a + b*x^n],x]

[Out]

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

Rule 266

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0617062, size = 67, normalized size = 1.26 \[ \frac{2 b \sqrt{a+b x^n} \left (\frac{\tanh ^{-1}\left (\sqrt{\frac{b x^n}{a}+1}\right )}{2 \sqrt{\frac{b x^n}{a}+1}}-\frac{a x^{-n}}{2 b}\right )}{a^2 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)/Sqrt[a + b*x^n],x]

[Out]

(2*b*Sqrt[a + b*x^n]*(-a/(2*b*x^n) + ArcTanh[Sqrt[1 + (b*x^n)/a]]/(2*Sqrt[1 + (b*x^n)/a])))/(a^2*n)

________________________________________________________________________________________

Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.06722, size = 265, normalized size = 5. \begin{align*} \left [\frac{\sqrt{a} b x^{n} \log \left (\frac{b x^{n} + 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) - 2 \, \sqrt{b x^{n} + a} a}{2 \, a^{2} n x^{n}}, -\frac{\sqrt{-a} b x^{n} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) + \sqrt{b x^{n} + a} a}{a^{2} n x^{n}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 171.165, size = 49, normalized size = 0.92 \begin{align*} - \frac{\sqrt{b} x^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{a n} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(b)*x**(-n/2)*sqrt(a*x**(-n)/b + 1)/(a*n) + b*asinh(sqrt(a)*x**(-n/2)/sqrt(b))/(a**(3/2)*n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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