3.2654 \(\int x^{-1-n} \sqrt{a+b x^n} \, dx\)

Optimal. Leaf size=51 \[ -\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

[Out]

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

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Rubi [A]  time = 0.0244006, antiderivative size = 51, 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, 47, 63, 208} \[ -\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)*Sqrt[a + b*x^n],x]

[Out]

-(Sqrt[a + b*x^n]/(n*x^n)) - (b*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(Sqrt[a]*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 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && 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 x^{-1-n} \sqrt{a+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{2 n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{n}\\ &=-\frac{x^{-n} \sqrt{a+b x^n}}{n}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{\sqrt{a} n}\\ \end{align*}

Mathematica [A]  time = 0.0394481, size = 62, normalized size = 1.22 \[ -\frac{x^{-n} \left (b x^n \sqrt{\frac{b x^n}{a}+1} \tanh ^{-1}\left (\sqrt{\frac{b x^n}{a}+1}\right )+a+b x^n\right )}{n \sqrt{a+b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1-n}\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)

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

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Fricas [A]  time = 1.15808, size = 258, normalized size = 5.06 \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 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 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*n*x^n), (sqrt
(-a)*b*x^n*arctan(sqrt(b*x^n + a)*sqrt(-a)/a) - sqrt(b*x^n + a)*a)/(a*n*x^n)]

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Sympy [A]  time = 131.743, size = 49, normalized size = 0.96 \begin{align*} - \frac{\sqrt{b} x^{- \frac{n}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{n} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{- \frac{n}{2}}}{\sqrt{b}} \right )}}{\sqrt{a} 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)/n - b*asinh(sqrt(a)*x**(-n/2)/sqrt(b))/(sqrt(a)*n)

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