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

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, 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 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]

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]]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [A]  time = 0.88, size = 119, normalized size = 2.33 \[ \left [\frac {\sqrt {a} b x^{n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, \sqrt {b x^{n} + a} a}{2 \, a n x^{n}}, \frac {\sqrt {-a} b x^{n} \arctan \left (\frac {\sqrt {b x^{n} + a} \sqrt {-a}}{a}\right ) - \sqrt {b x^{n} + a} a}{a n x^{n}}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{-n - 1}\,{d x} \]

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)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \sqrt {b \,x^{n}+a}\, x^{-n -1}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b x^{n} + a} x^{-n - 1}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,x^n}}{x^{n+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^(1/2)/x^(n + 1),x)

[Out]

int((a + b*x^n)^(1/2)/x^(n + 1), x)

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sympy [A]  time = 25.08, size = 49, normalized size = 0.96 \[ - \frac {\sqrt {b} x^{- \frac {n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{n} - \frac {b \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{\sqrt {a} n} \]

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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