3.2666 \(\int \frac{x^{-1-4 n}}{\sqrt{a+b x^n}} \, dx\)

Optimal. Leaf size=145 \[ -\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{9/2} n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n} \]

[Out]

-Sqrt[a + b*x^n]/(4*a*n*x^(4*n)) + (7*b*Sqrt[a + b*x^n])/(24*a^2*n*x^(3*n)) - (35*b^2*Sqrt[a + b*x^n])/(96*a^3
*n*x^(2*n)) + (35*b^3*Sqrt[a + b*x^n])/(64*a^4*n*x^n) - (35*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(64*a^(9/2)*
n)

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Rubi [A]  time = 0.0656875, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {266, 51, 63, 208} \[ -\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{9/2} n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]

[Out]

-Sqrt[a + b*x^n]/(4*a*n*x^(4*n)) + (7*b*Sqrt[a + b*x^n])/(24*a^2*n*x^(3*n)) - (35*b^2*Sqrt[a + b*x^n])/(96*a^3
*n*x^(2*n)) + (35*b^3*Sqrt[a + b*x^n])/(64*a^4*n*x^n) - (35*b^4*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(64*a^(9/2)*
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^{-1-4 n}}{\sqrt{a+b x^n}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^5 \sqrt{a+b x}} \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}-\frac{(7 b) \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{a+b x}} \, dx,x,x^n\right )}{8 a n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}+\frac{\left (35 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,x^n\right )}{48 a^2 n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}-\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,x^n\right )}{64 a^3 n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}+\frac{\left (35 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,x^n\right )}{128 a^4 n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}+\frac{\left (35 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b x^n}\right )}{64 a^4 n}\\ &=-\frac{x^{-4 n} \sqrt{a+b x^n}}{4 a n}+\frac{7 b x^{-3 n} \sqrt{a+b x^n}}{24 a^2 n}-\frac{35 b^2 x^{-2 n} \sqrt{a+b x^n}}{96 a^3 n}+\frac{35 b^3 x^{-n} \sqrt{a+b x^n}}{64 a^4 n}-\frac{35 b^4 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{64 a^{9/2} n}\\ \end{align*}

Mathematica [C]  time = 0.0084912, size = 40, normalized size = 0.28 \[ -\frac{2 b^4 \sqrt{a+b x^n} \, _2F_1\left (\frac{1}{2},5;\frac{3}{2};\frac{b x^n}{a}+1\right )}{a^5 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 4*n)/Sqrt[a + b*x^n],x]

[Out]

(-2*b^4*Sqrt[a + b*x^n]*Hypergeometric2F1[1/2, 5, 3/2, 1 + (b*x^n)/a])/(a^5*n)

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Maple [F]  time = 0.086, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-4\,n}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-4*n)/(a+b*x^n)^(1/2),x)

[Out]

int(x^(-1-4*n)/(a+b*x^n)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-4 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^(-4*n - 1)/sqrt(b*x^n + a), x)

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Fricas [A]  time = 1.03004, size = 482, normalized size = 3.32 \begin{align*} \left [\frac{105 \, \sqrt{a} b^{4} x^{4 \, n} \log \left (\frac{b x^{n} - 2 \, \sqrt{b x^{n} + a} \sqrt{a} + 2 \, a}{x^{n}}\right ) + 2 \,{\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt{b x^{n} + a}}{384 \, a^{5} n x^{4 \, n}}, \frac{105 \, \sqrt{-a} b^{4} x^{4 \, n} \arctan \left (\frac{\sqrt{b x^{n} + a} \sqrt{-a}}{a}\right ) +{\left (105 \, a b^{3} x^{3 \, n} - 70 \, a^{2} b^{2} x^{2 \, n} + 56 \, a^{3} b x^{n} - 48 \, a^{4}\right )} \sqrt{b x^{n} + a}}{192 \, a^{5} n x^{4 \, n}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")

[Out]

[1/384*(105*sqrt(a)*b^4*x^(4*n)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^n) + 2*(105*a*b^3*x^(3*n) - 70
*a^2*b^2*x^(2*n) + 56*a^3*b*x^n - 48*a^4)*sqrt(b*x^n + a))/(a^5*n*x^(4*n)), 1/192*(105*sqrt(-a)*b^4*x^(4*n)*ar
ctan(sqrt(b*x^n + a)*sqrt(-a)/a) + (105*a*b^3*x^(3*n) - 70*a^2*b^2*x^(2*n) + 56*a^3*b*x^n - 48*a^4)*sqrt(b*x^n
+ a))/(a^5*n*x^(4*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**(-1-4*n)/(a+b*x**n)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-4 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-4*n)/(a+b*x^n)^(1/2),x, algorithm="giac")

[Out]

integrate(x^(-4*n - 1)/sqrt(b*x^n + a), x)