3.541 \(\int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx\)

Optimal. Leaf size=289 \[ -\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{38 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{50 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{32 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 a^4 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

[Out]

(-8*a^4*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (32*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]
)/Sqrt[1 - 1/(a*x)] + (50*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a*x)]) - (38*a^4*(1 + 1/(a
*x))^(5/2)*Sqrt[c - c/(a*x)])/(5*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(7/2)*Sqrt[c - c/(a*x)])/Sqrt[1 - 1
/(a*x)] - (2*a^4*(1 + 1/(a*x))^(9/2)*Sqrt[c - c/(a*x)])/(9*Sqrt[1 - 1/(a*x)])

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Rubi [A]  time = 0.293352, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6182, 6180, 88} \[ -\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{38 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{50 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{32 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{8 a^4 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{\frac{1}{a x}+1}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5),x]

[Out]

(-8*a^4*Sqrt[c - c/(a*x)])/(Sqrt[1 - 1/(a*x)]*Sqrt[1 + 1/(a*x)]) - (32*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)]
)/Sqrt[1 - 1/(a*x)] + (50*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/(3*Sqrt[1 - 1/(a*x)]) - (38*a^4*(1 + 1/(a
*x))^(5/2)*Sqrt[c - c/(a*x)])/(5*Sqrt[1 - 1/(a*x)]) + (2*a^4*(1 + 1/(a*x))^(7/2)*Sqrt[c - c/(a*x)])/Sqrt[1 - 1
/(a*x)] - (2*a^4*(1 + 1/(a*x))^(9/2)*Sqrt[c - c/(a*x)])/(9*Sqrt[1 - 1/(a*x)])

Rule 6182

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Dist[(c + d/x)^p/(1 + d/(c*x))^
p, Int[u*(1 + d/(c*x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&
!IntegerQ[n/2] &&  !(IntegerQ[p] || GtQ[c, 0])

Rule 6180

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> -Dist[c^p, Subst[Int[((1
+ (d*x)/c)^p*(1 + x/a)^(n/2))/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ
[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx &=\frac{\sqrt{c-\frac{c}{a x}} \int \frac{e^{-3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^5} \, dx}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \frac{x^3 \left (1-\frac{x}{a}\right )^2}{\left (1+\frac{x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \left (-\frac{4 a^3}{\left (1+\frac{x}{a}\right )^{3/2}}+\frac{16 a^3}{\sqrt{1+\frac{x}{a}}}-25 a^3 \sqrt{1+\frac{x}{a}}+19 a^3 \left (1+\frac{x}{a}\right )^{3/2}-7 a^3 \left (1+\frac{x}{a}\right )^{5/2}+a^3 \left (1+\frac{x}{a}\right )^{7/2}\right ) \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{8 a^4 \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}} \sqrt{1+\frac{1}{a x}}}-\frac{32 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{50 a^4 \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}-\frac{38 a^4 \left (1+\frac{1}{a x}\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (1+\frac{1}{a x}\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.167155, size = 86, normalized size = 0.3 \[ -\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (656 a^5 x^5+328 a^4 x^4-82 a^3 x^3+41 a^2 x^2-20 a x+5\right ) \sqrt{c-\frac{c}{a x}}}{45 x^3 \left (a^2 x^2-1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c - c/(a*x)]/(E^(3*ArcCoth[a*x])*x^5),x]

[Out]

(-2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(5 - 20*a*x + 41*a^2*x^2 - 82*a^3*x^3 + 328*a^4*x^4 + 656*a^5*x^
5))/(45*x^3*(-1 + a^2*x^2))

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Maple [A]  time = 0.118, size = 86, normalized size = 0.3 \begin{align*} -{\frac{ \left ( 2\,ax+2 \right ) \left ( 656\,{x}^{5}{a}^{5}+328\,{x}^{4}{a}^{4}-82\,{x}^{3}{a}^{3}+41\,{a}^{2}{x}^{2}-20\,ax+5 \right ) }{45\,{x}^{4} \left ( ax-1 \right ) ^{2}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x)

[Out]

-2/45*(a*x+1)*(656*a^5*x^5+328*a^4*x^4-82*a^3*x^3+41*a^2*x^2-20*a*x+5)*(c*(a*x-1)/a/x)^(1/2)*((a*x-1)/(a*x+1))
^(3/2)/x^4/(a*x-1)^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)/x^5, x)

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Fricas [A]  time = 1.65585, size = 186, normalized size = 0.64 \begin{align*} -\frac{2 \,{\left (656 \, a^{5} x^{5} + 328 \, a^{4} x^{4} - 82 \, a^{3} x^{3} + 41 \, a^{2} x^{2} - 20 \, a x + 5\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{45 \,{\left (a x^{5} - x^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="fricas")

[Out]

-2/45*(656*a^5*x^5 + 328*a^4*x^4 - 82*a^3*x^3 + 41*a^2*x^2 - 20*a*x + 5)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x
- c)/(a*x))/(a*x^5 - x^4)

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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((c-c/a/x)**(1/2)*((a*x-1)/(a*x+1))**(3/2)/x**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-c/a/x)^(1/2)*((a*x-1)/(a*x+1))^(3/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(c - c/(a*x))*((a*x - 1)/(a*x + 1))^(3/2)/x^5, x)