3.515 $$\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}}}{x^5} \, dx$$

Optimal. Leaf size=303 $\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}$

[Out]

(4*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/
(3*Sqrt[1 - 1/(a*x)]) + (2*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)])/(7*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)]) - (4*Sqrt[2]*a^4*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.320336, antiderivative size = 303, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.222, Rules used = {6182, 6180, 88, 50, 63, 206} $\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{9/2} \sqrt{c-\frac{c}{a x}}}{9 \sqrt{1-\frac{1}{a x}}}-\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{7/2} \sqrt{c-\frac{c}{a x}}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{5/2} \sqrt{c-\frac{c}{a x}}}{5 \sqrt{1-\frac{1}{a x}}}+\frac{2 a^4 \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-\frac{c}{a x}}}{3 \sqrt{1-\frac{1}{a x}}}+\frac{4 a^4 \sqrt{\frac{1}{a x}+1} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{\frac{1}{a x}+1}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(4*a^4*Sqrt[1 + 1/(a*x)]*Sqrt[c - c/(a*x)])/Sqrt[1 - 1/(a*x)] + (2*a^4*(1 + 1/(a*x))^(3/2)*Sqrt[c - c/(a*x)])/
(3*Sqrt[1 - 1/(a*x)]) + (2*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)])/(7*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)]) - (4*Sqrt[2]*a^4*Sqrt[c - c/(a*x)]*ArcTanh[Sqrt[1 + 1/(a*x)]/Sqrt[2]])/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]))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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 )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=-\frac{\sqrt{c-\frac{c}{a x}} \operatorname{Subst}\left (\int \left (-a^3 \left (1+\frac{x}{a}\right )^{3/2}+\frac{a^3 \left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}}+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{2 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}}}{7 \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}}}-\frac{\left (a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^{3/2}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{2 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{2 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}}}{7 \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}}}-\frac{\left (2 a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}}}{1-\frac{x}{a}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 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{2 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}}}{7 \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}}}-\frac{\left (4 a^3 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 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{2 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}}}{7 \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}}}-\frac{\left (8 a^4 \sqrt{c-\frac{c}{a x}}\right ) \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\frac{1}{a x}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{4 a^4 \sqrt{1+\frac{1}{a x}} \sqrt{c-\frac{c}{a x}}}{\sqrt{1-\frac{1}{a x}}}+\frac{2 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{2 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}}}{7 \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}}}-\frac{4 \sqrt{2} a^4 \sqrt{c-\frac{c}{a x}} \tanh ^{-1}\left (\frac{\sqrt{1+\frac{1}{a x}}}{\sqrt{2}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.297134, size = 178, normalized size = 0.59 $\frac{2 a \sqrt{1-\frac{1}{a^2 x^2}} \left (788 a^4 x^4+236 a^3 x^3+138 a^2 x^2+95 a x+35\right ) \sqrt{c-\frac{c}{a x}}}{315 x^3 (a x-1)}-2 \sqrt{2} a^4 \sqrt{c} \log \left (2 \sqrt{2} a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (3 a^2 x^2-2 a x-1\right )\right )+2 \sqrt{2} a^4 \sqrt{c} \log \left ((a x-1)^2\right )$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(2*a*Sqrt[1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*(35 + 95*a*x + 138*a^2*x^2 + 236*a^3*x^3 + 788*a^4*x^4))/(315*x^3
*(-1 + a*x)) + 2*Sqrt[2]*a^4*Sqrt[c]*Log[(-1 + a*x)^2] - 2*Sqrt[2]*a^4*Sqrt[c]*Log[2*Sqrt[2]*a^2*Sqrt[c]*Sqrt[
1 - 1/(a^2*x^2)]*Sqrt[c - c/(a*x)]*x^2 + c*(-1 - 2*a*x + 3*a^2*x^2)]

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Maple [A]  time = 0.195, size = 209, normalized size = 0.7 \begin{align*} -{\frac{2\,ax-2}{ \left ( 315\,ax+315 \right ){x}^{4}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 315\,{a}^{4}\sqrt{2}\ln \left ({\frac{2\,\sqrt{2}\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x}a+3\,ax+1}{ax-1}} \right ){x}^{5}-788\,{a}^{4}\sqrt{{a}^{-1}}{x}^{4}\sqrt{ \left ( ax+1 \right ) x}-236\,{a}^{3}\sqrt{{a}^{-1}}{x}^{3}\sqrt{ \left ( ax+1 \right ) x}-138\,{a}^{2}\sqrt{{a}^{-1}}{x}^{2}\sqrt{ \left ( ax+1 \right ) x}-95\,a\sqrt{{a}^{-1}}x\sqrt{ \left ( ax+1 \right ) x}-35\,\sqrt{{a}^{-1}}\sqrt{ \left ( ax+1 \right ) x} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}{\frac{1}{\sqrt{{a}^{-1}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/315/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(c*(a*x-1)/a/x)^(1/2)/x^4*(315*a^4*2^(1/2)*ln((2*2^(1/2)*(1/a)^
(1/2)*((a*x+1)*x)^(1/2)*a+3*a*x+1)/(a*x-1))*x^5-788*a^4*(1/a)^(1/2)*x^4*((a*x+1)*x)^(1/2)-236*a^3*(1/a)^(1/2)*
x^3*((a*x+1)*x)^(1/2)-138*a^2*(1/a)^(1/2)*x^2*((a*x+1)*x)^(1/2)-95*a*(1/a)^(1/2)*x*((a*x+1)*x)^(1/2)-35*(1/a)^
(1/2)*((a*x+1)*x)^(1/2))/((a*x+1)*x)^(1/2)/(1/a)^(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}}}{x^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.94215, size = 926, normalized size = 3.06 \begin{align*} \left [\frac{315 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{c} \log \left (-\frac{17 \, a^{3} c x^{3} - 3 \, a^{2} c x^{2} - 13 \, a c x - 4 \, \sqrt{2}{\left (3 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1}\right ) + 2 \,{\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{315 \,{\left (a x^{5} - x^{4}\right )}}, \frac{2 \,{\left (315 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{2}{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{3 \, a^{2} c x^{2} - 2 \, a c x - c}\right ) +{\left (788 \, a^{5} x^{5} + 1024 \, a^{4} x^{4} + 374 \, a^{3} x^{3} + 233 \, a^{2} x^{2} + 130 \, a x + 35\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}\right )}}{315 \,{\left (a x^{5} - x^{4}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(17*a^3*c*x^3 - 3*a^2*c*x^2 - 13*a*c*x - 4*sqrt(2)*(3*a^3
*x^3 + 4*a^2*x^2 + a*x)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x)) - c)/(a^3*x^3 - 3*a^2*x^2 +
3*a*x - 1)) + 2*(788*a^5*x^5 + 1024*a^4*x^4 + 374*a^3*x^3 + 233*a^2*x^2 + 130*a*x + 35)*sqrt((a*x - 1)/(a*x +
1))*sqrt((a*c*x - c)/(a*x)))/(a*x^5 - x^4), 2/315*(315*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arctan(2*sqrt(2)*(
a^2*x^2 + a*x)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1))*sqrt((a*c*x - c)/(a*x))/(3*a^2*c*x^2 - 2*a*c*x - c)) + (788*
a^5*x^5 + 1024*a^4*x^4 + 374*a^3*x^3 + 233*a^2*x^2 + 130*a*x + 35)*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(1/((a*x-1)/(a*x+1))**(3/2)*(c-c/a/x)**(1/2)/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}}}{x^{5} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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