### 3.316 $$\int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{c-a c x}}{x^3} \, dx$$

Optimal. Leaf size=224 $\frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{2 x \sqrt{1-\frac{1}{a x}}}+\frac{7 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}$

[Out]

(7*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(2*S
qrt[1 - 1/(a*x)]*x) + (23*a^(3/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(4*Sqrt[1 - 1/(a
*x)]) - (4*Sqrt[2]*a^(3/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*
x)])])/Sqrt[1 - 1/(a*x)]

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Rubi [A]  time = 0.264502, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.391, Rules used = {6176, 6181, 101, 154, 157, 54, 215, 93, 206} $\frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\sqrt{1-\frac{1}{a x}}}+\frac{a \left (\frac{1}{a x}+1\right )^{3/2} \sqrt{c-a c x}}{2 x \sqrt{1-\frac{1}{a x}}}+\frac{7 a \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{4 x \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(7*a*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(4*Sqrt[1 - 1/(a*x)]*x) + (a*(1 + 1/(a*x))^(3/2)*Sqrt[c - a*c*x])/(2*S
qrt[1 - 1/(a*x)]*x) + (23*a^(3/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(4*Sqrt[1 - 1/(a
*x)]) - (4*Sqrt[2]*a^(3/2)*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*
x)])])/Sqrt[1 - 1/(a*x)]

Rule 6176

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

Rule 6181

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

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*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-a c x}}{x^3} \, dx &=\frac{\sqrt{c-a c x} \int \frac{e^{3 \coth ^{-1}(a x)} \sqrt{1-\frac{1}{a x}}}{x^{5/2}} \, dx}{\sqrt{1-\frac{1}{a x}} \sqrt{x}}\\ &=-\frac{\left (\sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x} \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{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}-\frac{\left (a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{x}{a}} \left (\frac{1}{2}+\frac{7 x}{2 a}\right )}{\sqrt{x} \left (1-\frac{x}{a}\right )} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (a^2 \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{-\frac{9}{4 a}-\frac{23 x}{4 a^2}}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 \sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (23 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{8 \sqrt{1-\frac{1}{a x}}}-\frac{\left (4 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (1-\frac{x}{a}\right ) \sqrt{1+\frac{x}{a}}} \, dx,x,\frac{1}{x}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{\left (23 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,\sqrt{\frac{1}{x}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{\left (8 a \sqrt{\frac{1}{x}} \sqrt{c-a c x}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{2 x^2}{a}} \, dx,x,\frac{\sqrt{\frac{1}{x}}}{\sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ &=\frac{7 a \sqrt{1+\frac{1}{a x}} \sqrt{c-a c x}}{4 \sqrt{1-\frac{1}{a x}} x}+\frac{a \left (1+\frac{1}{a x}\right )^{3/2} \sqrt{c-a c x}}{2 \sqrt{1-\frac{1}{a x}} x}+\frac{23 a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{4 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} a^{3/2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{1+\frac{1}{a x}}}\right )}{\sqrt{1-\frac{1}{a x}}}\\ \end{align*}

Mathematica [A]  time = 0.14794, size = 132, normalized size = 0.59 $\frac{\sqrt{c-a c x} \left (\frac{23 a^{3/2} \sinh ^{-1}\left (\frac{\sqrt{\frac{1}{x}}}{\sqrt{a}}\right )}{\left (\frac{1}{x}\right )^{3/2}}-\frac{16 \sqrt{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{\left (\frac{1}{x}\right )^{3/2}}+\sqrt{\frac{1}{a x}+1} (9 a x+2)\right )}{4 x^2 \sqrt{1-\frac{1}{a x}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(Sqrt[c - a*c*x]*(Sqrt[1 + 1/(a*x)]*(2 + 9*a*x) + (23*a^(3/2)*ArcSinh[Sqrt[x^(-1)]/Sqrt[a]])/(x^(-1))^(3/2) -
(16*Sqrt[2]*a^(3/2)*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/(x^(-1))^(3/2)))/(4*Sqrt[1 -
1/(a*x)]*x^2)

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Maple [A]  time = 0.211, size = 144, normalized size = 0.6 \begin{align*}{\frac{ax-1}{ \left ( 4\,ax+4 \right ){x}^{2}}\sqrt{-c \left ( ax-1 \right ) } \left ( -16\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ){x}^{2}{a}^{2}c+23\,c\arctan \left ({\frac{\sqrt{-c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{2}{a}^{2}+9\,xa\sqrt{-c \left ( ax+1 \right ) }\sqrt{c}+2\,\sqrt{-c \left ( ax+1 \right ) }\sqrt{c} \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{c}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(-16*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1
/2)/c^(1/2))*x^2*a^2*c+23*c*arctan((-c*(a*x+1))^(1/2)/c^(1/2))*x^2*a^2+9*x*a*(-c*(a*x+1))^(1/2)*c^(1/2)+2*(-c*
(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(-c*(a*x+1))^(1/2)/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}}{x^{3} \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)*(-a*c*x+c)^(1/2)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.69091, size = 979, normalized size = 4.37 \begin{align*} \left [\frac{16 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt{2} \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 23 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{-c} \log \left (-\frac{a^{2} c x^{2} + a c x - 2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (9 \, a^{2} x^{2} + 11 \, a x + 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{8 \,{\left (a x^{3} - x^{2}\right )}}, -\frac{16 \, \sqrt{2}{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) - 23 \,{\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a c x + c} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}}}{a c x - c}\right ) -{\left (9 \, a^{2} x^{2} + 11 \, a x + 2\right )} \sqrt{-a c x + c} \sqrt{\frac{a x - 1}{a x + 1}}}{4 \,{\left (a x^{3} - x^{2}\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)*(-a*c*x+c)^(1/2)/x^3,x, algorithm="fricas")

[Out]

[1/8*(16*sqrt(2)*(a^3*x^3 - a^2*x^2)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)*(a*x + 1)
*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 23*(a^3*x^3 - a^2*x^2)*sqrt(-c)*log(-(a^2*
c*x^2 + a*c*x - 2*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 2*c)/(a*x^2 - x)) + 2*(9*a^2
*x^2 + 11*a*x + 2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^3 - x^2), -1/4*(16*sqrt(2)*(a^3*x^3 - a^2*
x^2)*sqrt(c)*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - 23*(a^3*x^3 - a^
2*x^2)*sqrt(c)*arctan(sqrt(-a*c*x + c)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (9*a^2*x^2 + 11*a*x +
2)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a*x^3 - x^2)]

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

[Out]

Timed out

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Giac [C]  time = 1.31402, size = 248, normalized size = 1.11 \begin{align*} -\frac{1}{4} \, a^{2} c^{2}{\left (\frac{16 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{-a c x - c}}{2 \, \sqrt{c}}\right )}{c^{\frac{3}{2}} \mathrm{sgn}\left (-a c x - c\right )} - \frac{23 \, \arctan \left (\frac{\sqrt{-a c x - c}}{\sqrt{c}}\right )}{c^{\frac{3}{2}} \mathrm{sgn}\left (-a c x - c\right )} + \frac{9 \,{\left (-a c x - c\right )}^{\frac{3}{2}} + 7 \, \sqrt{-a c x - c} c}{a^{2} c^{3} x^{2} \mathrm{sgn}\left (-a c x - c\right )}\right )} - \frac{16 i \, \sqrt{2} a^{2} \sqrt{-c} \arctan \left (-i\right ) - 23 i \, a^{2} \sqrt{-c} \arctan \left (-i \, \sqrt{2}\right ) - 11 \, \sqrt{2} a^{2} \sqrt{-c}}{4 \, \mathrm{sgn}\left (c\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a^2*c^2*(16*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c))/(c^(3/2)*sgn(-a*c*x - c)) - 23*arctan(sq
rt(-a*c*x - c)/sqrt(c))/(c^(3/2)*sgn(-a*c*x - c)) + (9*(-a*c*x - c)^(3/2) + 7*sqrt(-a*c*x - c)*c)/(a^2*c^3*x^2
*sgn(-a*c*x - c))) - 1/4*(16*I*sqrt(2)*a^2*sqrt(-c)*arctan(-I) - 23*I*a^2*sqrt(-c)*arctan(-I*sqrt(2)) - 11*sqr
t(2)*a^2*sqrt(-c))/sgn(c)