\(\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx\) [460]

Optimal result

Integrand size = 23, antiderivative size = 127 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {\sqrt {c-a c x}}{3 x^3}+\frac {13 a \sqrt {c-a c x}}{12 x^2}-\frac {19 a^2 \sqrt {c-a c x}}{8 x}+\frac {45}{8} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

-1/3*(-a*c*x+c)^(1/2)/x^3+13/12*a*(-a*c*x+c)^(1/2)/x^2-19/8*a^2*(-a*c*x+c) 
^(1/2)/x+45/8*a^3*c^(1/2)*arctanh((-a*c*x+c)^(1/2)/c^(1/2))-4*2^(1/2)*a^3* 
c^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c-a c x} \left (-8+26 a x-57 a^2 x^2\right )}{24 x^3}+\frac {45}{8} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^3 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

Integrate[Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^4),x]
 

Output:

(Sqrt[c - a*c*x]*(-8 + 26*a*x - 57*a^2*x^2))/(24*x^3) + (45*a^3*Sqrt[c]*Ar 
cTanh[Sqrt[c - a*c*x]/Sqrt[c]])/8 - 4*Sqrt[2]*a^3*Sqrt[c]*ArcTanh[Sqrt[c - 
 a*c*x]/(Sqrt[2]*Sqrt[c])]
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6680, 35, 109, 27, 168, 27, 168, 27, 174, 73, 219, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx\)

\(\Big \downarrow \) 6680

\(\displaystyle \int \frac {(1-a x) \sqrt {c-a c x}}{x^4 (a x+1)}dx\)

\(\Big \downarrow \) 35

\(\displaystyle \frac {\int \frac {(c-a c x)^{3/2}}{x^4 (a x+1)}dx}{c}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {-\frac {1}{3} \int \frac {a c^2 (13-11 a x)}{2 x^3 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{6} a c^2 \int \frac {13-11 a x}{x^3 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {\int \frac {3 a c (19-13 a x)}{2 x^2 (a x+1) \sqrt {c-a c x}}dx}{2 c}-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \int \frac {19-13 a x}{x^2 (a x+1) \sqrt {c-a c x}}dx-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {\int \frac {a c (45-19 a x)}{2 x (a x+1) \sqrt {c-a c x}}dx}{c}-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {1}{2} a \int \frac {45-19 a x}{x (a x+1) \sqrt {c-a c x}}dx-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (45 \int \frac {1}{x \sqrt {c-a c x}}dx-64 a \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx\right )-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {128 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{c}-\frac {90 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {1}{6} a c^2 \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {64 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {19 \sqrt {c-a c x}}{c x}\right )-\frac {13 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {c \sqrt {c-a c x}}{3 x^3}}{c}\)

Input:

Int[Sqrt[c - a*c*x]/(E^(2*ArcTanh[a*x])*x^4),x]
 

Output:

(-1/3*(c*Sqrt[c - a*c*x])/x^3 - (a*c^2*((-13*Sqrt[c - a*c*x])/(2*c*x^2) - 
(3*a*((-19*Sqrt[c - a*c*x])/(c*x) - (a*((-90*ArcTanh[Sqrt[c - a*c*x]/Sqrt[ 
c]])/Sqrt[c] + (64*Sqrt[2]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqr 
t[c]))/2))/4))/6)/c
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 109

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 174

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

rule 219

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

rule 221

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

rule 6680

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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.70

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {-c \left (a x -1\right )}\, \left (57 a^{2} x^{2}-26 a x +8\right ) \sqrt {c}}{3}+a^{3} c \,x^{3} \left (32 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-45 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )\right )}{8 \sqrt {c}\, x^{3}}\)	\(89\)
risch	\(\frac {\left (57 a^{3} x^{3}-83 a^{2} x^{2}+34 a x -8\right ) c}{24 x^{3} \sqrt {-c \left (a x -1\right )}}-\frac {a^{3} \left (\frac {64 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {90 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c}{16}\)	\(92\)
derivativedivides	\(2 a^{3} c^{3} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {5}{2}}}+\frac {\frac {-\frac {19 \left (-a c x +c \right )^{\frac {5}{2}}}{16}+\frac {11 \left (-a c x +c \right )^{\frac {3}{2}} c}{6}-\frac {13 \sqrt {-a c x +c}\, c^{2}}{16}}{a^{3} c^{3} x^{3}}+\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{c^{2}}\right )\)	\(108\)
default	\(-2 a^{3} c^{3} \left (\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {5}{2}}}-\frac {\frac {-\frac {19 \left (-a c x +c \right )^{\frac {5}{2}}}{16}+\frac {11 \left (-a c x +c \right )^{\frac {3}{2}} c}{6}-\frac {13 \sqrt {-a c x +c}\, c^{2}}{16}}{a^{3} c^{3} x^{3}}+\frac {45 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{16 \sqrt {c}}}{c^{2}}\right )\)	\(109\)

Input:

int((-a*c*x+c)^(1/2)/(a*x+1)^2*(-a^2*x^2+1)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/8/c^(1/2)*(1/3*(-c*(a*x-1))^(1/2)*(57*a^2*x^2-26*a*x+8)*c^(1/2)+a^3*c*x 
^3*(32*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^(1/2))-45*arctanh( 
(-c*(a*x-1))^(1/2)/c^(1/2))))/x^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.83 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\left [\frac {96 \, \sqrt {2} a^{3} \sqrt {c} x^{3} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 135 \, a^{3} \sqrt {c} x^{3} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a c x + c}}{48 \, x^{3}}, -\frac {96 \, \sqrt {2} a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) - 135 \, a^{3} \sqrt {-c} x^{3} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (57 \, a^{2} x^{2} - 26 \, a x + 8\right )} \sqrt {-a c x + c}}{24 \, x^{3}}\right ] \] Input:

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

Output:

[1/48*(96*sqrt(2)*a^3*sqrt(c)*x^3*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c)* 
sqrt(c) - 3*c)/(a*x + 1)) + 135*a^3*sqrt(c)*x^3*log((a*c*x - 2*sqrt(-a*c*x 
 + c)*sqrt(c) - 2*c)/x) - 2*(57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/x^ 
3, -1/24*(96*sqrt(2)*a^3*sqrt(-c)*x^3*arctan(sqrt(2)*sqrt(-a*c*x + c)*sqrt 
(-c)/(a*c*x - c)) - 135*a^3*sqrt(-c)*x^3*arctan(sqrt(-a*c*x + c)*sqrt(-c)/ 
(a*c*x - c)) + (57*a^2*x^2 - 26*a*x + 8)*sqrt(-a*c*x + c))/x^3]
 

Sympy [F]

\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=- \int \left (- \frac {\sqrt {- a c x + c}}{a x^{5} + x^{4}}\right )\, dx - \int \frac {a x \sqrt {- a c x + c}}{a x^{5} + x^{4}}\, dx \] Input:

integrate((-a*c*x+c)**(1/2)/(a*x+1)**2*(-a**2*x**2+1)/x**4,x)
 

Output:

-Integral(-sqrt(-a*c*x + c)/(a*x**5 + x**4), x) - Integral(a*x*sqrt(-a*c*x 
 + c)/(a*x**5 + x**4), x)
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.44 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=-\frac {1}{48} \, a^{3} c^{3} {\left (\frac {2 \, {\left (57 \, {\left (-a c x + c\right )}^{\frac {5}{2}} - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c + 39 \, \sqrt {-a c x + c} c^{2}\right )}}{{\left (a c x - c\right )}^{3} c^{2} + 3 \, {\left (a c x - c\right )}^{2} c^{3} + 3 \, {\left (a c x - c\right )} c^{4} + c^{5}} - \frac {96 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {5}{2}}} + \frac {135 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {5}{2}}}\right )} \] Input:

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

Output:

-1/48*a^3*c^3*(2*(57*(-a*c*x + c)^(5/2) - 88*(-a*c*x + c)^(3/2)*c + 39*sqr 
t(-a*c*x + c)*c^2)/((a*c*x - c)^3*c^2 + 3*(a*c*x - c)^2*c^3 + 3*(a*c*x - c 
)*c^4 + c^5) - 96*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt( 
2)*sqrt(c) + sqrt(-a*c*x + c)))/c^(5/2) + 135*log((sqrt(-a*c*x + c) - sqrt 
(c))/(sqrt(-a*c*x + c) + sqrt(c)))/c^(5/2))
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.05 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {4 \, \sqrt {2} a^{3} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {45 \, a^{3} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{8 \, \sqrt {-c}} - \frac {57 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{3} c - 88 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{3} c^{2} + 39 \, \sqrt {-a c x + c} a^{3} c^{3}}{24 \, a^{3} c^{3} x^{3}} \] Input:

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

Output:

4*sqrt(2)*a^3*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 4 
5/8*a^3*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 1/24*(57*(a*c*x - c 
)^2*sqrt(-a*c*x + c)*a^3*c - 88*(-a*c*x + c)^(3/2)*a^3*c^2 + 39*sqrt(-a*c* 
x + c)*a^3*c^3)/(a^3*c^3*x^3)
 

Mupad [B] (verification not implemented)

Time = 23.15 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {11\,{\left (c-a\,c\,x\right )}^{3/2}}{3\,c\,x^3}-\frac {a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,45{}\mathrm {i}}{8}-\frac {13\,\sqrt {c-a\,c\,x}}{8\,x^3}-\frac {19\,{\left (c-a\,c\,x\right )}^{5/2}}{8\,c^2\,x^3}+\sqrt {2}\,a^3\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \] Input:

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

Output:

(11*(c - a*c*x)^(3/2))/(3*c*x^3) - (a^3*c^(1/2)*atan(((c - a*c*x)^(1/2)*1i 
)/c^(1/2))*45i)/8 - (13*(c - a*c*x)^(1/2))/(8*x^3) - (19*(c - a*c*x)^(5/2) 
)/(8*c^2*x^3) + 2^(1/2)*a^3*c^(1/2)*atan((2^(1/2)*(c - a*c*x)^(1/2)*1i)/(2 
*c^(1/2)))*4i
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^4} \, dx=\frac {\sqrt {c}\, \left (-114 \sqrt {-a x +1}\, a^{2} x^{2}+52 \sqrt {-a x +1}\, a x -16 \sqrt {-a x +1}+96 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{3} x^{3}-96 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{3} x^{3}-135 \,\mathrm {log}\left (\sqrt {-a x +1}-1\right ) a^{3} x^{3}+135 \,\mathrm {log}\left (\sqrt {-a x +1}+1\right ) a^{3} x^{3}\right )}{48 x^{3}} \] Input:

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

Output:

(sqrt(c)*( - 114*sqrt( - a*x + 1)*a**2*x**2 + 52*sqrt( - a*x + 1)*a*x - 16 
*sqrt( - a*x + 1) + 96*sqrt(2)*log(sqrt( - a*x + 1) - sqrt(2))*a**3*x**3 - 
 96*sqrt(2)*log(sqrt( - a*x + 1) + sqrt(2))*a**3*x**3 - 135*log(sqrt( - a* 
x + 1) - 1)*a**3*x**3 + 135*log(sqrt( - a*x + 1) + 1)*a**3*x**3))/(48*x**3 
)