\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx\) [371]

Optimal result

Integrand size = 23, antiderivative size = 148 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Output:

1/4*(-a*c*x+c)^(1/2)/x^4-17/24*a*(-a*c*x+c)^(1/2)/x^3+107/96*a^2*(-a*c*x+c 
)^(1/2)/x^2-149/64*a^3*(-a*c*x+c)^(1/2)/x+363/64*a^4*c^(1/2)*arctanh((-a*c 
*x+c)^(1/2)/c^(1/2))-4*2^(1/2)*a^4*c^(1/2)*arctanh(1/2*(-a*c*x+c)^(1/2)*2^ 
(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (48-136 a x+214 a^2 x^2-447 a^3 x^3\right )}{192 x^4}+\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \] Input:

Integrate[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^5),x]
 

Output:

(Sqrt[c - a*c*x]*(48 - 136*a*x + 214*a^2*x^2 - 447*a^3*x^3))/(192*x^4) + ( 
363*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64 - 4*Sqrt[2]*a^4*Sqrt[ 
c]*ArcTanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {6717, 6680, 35, 109, 27, 168, 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.

\(\displaystyle \int \frac {\sqrt {c-a c x} e^{-2 \coth ^{-1}(a x)}}{x^5} \, dx\)

\(\Big \downarrow \) 6717

\(\displaystyle -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5}dx\)

\(\Big \downarrow \) 6680

\(\displaystyle -\int \frac {(1-a x) \sqrt {c-a c x}}{x^5 (a x+1)}dx\)

\(\Big \downarrow \) 35

\(\displaystyle -\frac {\int \frac {(c-a c x)^{3/2}}{x^5 (a x+1)}dx}{c}\)

\(\Big \downarrow \) 109

\(\displaystyle -\frac {-\frac {1}{4} \int \frac {a c^2 (17-15 a x)}{2 x^4 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \int \frac {17-15 a x}{x^4 (a x+1) \sqrt {c-a c x}}dx-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {\int \frac {a c (107-85 a x)}{2 x^3 (a x+1) \sqrt {c-a c x}}dx}{3 c}-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \int \frac {107-85 a x}{x^3 (a x+1) \sqrt {c-a c x}}dx-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {\int \frac {3 a c (149-107 a x)}{2 x^2 (a x+1) \sqrt {c-a c x}}dx}{2 c}-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \int \frac {149-107 a x}{x^2 (a x+1) \sqrt {c-a c x}}dx-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 168

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {\int \frac {a c (363-149 a x)}{2 x (a x+1) \sqrt {c-a c x}}dx}{c}-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \int \frac {363-149 a x}{x (a x+1) \sqrt {c-a c x}}dx-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 174

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (363 \int \frac {1}{x \sqrt {c-a c x}}dx-512 a \int \frac {1}{(a x+1) \sqrt {c-a c x}}dx\right )-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 73

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {1024 \int \frac {1}{2-\frac {c-a c x}{c}}d\sqrt {c-a c x}}{c}-\frac {726 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {512 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {726 \int \frac {1}{\frac {1}{a}-\frac {c-a c x}{a c}}d\sqrt {c-a c x}}{a c}\right )-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {512 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {726 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\)

Input:

Int[Sqrt[c - a*c*x]/(E^(2*ArcCoth[a*x])*x^5),x]
 

Output:

-((-1/4*(c*Sqrt[c - a*c*x])/x^4 - (a*c^2*((-17*Sqrt[c - a*c*x])/(3*c*x^3) 
- (a*((-107*Sqrt[c - a*c*x])/(2*c*x^2) - (3*a*((-149*Sqrt[c - a*c*x])/(c*x 
) - (a*((-726*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/Sqrt[c] + (512*Sqrt[2]*Arc 
Tanh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]))/2))/4))/6))/8)/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])
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.66

method	result	size
pseudoelliptic	\(-\frac {\frac {\sqrt {-c \left (a x -1\right )}\, \left (447 a^{3} x^{3}-214 a^{2} x^{2}+136 a x -48\right ) \sqrt {c}}{3}+a^{4} c \,x^{4} \left (256 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-363 \,\operatorname {arctanh}\left (\frac {\sqrt {-c \left (a x -1\right )}}{\sqrt {c}}\right )\right )}{64 \sqrt {c}\, x^{4}}\)	\(97\)
risch	\(\frac {\left (447 a^{4} x^{4}-661 a^{3} x^{3}+350 a^{2} x^{2}-184 a x +48\right ) c}{192 x^{4} \sqrt {-c \left (a x -1\right )}}-\frac {a^{4} \left (\frac {512 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}-\frac {726 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) c}{128}\)	\(100\)
derivativedivides	\(2 c^{4} a^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 \left (-a c x +c \right )^{\frac {5}{2}} c}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 \sqrt {-a c x +c}\, c^{3}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\)	\(122\)
default	\(2 c^{4} a^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 \left (-a c x +c \right )^{\frac {5}{2}} c}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 \sqrt {-a c x +c}\, c^{3}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\)	\(122\)

Input:

int((-a*c*x+c)^(1/2)*(a*x-1)/(a*x+1)/x^5,x,method=_RETURNVERBOSE)
 

Output:

-1/64/c^(1/2)*(1/3*(-c*(a*x-1))^(1/2)*(447*a^3*x^3-214*a^2*x^2+136*a*x-48) 
*c^(1/2)+a^4*c*x^4*(256*2^(1/2)*arctanh(1/2*(-c*(a*x-1))^(1/2)*2^(1/2)/c^( 
1/2))-363*arctanh((-c*(a*x-1))^(1/2)/c^(1/2))))/x^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.68 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {768 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 1089 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{384 \, x^{4}}, -\frac {768 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) - 1089 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{a c x - c}\right ) + {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{192 \, x^{4}}\right ] \] Input:

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

Output:

[1/384*(768*sqrt(2)*a^4*sqrt(c)*x^4*log((a*c*x + 2*sqrt(2)*sqrt(-a*c*x + c 
)*sqrt(c) - 3*c)/(a*x + 1)) + 1089*a^4*sqrt(c)*x^4*log((a*c*x - 2*sqrt(-a* 
c*x + c)*sqrt(c) - 2*c)/x) - 2*(447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48)* 
sqrt(-a*c*x + c))/x^4, -1/192*(768*sqrt(2)*a^4*sqrt(-c)*x^4*arctan(sqrt(2) 
*sqrt(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) - 1089*a^4*sqrt(-c)*x^4*arctan(sqr 
t(-a*c*x + c)*sqrt(-c)/(a*c*x - c)) + (447*a^3*x^3 - 214*a^2*x^2 + 136*a*x 
 - 48)*sqrt(-a*c*x + c))/x^4]
 

Sympy [F]

\[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\int \frac {\sqrt {- c \left (a x - 1\right )} \left (a x - 1\right )}{x^{5} \left (a x + 1\right )}\, dx \] Input:

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

Output:

Integral(sqrt(-c*(a*x - 1))*(a*x - 1)/(x**5*(a*x + 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {1}{384} \, a^{4} c^{4} {\left (\frac {2 \, {\left (447 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 1127 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 321 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {768 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} - \frac {1089 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \] Input:

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

Output:

1/384*a^4*c^4*(2*(447*(-a*c*x + c)^(7/2) - 1127*(-a*c*x + c)^(5/2)*c + 104 
9*(-a*c*x + c)^(3/2)*c^2 - 321*sqrt(-a*c*x + c)*c^3)/((a*c*x - c)^4*c^3 + 
4*(a*c*x - c)^3*c^4 + 6*(a*c*x - c)^2*c^5 + 4*(a*c*x - c)*c^6 + c^7) + 768 
*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqrt 
(-a*c*x + c)))/c^(7/2) - 1089*log((sqrt(-a*c*x + c) - sqrt(c))/(sqrt(-a*c* 
x + c) + sqrt(c)))/c^(7/2))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {4 \, \sqrt {2} a^{4} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {363 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c}} - \frac {447 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{4} c + 1127 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{2} - 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{3} + 321 \, \sqrt {-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \] Input:

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

Output:

4*sqrt(2)*a^4*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 3 
63/64*a^4*c*arctan(sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) - 1/192*(447*(a*c*x 
 - c)^3*sqrt(-a*c*x + c)*a^4*c + 1127*(a*c*x - c)^2*sqrt(-a*c*x + c)*a^4*c 
^2 - 1049*(-a*c*x + c)^(3/2)*a^4*c^3 + 321*sqrt(-a*c*x + c)*a^4*c^4)/(a^4* 
c^4*x^4)
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {1049\,{\left (c-a\,c\,x\right )}^{3/2}}{192\,c\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,363{}\mathrm {i}}{64}-\frac {107\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {1127\,{\left (c-a\,c\,x\right )}^{5/2}}{192\,c^2\,x^4}+\frac {149\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}+\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \] Input:

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

Output:

(1049*(c - a*c*x)^(3/2))/(192*c*x^4) - (a^4*c^(1/2)*atan(((c - a*c*x)^(1/2 
)*1i)/c^(1/2))*363i)/64 - (107*(c - a*c*x)^(1/2))/(64*x^4) - (1127*(c - a* 
c*x)^(5/2))/(192*c^2*x^4) + (149*(c - a*c*x)^(7/2))/(64*c^3*x^4) + 2^(1/2) 
*a^4*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 = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.93 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c}\, \left (-894 \sqrt {-a x +1}\, a^{3} x^{3}+428 \sqrt {-a x +1}\, a^{2} x^{2}-272 \sqrt {-a x +1}\, a x +96 \sqrt {-a x +1}+768 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}-\sqrt {2}\right ) a^{4} x^{4}-768 \sqrt {2}\, \mathrm {log}\left (\sqrt {-a x +1}+\sqrt {2}\right ) a^{4} x^{4}-1089 \,\mathrm {log}\left (\sqrt {-a x +1}-1\right ) a^{4} x^{4}+1089 \,\mathrm {log}\left (\sqrt {-a x +1}+1\right ) a^{4} x^{4}\right )}{384 x^{4}} \] Input:

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

Output:

(sqrt(c)*( - 894*sqrt( - a*x + 1)*a**3*x**3 + 428*sqrt( - a*x + 1)*a**2*x* 
*2 - 272*sqrt( - a*x + 1)*a*x + 96*sqrt( - a*x + 1) + 768*sqrt(2)*log(sqrt 
( - a*x + 1) - sqrt(2))*a**4*x**4 - 768*sqrt(2)*log(sqrt( - a*x + 1) + sqr 
t(2))*a**4*x**4 - 1089*log(sqrt( - a*x + 1) - 1)*a**4*x**4 + 1089*log(sqrt 
( - a*x + 1) + 1)*a**4*x**4))/(384*x**4)