Integrand size = 23, antiderivative size = 148 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {\sqrt {c-a c x}}{4 x^4}+\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {149 a^3 \sqrt {c-a c x}}{64 x}-\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
-363/64*a^4*arctanh((-a*c*x+c)^(1/2)/c^(1/2))*c^(1/2)+4*a^4*arctanh(1/2*(- a*c*x+c)^(1/2)*2^(1/2)/c^(1/2))*2^(1/2)*c^(1/2)-1/4*(-a*c*x+c)^(1/2)/x^4+1 7/24*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
Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (-48+136 a x-214 a^2 x^2+447 a^3 x^3\right )}{192 x^4}-\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )+4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]
(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])]
Time = 0.44 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.16, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {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.
| \(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle \frac {-\frac {1}{8} a c^2 \left (-\frac {1}{6} a \left (-\frac {3}{4} a \left (-\frac {1}{2} a \left (\frac {512 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {c}}-\frac {726 \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {149 \sqrt {c-a c x}}{c x}\right )-\frac {107 \sqrt {c-a c x}}{2 c x^2}\right )-\frac {17 \sqrt {c-a c x}}{3 c x^3}\right )-\frac {c \sqrt {c-a c x}}{4 x^4}}{c}\) |
(-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]*ArcTa nh[Sqrt[c - a*c*x]/(Sqrt[2]*Sqrt[c])])/Sqrt[c]))/2))/4))/6))/8)/c
3.5.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
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])
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]
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]
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])
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]
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])
Time = 0.17 (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 | \(-2 a^{4} c^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 c^{3} \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\) | \(122\) |
| default | \(-2 a^{4} c^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 c^{3} \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\) | \(122\) |
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
Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.59 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {768 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 1089 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) + 2 \, {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{384 \, x^{4}}, -\frac {768 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 1089 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{192 \, x^{4}}\right ] \]
[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(1/2*sqr t(2)*sqrt(-a*c*x + c)*sqrt(-c)/c) - 1089*a^4*sqrt(-c)*x^4*arctan(sqrt(-a*c *x + c)*sqrt(-c)/c) - (447*a^3*x^3 - 214*a^2*x^2 + 136*a*x - 48)*sqrt(-a*c *x + c))/x^4]
\[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=- \int \left (- \frac {\sqrt {- a c x + c}}{a x^{6} + x^{5}}\right )\, dx - \int \frac {a x \sqrt {- a c x + c}}{a x^{6} + x^{5}}\, dx \]
-Integral(-sqrt(-a*c*x + c)/(a*x**6 + x**5), x) - Integral(a*x*sqrt(-a*c*x + c)/(a*x**6 + x**5), x)
Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {1}{384} \, a^{4} c^{4} {\left (\frac {2 \, {\left (447 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 1127 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 321 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {768 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} - \frac {1089 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]
-1/384*a^4*c^4*(2*(447*(-a*c*x + c)^(7/2) - 1127*(-a*c*x + c)^(5/2)*c + 10 49*(-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) + 76 8*sqrt(2)*log(-(sqrt(2)*sqrt(c) - sqrt(-a*c*x + c))/(sqrt(2)*sqrt(c) + sqr t(-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))
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=-\frac {4 \, \sqrt {2} a^{4} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} + \frac {363 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c}} + \frac {447 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{4} c + 1127 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{2} - 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{3} + 321 \, \sqrt {-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \]
-4*sqrt(2)*a^4*c*arctan(1/2*sqrt(2)*sqrt(-a*c*x + c)/sqrt(-c))/sqrt(-c) + 363/64*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)
Time = 0.15 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {107\,\sqrt {c-a\,c\,x}}{64\,x^4}+\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,363{}\mathrm {i}}{64}-\frac {1049\,{\left (c-a\,c\,x\right )}^{3/2}}{192\,c\,x^4}+\frac {1127\,{\left (c-a\,c\,x\right )}^{5/2}}{192\,c^2\,x^4}-\frac {149\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}-\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]