\(\int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx\) [349]

Optimal result

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

[Out]

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^4-17/24*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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6302, 6265, 21, 100, 156, 162, 65, 214, 212} \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right )-\frac {149 a^3 \sqrt {c-a c x}}{64 x}+\frac {107 a^2 \sqrt {c-a c x}}{96 x^2}+\frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3} \]

[In]

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

[Out]

Sqrt[c - a*c*x]/(4*x^4) - (17*a*Sqrt[c - a*c*x])/(24*x^3) + (107*a^2*Sqrt[c - a*c*x])/(96*x^2) - (149*a^3*Sqrt
[c - a*c*x])/(64*x) + (363*a^4*Sqrt[c]*ArcTanh[Sqrt[c - a*c*x]/Sqrt[c]])/64 - 4*Sqrt[2]*a^4*Sqrt[c]*ArcTanh[Sq
rt[c - a*c*x]/(Sqrt[2]*Sqrt[c])]

Rule 21

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

Rule 65

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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 156

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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 6265

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 6302

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

Rubi steps \begin{align*} \text {integral}& = -\int \frac {e^{-2 \text {arctanh}(a x)} \sqrt {c-a c x}}{x^5} \, dx \\ & = -\int \frac {(1-a x) \sqrt {c-a c x}}{x^5 (1+a x)} \, dx \\ & = -\frac {\int \frac {(c-a c x)^{3/2}}{x^5 (1+a x)} \, dx}{c} \\ & = \frac {\sqrt {c-a c x}}{4 x^4}+\frac {\int \frac {\frac {17 a c^2}{2}-\frac {15}{2} a^2 c^2 x}{x^4 (1+a x) \sqrt {c-a c x}} \, dx}{4 c} \\ & = \frac {\sqrt {c-a c x}}{4 x^4}-\frac {17 a \sqrt {c-a c x}}{24 x^3}-\frac {\int \frac {\frac {107 a^2 c^3}{4}-\frac {85}{4} a^3 c^3 x}{x^3 (1+a x) \sqrt {c-a c x}} \, dx}{12 c^2} \\ & = \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 {\int \frac {\frac {447 a^3 c^4}{8}-\frac {321}{8} a^4 c^4 x}{x^2 (1+a x) \sqrt {c-a c x}} \, dx}{24 c^3} \\ & = \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 {\int \frac {\frac {1089 a^4 c^5}{16}-\frac {447}{16} a^5 c^5 x}{x (1+a x) \sqrt {c-a c x}} \, dx}{24 c^4} \\ & = \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 {1}{128} \left (363 a^4 c\right ) \int \frac {1}{x \sqrt {c-a c x}} \, dx+\left (4 a^5 c\right ) \int \frac {1}{(1+a x) \sqrt {c-a c x}} \, 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 {1}{64} \left (363 a^3\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a c}} \, dx,x,\sqrt {c-a c x}\right )-\left (8 a^4\right ) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{c}} \, dx,x,\sqrt {c-a c x}\right ) \\ & = \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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.74 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {\sqrt {c-a c x} \left (48-136 a x+214 a^2 x^2-447 a^3 x^3\right )}{192 x^4}+\frac {363}{64} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {c}}\right )-4 \sqrt {2} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a c x}}{\sqrt {2} \sqrt {c}}\right ) \]

[In]

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

[Out]

(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])]

Maple [A] (verified)

Time = 0.61 (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	\(\frac {\left (447 a^{4} x^{4}-661 a^{3} x^{3}+350 a^{2} x^{2}-184 a x +48\right ) c}{192 x^{4} \sqrt {-c \left (a x -1\right )}}-\frac {a^{4} \left (-\frac {726 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{\sqrt {c}}+\frac {512 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{\sqrt {c}}\right ) c}{128}\)	\(100\)
derivativedivides	\(2 c^{4} a^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 c^{3} \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\)	\(122\)
default	\(2 c^{4} a^{4} \left (-\frac {2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}\, \sqrt {2}}{2 \sqrt {c}}\right )}{c^{\frac {7}{2}}}+\frac {\frac {\frac {149 \left (-a c x +c \right )^{\frac {7}{2}}}{128}-\frac {1127 c \left (-a c x +c \right )^{\frac {5}{2}}}{384}+\frac {1049 c^{2} \left (-a c x +c \right )^{\frac {3}{2}}}{384}-\frac {107 c^{3} \sqrt {-a c x +c}}{128}}{a^{4} c^{4} x^{4}}+\frac {363 \,\operatorname {arctanh}\left (\frac {\sqrt {-a c x +c}}{\sqrt {c}}\right )}{128 \sqrt {c}}}{c^{3}}\right )\)	\(122\)

[In]

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

[Out]

-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)*arct
anh(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)

none

Time = 0.27 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.59 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\left [\frac {768 \, \sqrt {2} a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} \sqrt {c} - 3 \, c}{a x + 1}\right ) + 1089 \, a^{4} \sqrt {c} x^{4} \log \left (\frac {a c x - 2 \, \sqrt {-a c x + c} \sqrt {c} - 2 \, c}{x}\right ) - 2 \, {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{384 \, x^{4}}, \frac {768 \, \sqrt {2} a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {-c}}{2 \, c}\right ) - 1089 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a c x + c} \sqrt {-c}}{c}\right ) - {\left (447 \, a^{3} x^{3} - 214 \, a^{2} x^{2} + 136 \, a x - 48\right )} \sqrt {-a c x + c}}{192 \, x^{4}}\right ] \]

[In]

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

[Out]

[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*sqrt(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]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {1}{384} \, a^{4} c^{4} {\left (\frac {2 \, {\left (447 \, {\left (-a c x + c\right )}^{\frac {7}{2}} - 1127 \, {\left (-a c x + c\right )}^{\frac {5}{2}} c + 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} c^{2} - 321 \, \sqrt {-a c x + c} c^{3}\right )}}{{\left (a c x - c\right )}^{4} c^{3} + 4 \, {\left (a c x - c\right )}^{3} c^{4} + 6 \, {\left (a c x - c\right )}^{2} c^{5} + 4 \, {\left (a c x - c\right )} c^{6} + c^{7}} + \frac {768 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-a c x + c}}{\sqrt {2} \sqrt {c} + \sqrt {-a c x + c}}\right )}{c^{\frac {7}{2}}} - \frac {1089 \, \log \left (\frac {\sqrt {-a c x + c} - \sqrt {c}}{\sqrt {-a c x + c} + \sqrt {c}}\right )}{c^{\frac {7}{2}}}\right )} \]

[In]

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

[Out]

1/384*a^4*c^4*(2*(447*(-a*c*x + c)^(7/2) - 1127*(-a*c*x + c)^(5/2)*c + 1049*(-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) + 7
68*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)

none

Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {4 \, \sqrt {2} a^{4} c \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c}}{2 \, \sqrt {-c}}\right )}{\sqrt {-c}} - \frac {363 \, a^{4} c \arctan \left (\frac {\sqrt {-a c x + c}}{\sqrt {-c}}\right )}{64 \, \sqrt {-c}} - \frac {447 \, {\left (a c x - c\right )}^{3} \sqrt {-a c x + c} a^{4} c + 1127 \, {\left (a c x - c\right )}^{2} \sqrt {-a c x + c} a^{4} c^{2} - 1049 \, {\left (-a c x + c\right )}^{\frac {3}{2}} a^{4} c^{3} + 321 \, \sqrt {-a c x + c} a^{4} c^{4}}{192 \, a^{4} c^{4} x^{4}} \]

[In]

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

[Out]

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)

Mupad [B] (verification not implemented)

Time = 4.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.82 \[ \int \frac {e^{-2 \coth ^{-1}(a x)} \sqrt {c-a c x}}{x^5} \, dx=\frac {1049\,{\left (c-a\,c\,x\right )}^{3/2}}{192\,c\,x^4}-\frac {a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{\sqrt {c}}\right )\,363{}\mathrm {i}}{64}-\frac {107\,\sqrt {c-a\,c\,x}}{64\,x^4}-\frac {1127\,{\left (c-a\,c\,x\right )}^{5/2}}{192\,c^2\,x^4}+\frac {149\,{\left (c-a\,c\,x\right )}^{7/2}}{64\,c^3\,x^4}+\sqrt {2}\,a^4\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-a\,c\,x}\,1{}\mathrm {i}}{2\,\sqrt {c}}\right )\,4{}\mathrm {i} \]

[In]

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

[Out]

(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