\(\int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx\) [1107]

Optimal result

Integrand size = 27, antiderivative size = 130 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {\sqrt {c-a^2 c x^2}}{4 x^4}-\frac {2 a \sqrt {c-a^2 c x^2}}{3 x^3}-\frac {7 a^2 \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {4 a^3 \sqrt {c-a^2 c x^2}}{3 x}-\frac {7}{8} a^4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

-1/4*(-a^2*c*x^2+c)^(1/2)/x^4-2/3*a*(-a^2*c*x^2+c)^(1/2)/x^3-7/8*a^2*(-a^2 
*c*x^2+c)^(1/2)/x^2-4/3*a^3*(-a^2*c*x^2+c)^(1/2)/x-7/8*a^4*c^(1/2)*arctanh 
((-a^2*c*x^2+c)^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\frac {\sqrt {c-a^2 c x^2} \left (6+16 a x+21 a^2 x^2+32 a^3 x^3\right )}{24 x^4}+\frac {7}{8} a^4 \sqrt {c} \log (x)-\frac {7}{8} a^4 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

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

Output:

-1/24*(Sqrt[c - a^2*c*x^2]*(6 + 16*a*x + 21*a^2*x^2 + 32*a^3*x^3))/x^4 + ( 
7*a^4*Sqrt[c]*Log[x])/8 - (7*a^4*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^ 
2]])/8
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.15, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {6701, 540, 25, 27, 539, 25, 27, 539, 25, 27, 534, 243, 73, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

\(\displaystyle c \left (-\frac {\int -\frac {a c (7 a x+8)}{x^4 \sqrt {c-a^2 c x^2}}dx}{4 c}-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\int \frac {a c (7 a x+8)}{x^4 \sqrt {c-a^2 c x^2}}dx}{4 c}-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{4} a \int \frac {7 a x+8}{x^4 \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 539

\(\displaystyle c \left (\frac {1}{4} a \left (-\frac {\int -\frac {a c (16 a x+21)}{x^3 \sqrt {c-a^2 c x^2}}dx}{3 c}-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {\int \frac {a c (16 a x+21)}{x^3 \sqrt {c-a^2 c x^2}}dx}{3 c}-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \int \frac {16 a x+21}{x^3 \sqrt {c-a^2 c x^2}}dx-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 539

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (-\frac {\int -\frac {a c (21 a x+32)}{x^2 \sqrt {c-a^2 c x^2}}dx}{2 c}-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {\int \frac {a c (21 a x+32)}{x^2 \sqrt {c-a^2 c x^2}}dx}{2 c}-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \int \frac {21 a x+32}{x^2 \sqrt {c-a^2 c x^2}}dx-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 534

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (21 a \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (\frac {21}{2} a \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-\frac {21 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a c}-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {1}{3} a \left (\frac {1}{2} a \left (-\frac {21 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {32 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {21 \sqrt {c-a^2 c x^2}}{2 c x^2}\right )-\frac {8 \sqrt {c-a^2 c x^2}}{3 c x^3}\right )-\frac {\sqrt {c-a^2 c x^2}}{4 c x^4}\right )\)

Input:

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

Output:

c*(-1/4*Sqrt[c - a^2*c*x^2]/(c*x^4) + (a*((-8*Sqrt[c - a^2*c*x^2])/(3*c*x^ 
3) + (a*((-21*Sqrt[c - a^2*c*x^2])/(2*c*x^2) + (a*((-32*Sqrt[c - a^2*c*x^2 
])/(c*x) - (21*a*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/Sqrt[c]))/2))/3))/4 
)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 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 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 534

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d   Int[ 
x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 
0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
 

rule 539

Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[c*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] + Simp[1/(a*(m + 1)) 
   Int[x^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] 
/; FreeQ[{a, b, c, d, p}, x] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 540

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[(c + d*x)^n, x, x], R = PolynomialRemain 
der[(c + d*x)^n, x, x]}, Simp[R*x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))) 
, x] + Simp[1/(a*(m + 1))   Int[x^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*(m + 
1)*Qx - b*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IG 
tQ[n, 1] && ILtQ[m, -1] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 6701

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

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.73

method	result	size
risch	\(\frac {\left (32 a^{5} x^{5}+21 a^{4} x^{4}-16 a^{3} x^{3}-15 a^{2} x^{2}-16 a x -6\right ) c}{24 x^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {7 a^{4} \sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\)	\(95\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4 c \,x^{4}}+\frac {9 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{2 c \,x^{2}}-\frac {a^{2} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )}{2}\right )}{4}-\frac {2 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3 c \,x^{3}}+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{c x}-2 a^{2} \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )+2 a^{4} \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )-2 a^{4} \left (\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}-\frac {a c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{\sqrt {a^{2} c}}\right )\)	\(341\)

Input:

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

Output:

1/24*(32*a^5*x^5+21*a^4*x^4-16*a^3*x^3-15*a^2*x^2-16*a*x-6)/x^4/(-c*(a^2*x 
^2-1))^(1/2)*c-7/8*a^4*c^(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.31 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\left [\frac {21 \, a^{4} \sqrt {c} x^{4} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, \frac {21 \, a^{4} \sqrt {-c} x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - {\left (32 \, a^{3} x^{3} + 21 \, a^{2} x^{2} + 16 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \] Input:

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

Output:

[1/48*(21*a^4*sqrt(c)*x^4*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) 
 - 2*c)/x^2) - 2*(32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt(-a^2*c*x^2 + 
c))/x^4, 1/24*(21*a^4*sqrt(-c)*x^4*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) 
 - (32*a^3*x^3 + 21*a^2*x^2 + 16*a*x + 6)*sqrt(-a^2*c*x^2 + c))/x^4]
 

Sympy [F]

\[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=- \int \frac {\sqrt {- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx - \int \frac {a x \sqrt {- a^{2} c x^{2} + c}}{a x^{6} - x^{5}}\, dx \] Input:

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

Output:

-Integral(sqrt(-a**2*c*x**2 + c)/(a*x**6 - x**5), x) - Integral(a*x*sqrt(- 
a**2*c*x**2 + c)/(a*x**6 - x**5), x)
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 324 vs. \(2 (106) = 212\).

Time = 0.14 (sec) , antiderivative size = 324, normalized size of antiderivative = 2.49 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {7 \, a^{4} c \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{2} + 96 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 45 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{3} - 128 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} {\left | a \right |} + 21 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{4} + 32 \, a^{3} \sqrt {-c} c^{4} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \] Input:

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

Output:

7/4*a^4*c*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(- 
c) - 1/12*(21*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^7*a^4*c - 45*(sqrt(- 
a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^4*c^2 + 96*(sqrt(-a^2*c)*x - sqrt(-a^ 
2*c*x^2 + c))^4*a^3*sqrt(-c)*c^2*abs(a) - 45*(sqrt(-a^2*c)*x - sqrt(-a^2*c 
*x^2 + c))^3*a^4*c^3 - 128*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^3*s 
qrt(-c)*c^3*abs(a) + 21*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^4*c^4 + 
32*a^3*sqrt(-c)*c^4*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c 
)^4
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=-\int \frac {\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^2}{x^5\,\left (a^2\,x^2-1\right )} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.69 \[ \int \frac {e^{2 \text {arctanh}(a x)} \sqrt {c-a^2 c x^2}}{x^5} \, dx=\frac {\sqrt {c}\, \left (-32 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-21 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-16 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}+21 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}\right )}{24 x^{4}} \] Input:

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

Output:

(sqrt(c)*( - 32*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 21*sqrt( - a**2*x**2 + 
1)*a**2*x**2 - 16*sqrt( - a**2*x**2 + 1)*a*x - 6*sqrt( - a**2*x**2 + 1) + 
21*log(tan(asin(a*x)/2))*a**4*x**4))/(24*x**4)