\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{5/2}}{x^3} \, dx\) [1141]

Optimal result

Integrand size = 27, antiderivative size = 151 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=-\frac {1}{2} a^2 c^2 (1+6 a x) \sqrt {c-a^2 c x^2}-\frac {a c (12+a x) \left (c-a^2 c x^2\right )^{3/2}}{6 x}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 x^2}-3 a^2 c^{5/2} \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )+\frac {1}{2} a^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

-1/2*a^2*c^2*(6*a*x+1)*(-a^2*c*x^2+c)^(1/2)-1/6*a*c*(a*x+12)*(-a^2*c*x^2+c 
)^(3/2)/x-1/2*(-a^2*c*x^2+c)^(5/2)/x^2-3*a^2*c^(5/2)*arctan(a*c^(1/2)*x/(- 
a^2*c*x^2+c)^(1/2))+1/2*a^2*c^(5/2)*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=-\frac {c^2 \sqrt {c-a^2 c x^2} \left (3+12 a x-2 a^2 x^2+6 a^3 x^3+2 a^4 x^4\right )}{6 x^2}+3 a^2 c^{5/2} \arctan \left (\frac {a x \sqrt {c-a^2 c x^2}}{\sqrt {c} \left (-1+a^2 x^2\right )}\right )-\frac {1}{2} a^2 c^{5/2} \log (x)+\frac {1}{2} a^2 c^{5/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^3,x]
 

Output:

-1/6*(c^2*Sqrt[c - a^2*c*x^2]*(3 + 12*a*x - 2*a^2*x^2 + 6*a^3*x^3 + 2*a^4* 
x^4))/x^2 + 3*a^2*c^(5/2)*ArcTan[(a*x*Sqrt[c - a^2*c*x^2])/(Sqrt[c]*(-1 + 
a^2*x^2))] - (a^2*c^(5/2)*Log[x])/2 + (a^2*c^(5/2)*Log[c + Sqrt[c]*Sqrt[c 
- a^2*c*x^2]])/2
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {6701, 540, 25, 27, 536, 535, 27, 538, 224, 216, 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.

\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx\)

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

\(\displaystyle c \left (-\frac {\int -\frac {a c (4-a x) \left (c-a^2 c x^2\right )^{3/2}}{x^2}dx}{2 c}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\int \frac {a c (4-a x) \left (c-a^2 c x^2\right )^{3/2}}{x^2}dx}{2 c}-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{2} a \int \frac {(4-a x) \left (c-a^2 c x^2\right )^{3/2}}{x^2}dx-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 536

\(\displaystyle c \left (\frac {1}{2} a \left (\int \frac {\left (-12 c x a^2-c a\right ) \sqrt {c-a^2 c x^2}}{x}dx-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 535

\(\displaystyle c \left (\frac {1}{2} a \left (\frac {1}{2} c \int -\frac {2 a c (6 a x+1)}{x \sqrt {c-a^2 c x^2}}dx-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}-a c (6 a x+1) \sqrt {c-a^2 c x^2}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \int \frac {6 a x+1}{x \sqrt {c-a^2 c x^2}}dx-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 538

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (6 a \int \frac {1}{\sqrt {c-a^2 c x^2}}dx+\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx+6 a \int \frac {1}{\frac {a^2 c x^2}{c-a^2 c x^2}+1}d\frac {x}{\sqrt {c-a^2 c x^2}}\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 216

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (\int \frac {1}{x \sqrt {c-a^2 c x^2}}dx+\frac {6 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2+\frac {6 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (\frac {6 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a^2 c}\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (\frac {1}{2} a \left (-a c^2 \left (\frac {6 \arctan \left (\frac {a \sqrt {c} x}{\sqrt {c-a^2 c x^2}}\right )}{\sqrt {c}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-a c (6 a x+1) \sqrt {c-a^2 c x^2}-\frac {(a x+12) \left (c-a^2 c x^2\right )^{3/2}}{3 x}\right )-\frac {\left (c-a^2 c x^2\right )^{5/2}}{2 c x^2}\right )\)

Input:

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(5/2))/x^3,x]
 

Output:

c*(-1/2*(c - a^2*c*x^2)^(5/2)/(c*x^2) + (a*(-(a*c*(1 + 6*a*x)*Sqrt[c - a^2 
*c*x^2]) - ((12 + a*x)*(c - a^2*c*x^2)^(3/2))/(3*x) - a*c^2*((6*ArcTan[(a* 
Sqrt[c]*x)/Sqrt[c - a^2*c*x^2]])/Sqrt[c] - ArcTanh[Sqrt[c - a^2*c*x^2]/Sqr 
t[c]]/Sqrt[c])))/2)
 

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 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

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 535

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_), x_Symbol] :> Sim 
p[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^p/(2*p*(2*p + 1))), x] + Simp[a/(2*p 
 + 1)   Int[(c*(2*p + 1) + 2*d*p*x)*((a + b*x^2)^(p - 1)/x), x], x] /; Free 
Q[{a, b, c, d}, x] && GtQ[p, 0] && IntegerQ[2*p]
 

rule 536

Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S 
imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( 
a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer 
Q[2*p]
 

rule 538

Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp 
[c   Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d   Int[1/Sqrt[a + b*x^2], x] 
, x] /; FreeQ[{a, b, c, d}, x]
 

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 = 0.21 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.38

method	result	size
risch	\(\frac {\left (4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1\right ) c^{3}}{2 x^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\left (\frac {a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2 \sqrt {c}}-\frac {3 a^{3} \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{\sqrt {a^{2} c}}+\frac {a^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}{c}-\frac {a^{3} x \sqrt {-a^{2} c \,x^{2}+c}}{c}-\frac {a^{4} x^{2} \sqrt {-a^{2} c \,x^{2}+c}}{3 c}-\frac {2 a^{2} \sqrt {-a^{2} c \,x^{2}+c}}{3 c}\right ) c^{3}\)	\(208\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{2 c \,x^{2}}-\frac {a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{5}+c \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )\right )}{2}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}{c x}-6 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6}+\frac {5 c \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )}{6}\right )\right )-2 a^{2} \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}{5}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{8 a^{2} c}+\frac {3 c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )\)	\(435\)

Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x,method=_RETURNVERBOS 
E)
 

Output:

1/2*(4*a^3*x^3+a^2*x^2-4*a*x-1)/x^2/(-c*(a^2*x^2-1))^(1/2)*c^3+(1/2*a^2/c^ 
(1/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-3*a^3/(a^2*c)^(1/2)*arcta 
n((a^2*c)^(1/2)*x/(-a^2*c*x^2+c)^(1/2))+a^2/c*(-c*(a^2*x^2-1))^(1/2)-a^3*x 
/c*(-a^2*c*x^2+c)^(1/2)-1/3*a^4*x^2/c*(-a^2*c*x^2+c)^(1/2)-2/3*a^2/c*(-a^2 
*c*x^2+c)^(1/2))*c^3
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.10 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=\left [\frac {36 \, a^{2} c^{\frac {5}{2}} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} a \sqrt {c} x}{a^{2} c x^{2} - c}\right ) + 3 \, a^{2} c^{\frac {5}{2}} x^{2} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, {\left (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{12 \, x^{2}}, -\frac {3 \, a^{2} \sqrt {-c} c^{2} x^{2} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - 9 \, a^{2} \sqrt {-c} c^{2} x^{2} \log \left (2 \, a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} a \sqrt {-c} x - c\right ) + {\left (2 \, a^{4} c^{2} x^{4} + 6 \, a^{3} c^{2} x^{3} - 2 \, a^{2} c^{2} x^{2} + 12 \, a c^{2} x + 3 \, c^{2}\right )} \sqrt {-a^{2} c x^{2} + c}}{6 \, x^{2}}\right ] \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x, algorithm="fr 
icas")
 

Output:

[1/12*(36*a^2*c^(5/2)*x^2*arctan(sqrt(-a^2*c*x^2 + c)*a*sqrt(c)*x/(a^2*c*x 
^2 - c)) + 3*a^2*c^(5/2)*x^2*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt 
(c) - 2*c)/x^2) - 2*(2*a^4*c^2*x^4 + 6*a^3*c^2*x^3 - 2*a^2*c^2*x^2 + 12*a* 
c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2, -1/6*(3*a^2*sqrt(-c)*c^2*x^2*arc 
tan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) - 9*a^2*sqrt(-c)*c^2*x^2*log(2*a^2*c* 
x^2 - 2*sqrt(-a^2*c*x^2 + c)*a*sqrt(-c)*x - c) + (2*a^4*c^2*x^4 + 6*a^3*c^ 
2*x^3 - 2*a^2*c^2*x^2 + 12*a*c^2*x + 3*c^2)*sqrt(-a^2*c*x^2 + c))/x^2]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.80 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.70 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=- a^{4} c^{2} \left (\begin {cases} \frac {x^{2} \sqrt {- a^{2} c x^{2} + c}}{3} - \frac {\sqrt {- a^{2} c x^{2} + c}}{3 a^{2}} & \text {for}\: a^{2} c \neq 0 \\\frac {\sqrt {c} x^{2}}{2} & \text {otherwise} \end {cases}\right ) - 2 a^{3} c^{2} \left (\begin {cases} \frac {c \left (\begin {cases} \frac {\log {\left (- 2 a^{2} c x + 2 \sqrt {- a^{2} c} \sqrt {- a^{2} c x^{2} + c} \right )}}{\sqrt {- a^{2} c}} & \text {for}\: c \neq 0 \\\frac {x \log {\left (x \right )}}{\sqrt {- a^{2} c x^{2}}} & \text {otherwise} \end {cases}\right )}{2} + \frac {x \sqrt {- a^{2} c x^{2} + c}}{2} & \text {for}\: a^{2} c \neq 0 \\\sqrt {c} x & \text {otherwise} \end {cases}\right ) + 2 a c^{2} \left (\begin {cases} - \frac {i a^{2} \sqrt {c} x}{\sqrt {a^{2} x^{2} - 1}} + i a \sqrt {c} \operatorname {acosh}{\left (a x \right )} + \frac {i \sqrt {c}}{x \sqrt {a^{2} x^{2} - 1}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {a^{2} \sqrt {c} x}{\sqrt {- a^{2} x^{2} + 1}} - a \sqrt {c} \operatorname {asin}{\left (a x \right )} - \frac {\sqrt {c}}{x \sqrt {- a^{2} x^{2} + 1}} & \text {otherwise} \end {cases}\right ) + c^{2} \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((a*x+1)**2/(-a**2*x**2+1)*(-a**2*c*x**2+c)**(5/2)/x**3,x)
 

Output:

-a**4*c**2*Piecewise((x**2*sqrt(-a**2*c*x**2 + c)/3 - sqrt(-a**2*c*x**2 + 
c)/(3*a**2), Ne(a**2*c, 0)), (sqrt(c)*x**2/2, True)) - 2*a**3*c**2*Piecewi 
se((c*Piecewise((log(-2*a**2*c*x + 2*sqrt(-a**2*c)*sqrt(-a**2*c*x**2 + c)) 
/sqrt(-a**2*c), Ne(c, 0)), (x*log(x)/sqrt(-a**2*c*x**2), True))/2 + x*sqrt 
(-a**2*c*x**2 + c)/2, Ne(a**2*c, 0)), (sqrt(c)*x, True)) + 2*a*c**2*Piecew 
ise((-I*a**2*sqrt(c)*x/sqrt(a**2*x**2 - 1) + I*a*sqrt(c)*acosh(a*x) + I*sq 
rt(c)/(x*sqrt(a**2*x**2 - 1)), Abs(a**2*x**2) > 1), (a**2*sqrt(c)*x/sqrt(- 
a**2*x**2 + 1) - a*sqrt(c)*asin(a*x) - sqrt(c)/(x*sqrt(-a**2*x**2 + 1)), T 
rue)) + c**2*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqr 
t(-1 + 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Ab 
s(a**2*x**2) > 1), (-I*a**2*sqrt(c)*asin(1/(a*x))/2 - I*a*sqrt(c)*sqrt(1 - 
 1/(a**2*x**2))/(2*x), True))
 

Maxima [F]

\[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=\int { -\frac {{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} {\left (a x + 1\right )}^{2}}{{\left (a^{2} x^{2} - 1\right )} x^{3}} \,d x } \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x, algorithm="ma 
xima")
 

Output:

-integrate((-a^2*c*x^2 + c)^(5/2)*(a*x + 1)^2/((a^2*x^2 - 1)*x^3), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (125) = 250\).

Time = 0.15 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.00 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=-\frac {a^{2} c^{3} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{\sqrt {-c}} - \frac {3 \, a^{3} \sqrt {-c} c^{2} \log \left ({\left | -\sqrt {-a^{2} c} x + \sqrt {-a^{2} c x^{2} + c} \right |}\right )}{{\left | a \right |}} + \frac {1}{3} \, \sqrt {-a^{2} c x^{2} + c} {\left (a^{2} c^{2} - {\left (a^{4} c^{2} x + 3 \, a^{3} c^{2}\right )} x\right )} - \frac {{\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{2} c^{3} {\left | a \right |} - 4 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{3} + {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{2} c^{4} {\left | a \right |} + 4 \, a^{3} \sqrt {-c} c^{4}}{{\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{2} {\left | a \right |}} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x, algorithm="gi 
ac")
 

Output:

-a^2*c^3*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqrt(-c 
) - 3*a^3*sqrt(-c)*c^2*log(abs(-sqrt(-a^2*c)*x + sqrt(-a^2*c*x^2 + c)))/ab 
s(a) + 1/3*sqrt(-a^2*c*x^2 + c)*(a^2*c^2 - (a^4*c^2*x + 3*a^3*c^2)*x) - (( 
sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^2*c^3*abs(a) - 4*(sqrt(-a^2*c)* 
x - sqrt(-a^2*c*x^2 + c))^2*a^3*sqrt(-c)*c^3 + (sqrt(-a^2*c)*x - sqrt(-a^2 
*c*x^2 + c))*a^2*c^4*abs(a) + 4*a^3*sqrt(-c)*c^4)/(((sqrt(-a^2*c)*x - sqrt 
(-a^2*c*x^2 + c))^2 - c)^2*abs(a))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{5/2}\,{\left (a\,x+1\right )}^2}{x^3\,\left (a^2\,x^2-1\right )} \,d x \] Input:

int(-((c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(x^3*(a^2*x^2 - 1)),x)
 

Output:

-int(((c - a^2*c*x^2)^(5/2)*(a*x + 1)^2)/(x^3*(a^2*x^2 - 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{5/2}}{x^3} \, dx=\frac {\sqrt {c}\, c^{2} \left (-72 \mathit {asin} \left (a x \right ) a^{2} x^{2}-8 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-24 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-48 \sqrt {-a^{2} x^{2}+1}\, a x -12 \sqrt {-a^{2} x^{2}+1}-12 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{2} x^{2}+a^{2} x^{2}\right )}{24 x^{2}} \] Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(5/2)/x^3,x)
 

Output:

(sqrt(c)*c**2*( - 72*asin(a*x)*a**2*x**2 - 8*sqrt( - a**2*x**2 + 1)*a**4*x 
**4 - 24*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 8*sqrt( - a**2*x**2 + 1)*a**2* 
x**2 - 48*sqrt( - a**2*x**2 + 1)*a*x - 12*sqrt( - a**2*x**2 + 1) - 12*log( 
tan(asin(a*x)/2))*a**2*x**2 + a**2*x**2))/(24*x**2)