\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{3/2}}{x^5} \, dx\) [1131]

Optimal result

Integrand size = 27, antiderivative size = 106 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=-\frac {5 a^2 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 x^4}-\frac {2 a \left (c-a^2 c x^2\right )^{3/2}}{3 x^3}+\frac {5}{8} a^4 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

-5/8*a^2*c*(-a^2*c*x^2+c)^(1/2)/x^2-1/4*(-a^2*c*x^2+c)^(3/2)/x^4-2/3*a*(-a 
^2*c*x^2+c)^(3/2)/x^3+5/8*a^4*c^(3/2)*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2) 
)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (-6-16 a x-9 a^2 x^2+16 a^3 x^3\right )}{24 x^4}-\frac {5}{8} a^4 c^{3/2} \log (x)+\frac {5}{8} a^4 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

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

Output:

(c*Sqrt[c - a^2*c*x^2]*(-6 - 16*a*x - 9*a^2*x^2 + 16*a^3*x^3))/(24*x^4) - 
(5*a^4*c^(3/2)*Log[x])/8 + (5*a^4*c^(3/2)*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x 
^2]])/8
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6701, 540, 25, 27, 534, 243, 51, 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.

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

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

\(\displaystyle c \left (-\frac {\int -\frac {a c (5 a x+8) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\int \frac {a c (5 a x+8) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{4} a \int \frac {(5 a x+8) \sqrt {c-a^2 c x^2}}{x^4}dx-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 534

\(\displaystyle c \left (\frac {1}{4} a \left (5 a \int \frac {\sqrt {c-a^2 c x^2}}{x^3}dx-\frac {8 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {5}{2} a \int \frac {\sqrt {c-a^2 c x^2}}{x^4}dx^2-\frac {8 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {5}{2} a \left (-\frac {1}{2} a^2 c \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {8 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {5}{2} a \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {8 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (\frac {1}{4} a \left (\frac {5}{2} a \left (a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {8 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )\)

Input:

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

Output:

c*(-1/4*(c - a^2*c*x^2)^(3/2)/(c*x^4) + (a*((-8*(c - a^2*c*x^2)^(3/2))/(3* 
c*x^3) + (5*a*(-(Sqrt[c - a^2*c*x^2]/x^2) + a^2*Sqrt[c]*ArcTanh[Sqrt[c - a 
^2*c*x^2]/Sqrt[c]]))/2))/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 51

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

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 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 = 97, normalized size of antiderivative = 0.92

method	result	size
risch	\(-\frac {\left (16 a^{5} x^{5}-9 a^{4} x^{4}-32 a^{3} x^{3}+3 a^{2} x^{2}+16 a x +6\right ) c^{2}}{24 x^{4} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {5 a^{4} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\)	\(97\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}+\frac {7 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{2 c \,x^{2}}-\frac {3 a^{2} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )}{2}\right )}{4}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{3 c \,x^{3}}-\frac {2 a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )}{3}\right )+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{c x}-4 a^{2} \left (\frac {x \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{4}+\frac {3 c \left (\frac {x \sqrt {-a^{2} c \,x^{2}+c}}{2}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-a^{2} c \,x^{2}+c}}\right )}{2 \sqrt {a^{2} c}}\right )}{4}\right )\right )+2 a^{4} \left (\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{3}+c \left (\sqrt {-a^{2} c \,x^{2}+c}-\sqrt {c}\, \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )\right )\right )-2 a^{4} \left (\frac {\left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}{3}-a c \left (-\frac {\left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}{4 a^{2} c}+\frac {c \arctan \left (\frac {\sqrt {a^{2} c}\, x}{\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{2 \sqrt {a^{2} c}}\right )\right )\)	\(563\)

Input:

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

Output:

-1/24*(16*a^5*x^5-9*a^4*x^4-32*a^3*x^3+3*a^2*x^2+16*a*x+6)/x^4/(-c*(a^2*x^ 
2-1))^(1/2)*c^2+5/8*a^4*c^(3/2)*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 = 181, normalized size of antiderivative = 1.71 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=\left [\frac {15 \, a^{4} c^{\frac {3}{2}} x^{4} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{48 \, x^{4}}, -\frac {15 \, a^{4} \sqrt {-c} c x^{4} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - {\left (16 \, a^{3} c x^{3} - 9 \, a^{2} c x^{2} - 16 \, a c x - 6 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{24 \, x^{4}}\right ] \] Input:

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

Output:

[1/48*(15*a^4*c^(3/2)*x^4*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) 
 - 2*c)/x^2) + 2*(16*a^3*c*x^3 - 9*a^2*c*x^2 - 16*a*c*x - 6*c)*sqrt(-a^2*c 
*x^2 + c))/x^4, -1/24*(15*a^4*sqrt(-c)*c*x^4*arctan(sqrt(-a^2*c*x^2 + c)*s 
qrt(-c)/c) - (16*a^3*c*x^3 - 9*a^2*c*x^2 - 16*a*c*x - 6*c)*sqrt(-a^2*c*x^2 
 + c))/x^4]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 5.23 (sec) , antiderivative size = 447, normalized size of antiderivative = 4.22 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=a^{2} c \left (\begin {cases} \frac {a^{2} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{2} + \frac {a \sqrt {c}}{2 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{2 a x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{2} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{2} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{3 x^{2}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {a^{4} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{8} - \frac {a^{3} \sqrt {c}}{8 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {3 a \sqrt {c}}{8 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{4 a x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{4} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{8} + \frac {i a^{3} \sqrt {c}}{8 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {3 i a \sqrt {c}}{8 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{4 a x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) \] Input:

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

Output:

a**2*c*Piecewise((a**2*sqrt(c)*acosh(1/(a*x))/2 + a*sqrt(c)/(2*x*sqrt(-1 + 
 1/(a**2*x**2))) - sqrt(c)/(2*a*x**3*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(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)) + 2*a*c*Piecewise((a**3*sqrt(c)*sqrt(-1 + 1/(a**2* 
x**2))/3 - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(3*x**2), 1/Abs(a**2*x**2) > 
 1), (I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/3 - I*a*sqrt(c)*sqrt(1 - 1/(a 
**2*x**2))/(3*x**2), True)) + c*Piecewise((a**4*sqrt(c)*acosh(1/(a*x))/8 - 
 a**3*sqrt(c)/(8*x*sqrt(-1 + 1/(a**2*x**2))) + 3*a*sqrt(c)/(8*x**3*sqrt(-1 
 + 1/(a**2*x**2))) - sqrt(c)/(4*a*x**5*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a* 
*2*x**2) > 1), (-I*a**4*sqrt(c)*asin(1/(a*x))/8 + I*a**3*sqrt(c)/(8*x*sqrt 
(1 - 1/(a**2*x**2))) - 3*I*a*sqrt(c)/(8*x**3*sqrt(1 - 1/(a**2*x**2))) + I* 
sqrt(c)/(4*a*x**5*sqrt(1 - 1/(a**2*x**2))), True))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

-integrate((-a^2*c*x^2 + c)^(3/2)*(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. 371 vs. \(2 (86) = 172\).

Time = 0.16 (sec) , antiderivative size = 371, normalized size of antiderivative = 3.50 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=-\frac {5 \, a^{4} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} - \frac {9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{7} a^{4} c^{2} + 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{3} \sqrt {-c} c^{2} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{4} c^{3} - 48 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{3} \sqrt {-c} c^{3} {\left | a \right |} - 33 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{4} c^{4} + 16 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{3} \sqrt {-c} c^{4} {\left | a \right |} + 9 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{4} c^{5} - 16 \, a^{3} \sqrt {-c} c^{5} {\left | a \right |}}{12 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{4}} \] Input:

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

Output:

-5/4*a^4*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqr 
t(-c) - 1/12*(9*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^7*a^4*c^2 + 48*(sq 
rt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^6*a^3*sqrt(-c)*c^2*abs(a) - 33*(sqrt( 
-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^5*a^4*c^3 - 48*(sqrt(-a^2*c)*x - sqrt(-a 
^2*c*x^2 + c))^4*a^3*sqrt(-c)*c^3*abs(a) - 33*(sqrt(-a^2*c)*x - sqrt(-a^2* 
c*x^2 + c))^3*a^4*c^4 + 16*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^3*s 
qrt(-c)*c^4*abs(a) + 9*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^4*c^5 - 1 
6*a^3*sqrt(-c)*c^5*abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c) 
^4
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^5} \, dx=\frac {\sqrt {c}\, c \left (16 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-9 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-16 \sqrt {-a^{2} x^{2}+1}\, a x -6 \sqrt {-a^{2} x^{2}+1}-15 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{4} x^{4}\right )}{24 x^{4}} \] Input:

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

Output:

(sqrt(c)*c*(16*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 9*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) - 15 
*log(tan(asin(a*x)/2))*a**4*x**4))/(24*x**4)