\(\int \frac {e^{2 \text {arctanh}(a x)} (c-a^2 c x^2)^{3/2}}{x^8} \, dx\) [1134]

Optimal result

Integrand size = 27, antiderivative size = 183 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=-\frac {a c \sqrt {c-a^2 c x^2}}{3 x^6}+\frac {a^3 c \sqrt {c-a^2 c x^2}}{12 x^4}+\frac {a^5 c \sqrt {c-a^2 c x^2}}{8 x^2}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 x^7}-\frac {11 a^2 \left (c-a^2 c x^2\right )^{3/2}}{35 x^5}-\frac {22 a^4 \left (c-a^2 c x^2\right )^{3/2}}{105 x^3}+\frac {1}{8} a^7 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right ) \] Output:

-1/3*a*c*(-a^2*c*x^2+c)^(1/2)/x^6+1/12*a^3*c*(-a^2*c*x^2+c)^(1/2)/x^4+1/8* 
a^5*c*(-a^2*c*x^2+c)^(1/2)/x^2-1/7*(-a^2*c*x^2+c)^(3/2)/x^7-11/35*a^2*(-a^ 
2*c*x^2+c)^(3/2)/x^5-22/105*a^4*(-a^2*c*x^2+c)^(3/2)/x^3+1/8*a^7*c^(3/2)*a 
rctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.66 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (-120-280 a x-144 a^2 x^2+70 a^3 x^3+88 a^4 x^4+105 a^5 x^5+176 a^6 x^6\right )}{840 x^7}-\frac {1}{8} a^7 c^{3/2} \log (x)+\frac {1}{8} a^7 c^{3/2} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right ) \] Input:

Integrate[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^8,x]
 

Output:

(c*Sqrt[c - a^2*c*x^2]*(-120 - 280*a*x - 144*a^2*x^2 + 70*a^3*x^3 + 88*a^4 
*x^4 + 105*a^5*x^5 + 176*a^6*x^6))/(840*x^7) - (a^7*c^(3/2)*Log[x])/8 + (a 
^7*c^(3/2)*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/8
 

Rubi [A] (verified)

Time = 0.85 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.13, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.630, Rules used = {6701, 540, 25, 27, 539, 27, 539, 25, 27, 539, 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.

\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx\)

\(\Big \downarrow \) 6701

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

\(\Big \downarrow \) 540

\(\displaystyle c \left (-\frac {\int -\frac {a c (11 a x+14) \sqrt {c-a^2 c x^2}}{x^7}dx}{7 c}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {\int \frac {a c (11 a x+14) \sqrt {c-a^2 c x^2}}{x^7}dx}{7 c}-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{7} a \int \frac {(11 a x+14) \sqrt {c-a^2 c x^2}}{x^7}dx-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 539

\(\displaystyle c \left (\frac {1}{7} a \left (-\frac {\int -\frac {6 a c (7 a x+11) \sqrt {c-a^2 c x^2}}{x^6}dx}{6 c}-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{7} a \left (a \int \frac {(7 a x+11) \sqrt {c-a^2 c x^2}}{x^6}dx-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 539

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (-\frac {\int -\frac {a c (22 a x+35) \sqrt {c-a^2 c x^2}}{x^5}dx}{5 c}-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {\int \frac {a c (22 a x+35) \sqrt {c-a^2 c x^2}}{x^5}dx}{5 c}-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \int \frac {(22 a x+35) \sqrt {c-a^2 c x^2}}{x^5}dx-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 539

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (-\frac {\int -\frac {a c (35 a x+88) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {\int \frac {a c (35 a x+88) \sqrt {c-a^2 c x^2}}{x^4}dx}{4 c}-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \int \frac {(35 a x+88) \sqrt {c-a^2 c x^2}}{x^4}dx-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 534

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (35 a \int \frac {\sqrt {c-a^2 c x^2}}{x^3}dx-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 243

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {35}{2} a \int \frac {\sqrt {c-a^2 c x^2}}{x^4}dx^2-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 51

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {35}{2} a \left (-\frac {1}{2} a^2 c \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 73

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {35}{2} a \left (\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle c \left (\frac {1}{7} a \left (a \left (\frac {1}{5} a \left (\frac {1}{4} a \left (\frac {35}{2} a \left (a^2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )-\frac {\sqrt {c-a^2 c x^2}}{x^2}\right )-\frac {88 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^3}\right )-\frac {35 \left (c-a^2 c x^2\right )^{3/2}}{4 c x^4}\right )-\frac {11 \left (c-a^2 c x^2\right )^{3/2}}{5 c x^5}\right )-\frac {7 \left (c-a^2 c x^2\right )^{3/2}}{3 c x^6}\right )-\frac {\left (c-a^2 c x^2\right )^{3/2}}{7 c x^7}\right )\)

Input:

Int[(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(3/2))/x^8,x]
 

Output:

c*(-1/7*(c - a^2*c*x^2)^(3/2)/(c*x^7) + (a*((-7*(c - a^2*c*x^2)^(3/2))/(3* 
c*x^6) + a*((-11*(c - a^2*c*x^2)^(3/2))/(5*c*x^5) + (a*((-35*(c - a^2*c*x^ 
2)^(3/2))/(4*c*x^4) + (a*((-88*(c - a^2*c*x^2)^(3/2))/(3*c*x^3) + (35*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))/5)))/7)
 

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 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 = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.66

method	result	size
risch	\(-\frac {\left (176 a^{8} x^{8}+105 a^{7} x^{7}-88 x^{6} a^{6}-35 a^{5} x^{5}-232 a^{4} x^{4}-350 a^{3} x^{3}+24 a^{2} x^{2}+280 a x +120\right ) c^{2}}{840 x^{7} \sqrt {-c \left (a^{2} x^{2}-1\right )}}+\frac {a^{7} c^{\frac {3}{2}} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{8}\)	\(121\)
default	\(-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{7 c \,x^{7}}-\frac {16 a^{2} \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{35 c \,x^{5}}+2 a \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{6 c \,x^{6}}+\frac {a^{2} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}-\frac {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}\right )}{6}\right )+2 a^{3} \left (-\frac {\left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}{4 c \,x^{4}}-\frac {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}\right )+2 a^{4} \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^{5} \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 )+2 a^{6} \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^{7} \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^{7} \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 )\)	\(860\)

Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x,method=_RETURNVERBOS 
E)
 

Output:

-1/840*(176*a^8*x^8+105*a^7*x^7-88*a^6*x^6-35*a^5*x^5-232*a^4*x^4-350*a^3* 
x^3+24*a^2*x^2+280*a*x+120)/x^7/(-c*(a^2*x^2-1))^(1/2)*c^2+1/8*a^7*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 = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.28 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=\left [\frac {105 \, a^{7} c^{\frac {3}{2}} x^{7} \log \left (-\frac {a^{2} c x^{2} - 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) + 2 \, {\left (176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{1680 \, x^{7}}, -\frac {105 \, a^{7} \sqrt {-c} c x^{7} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - {\left (176 \, a^{6} c x^{6} + 105 \, a^{5} c x^{5} + 88 \, a^{4} c x^{4} + 70 \, a^{3} c x^{3} - 144 \, a^{2} c x^{2} - 280 \, a c x - 120 \, c\right )} \sqrt {-a^{2} c x^{2} + c}}{840 \, x^{7}}\right ] \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="fr 
icas")
 

Output:

[1/1680*(105*a^7*c^(3/2)*x^7*log(-(a^2*c*x^2 - 2*sqrt(-a^2*c*x^2 + c)*sqrt 
(c) - 2*c)/x^2) + 2*(176*a^6*c*x^6 + 105*a^5*c*x^5 + 88*a^4*c*x^4 + 70*a^3 
*c*x^3 - 144*a^2*c*x^2 - 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7, -1/ 
840*(105*a^7*sqrt(-c)*c*x^7*arctan(sqrt(-a^2*c*x^2 + c)*sqrt(-c)/c) - (176 
*a^6*c*x^6 + 105*a^5*c*x^5 + 88*a^4*c*x^4 + 70*a^3*c*x^3 - 144*a^2*c*x^2 - 
 280*a*c*x - 120*c)*sqrt(-a^2*c*x^2 + c))/x^7]
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.17 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.61 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=a^{2} c \left (\begin {cases} \frac {2 i a^{4} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x} + \frac {i a^{2} \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{15 x^{3}} - \frac {i \sqrt {c} \sqrt {a^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{a^{2} x^{2}}\right | > 1 \\\frac {2 a^{4} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x} + \frac {a^{2} \sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{15 x^{3}} - \frac {\sqrt {c} \sqrt {- a^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right ) + 2 a c \left (\begin {cases} \frac {a^{6} \sqrt {c} \operatorname {acosh}{\left (\frac {1}{a x} \right )}}{16} - \frac {a^{5} \sqrt {c}}{16 x \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {a^{3} \sqrt {c}}{48 x^{3} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} + \frac {5 a \sqrt {c}}{24 x^{5} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} - \frac {\sqrt {c}}{6 a x^{7} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\- \frac {i a^{6} \sqrt {c} \operatorname {asin}{\left (\frac {1}{a x} \right )}}{16} + \frac {i a^{5} \sqrt {c}}{16 x \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {i a^{3} \sqrt {c}}{48 x^{3} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} - \frac {5 i a \sqrt {c}}{24 x^{5} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} + \frac {i \sqrt {c}}{6 a x^{7} \sqrt {1 - \frac {1}{a^{2} x^{2}}}} & \text {otherwise} \end {cases}\right ) + c \left (\begin {cases} \frac {8 a^{7} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{105} + \frac {4 a^{5} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{105 x^{2}} + \frac {a^{3} \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {a \sqrt {c} \sqrt {-1 + \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \text {for}\: \frac {1}{\left |{a^{2} x^{2}}\right |} > 1 \\\frac {8 i a^{7} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{105} + \frac {4 i a^{5} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{105 x^{2}} + \frac {i a^{3} \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{35 x^{4}} - \frac {i a \sqrt {c} \sqrt {1 - \frac {1}{a^{2} x^{2}}}}{7 x^{6}} & \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**8,x)
 

Output:

a**2*c*Piecewise((2*I*a**4*sqrt(c)*sqrt(a**2*x**2 - 1)/(15*x) + I*a**2*sqr 
t(c)*sqrt(a**2*x**2 - 1)/(15*x**3) - I*sqrt(c)*sqrt(a**2*x**2 - 1)/(5*x**5 
), Abs(a**2*x**2) > 1), (2*a**4*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x) + a**2 
*sqrt(c)*sqrt(-a**2*x**2 + 1)/(15*x**3) - sqrt(c)*sqrt(-a**2*x**2 + 1)/(5* 
x**5), True)) + 2*a*c*Piecewise((a**6*sqrt(c)*acosh(1/(a*x))/16 - a**5*sqr 
t(c)/(16*x*sqrt(-1 + 1/(a**2*x**2))) + a**3*sqrt(c)/(48*x**3*sqrt(-1 + 1/( 
a**2*x**2))) + 5*a*sqrt(c)/(24*x**5*sqrt(-1 + 1/(a**2*x**2))) - sqrt(c)/(6 
*a*x**7*sqrt(-1 + 1/(a**2*x**2))), 1/Abs(a**2*x**2) > 1), (-I*a**6*sqrt(c) 
*asin(1/(a*x))/16 + I*a**5*sqrt(c)/(16*x*sqrt(1 - 1/(a**2*x**2))) - I*a**3 
*sqrt(c)/(48*x**3*sqrt(1 - 1/(a**2*x**2))) - 5*I*a*sqrt(c)/(24*x**5*sqrt(1 
 - 1/(a**2*x**2))) + I*sqrt(c)/(6*a*x**7*sqrt(1 - 1/(a**2*x**2))), True)) 
+ c*Piecewise((8*a**7*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/105 + 4*a**5*sqrt(c 
)*sqrt(-1 + 1/(a**2*x**2))/(105*x**2) + a**3*sqrt(c)*sqrt(-1 + 1/(a**2*x** 
2))/(35*x**4) - a*sqrt(c)*sqrt(-1 + 1/(a**2*x**2))/(7*x**6), 1/Abs(a**2*x* 
*2) > 1), (8*I*a**7*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/105 + 4*I*a**5*sqrt(c) 
*sqrt(1 - 1/(a**2*x**2))/(105*x**2) + I*a**3*sqrt(c)*sqrt(1 - 1/(a**2*x**2 
))/(35*x**4) - I*a*sqrt(c)*sqrt(1 - 1/(a**2*x**2))/(7*x**6), True))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.57 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=-\frac {3 \, a^{8} c^{\frac {5}{2}} \log \left (\frac {\sqrt {-a^{2} c x^{2} + c} - \sqrt {c}}{\sqrt {-a^{2} c x^{2} + c} + \sqrt {c}}\right ) - \frac {2 \, {\left (3 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{8} c^{3} - 8 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} a^{8} c^{4} - 3 \, \sqrt {-a^{2} c x^{2} + c} a^{8} c^{5}\right )}}{{\left (a^{2} c x^{2} - c\right )}^{3} + 3 \, {\left (a^{2} c x^{2} - c\right )}^{2} c + 3 \, {\left (a^{2} c x^{2} - c\right )} c^{2} + c^{3}}}{48 \, a c} + \frac {{\left (2 \, a^{4} c^{\frac {3}{2}} x^{4} + a^{2} c^{\frac {3}{2}} x^{2} - 3 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1} a^{2}}{15 \, x^{5}} + \frac {{\left (8 \, a^{6} c^{\frac {3}{2}} x^{6} + 4 \, a^{4} c^{\frac {3}{2}} x^{4} + 3 \, a^{2} c^{\frac {3}{2}} x^{2} - 15 \, c^{\frac {3}{2}}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{105 \, x^{7}} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="ma 
xima")
 

Output:

-1/48*(3*a^8*c^(5/2)*log((sqrt(-a^2*c*x^2 + c) - sqrt(c))/(sqrt(-a^2*c*x^2 
 + c) + sqrt(c))) - 2*(3*(-a^2*c*x^2 + c)^(5/2)*a^8*c^3 - 8*(-a^2*c*x^2 + 
c)^(3/2)*a^8*c^4 - 3*sqrt(-a^2*c*x^2 + c)*a^8*c^5)/((a^2*c*x^2 - c)^3 + 3* 
(a^2*c*x^2 - c)^2*c + 3*(a^2*c*x^2 - c)*c^2 + c^3))/(a*c) + 1/15*(2*a^4*c^ 
(3/2)*x^4 + a^2*c^(3/2)*x^2 - 3*c^(3/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)*a^2/ 
x^5 + 1/105*(8*a^6*c^(3/2)*x^6 + 4*a^4*c^(3/2)*x^4 + 3*a^2*c^(3/2)*x^2 - 1 
5*c^(3/2))*sqrt(a*x + 1)*sqrt(-a*x + 1)/x^7
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (151) = 302\).

Time = 0.16 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.89 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=-\frac {a^{7} c^{2} \arctan \left (-\frac {\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}}{\sqrt {-c}}\right )}{4 \, \sqrt {-c}} + \frac {105 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{13} a^{7} c^{2} - 700 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{11} a^{7} c^{3} + 1680 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{10} a^{6} \sqrt {-c} c^{3} {\left | a \right |} - 3395 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{9} a^{7} c^{4} - 7280 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{8} a^{6} \sqrt {-c} c^{4} {\left | a \right |} - 1120 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{6} a^{6} \sqrt {-c} c^{5} {\left | a \right |} + 3395 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{5} a^{7} c^{6} - 2016 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{4} a^{6} \sqrt {-c} c^{6} {\left | a \right |} + 700 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{3} a^{7} c^{7} + 1232 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} a^{6} \sqrt {-c} c^{7} {\left | a \right |} - 105 \, {\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )} a^{7} c^{8} - 176 \, a^{6} \sqrt {-c} c^{8} {\left | a \right |}}{420 \, {\left ({\left (\sqrt {-a^{2} c} x - \sqrt {-a^{2} c x^{2} + c}\right )}^{2} - c\right )}^{7}} \] Input:

integrate((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x, algorithm="gi 
ac")
 

Output:

-1/4*a^7*c^2*arctan(-(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))/sqrt(-c))/sqr 
t(-c) + 1/420*(105*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^13*a^7*c^2 - 70 
0*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^11*a^7*c^3 + 1680*(sqrt(-a^2*c)* 
x - sqrt(-a^2*c*x^2 + c))^10*a^6*sqrt(-c)*c^3*abs(a) - 3395*(sqrt(-a^2*c)* 
x - sqrt(-a^2*c*x^2 + c))^9*a^7*c^4 - 7280*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x 
^2 + c))^8*a^6*sqrt(-c)*c^4*abs(a) - 1120*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^ 
2 + c))^6*a^6*sqrt(-c)*c^5*abs(a) + 3395*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 
 + c))^5*a^7*c^6 - 2016*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^4*a^6*sqrt 
(-c)*c^6*abs(a) + 700*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^3*a^7*c^7 + 
1232*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2*a^6*sqrt(-c)*c^7*abs(a) - 1 
05*(sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))*a^7*c^8 - 176*a^6*sqrt(-c)*c^8* 
abs(a))/((sqrt(-a^2*c)*x - sqrt(-a^2*c*x^2 + c))^2 - c)^7
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=-\int \frac {{\left (c-a^2\,c\,x^2\right )}^{3/2}\,{\left (a\,x+1\right )}^2}{x^8\,\left (a^2\,x^2-1\right )} \,d x \] Input:

int(-((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^8*(a^2*x^2 - 1)),x)
 

Output:

-int(((c - a^2*c*x^2)^(3/2)*(a*x + 1)^2)/(x^8*(a^2*x^2 - 1)), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.81 \[ \int \frac {e^{2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{3/2}}{x^8} \, dx=\frac {\sqrt {c}\, c \left (176 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}+105 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+88 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+70 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-144 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-280 \sqrt {-a^{2} x^{2}+1}\, a x -120 \sqrt {-a^{2} x^{2}+1}-105 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) a^{7} x^{7}\right )}{840 x^{7}} \] Input:

int((a*x+1)^2/(-a^2*x^2+1)*(-a^2*c*x^2+c)^(3/2)/x^8,x)
 

Output:

(sqrt(c)*c*(176*sqrt( - a**2*x**2 + 1)*a**6*x**6 + 105*sqrt( - a**2*x**2 + 
 1)*a**5*x**5 + 88*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 70*sqrt( - a**2*x**2 
 + 1)*a**3*x**3 - 144*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 280*sqrt( - a**2* 
x**2 + 1)*a*x - 120*sqrt( - a**2*x**2 + 1) - 105*log(tan(asin(a*x)/2))*a** 
7*x**7))/(840*x**7)