\(\int \frac {(a+c x^2)^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 28, antiderivative size = 137 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=-\frac {\left (c+a e^2\right )^3 x}{e^6 \sqrt {-1+e x} \sqrt {1+e x}}+\frac {c^2 \left (7 c+12 a e^2\right ) x \sqrt {-1+e x} \sqrt {1+e x}}{8 e^6}+\frac {c^3 x^3 \sqrt {-1+e x} \sqrt {1+e x}}{4 e^4}+\frac {3 c \left (5 c^2+12 a c e^2+8 a^2 e^4\right ) \text {arccosh}(e x)}{8 e^7} \] Output:

-(a*e^2+c)^3*x/e^6/(e*x-1)^(1/2)/(e*x+1)^(1/2)+1/8*c^2*(12*a*e^2+7*c)*x*(e 
*x-1)^(1/2)*(e*x+1)^(1/2)/e^6+1/4*c^3*x^3*(e*x-1)^(1/2)*(e*x+1)^(1/2)/e^4+ 
3/8*c*(8*a^2*e^4+12*a*c*e^2+5*c^2)*arccosh(e*x)/e^7
 

Mathematica [A] (warning: unable to verify)

Time = 0.50 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {\frac {e x \left (-24 a^2 c e^4-8 a^3 e^6+12 a c^2 e^2 \left (-3+e^2 x^2\right )+c^3 \left (-15+5 e^2 x^2+2 e^4 x^4\right )\right )}{\sqrt {-1+e x} \sqrt {1+e x}}+6 c \left (5 c^2+12 a c e^2+8 a^2 e^4\right ) \text {arctanh}\left (\sqrt {\frac {-1+e x}{1+e x}}\right )}{8 e^7} \] Input:

Integrate[(a + c*x^2)^3/((-1 + e*x)^(3/2)*(1 + e*x)^(3/2)),x]
 

Output:

((e*x*(-24*a^2*c*e^4 - 8*a^3*e^6 + 12*a*c^2*e^2*(-3 + e^2*x^2) + c^3*(-15 
+ 5*e^2*x^2 + 2*e^4*x^4)))/(Sqrt[-1 + e*x]*Sqrt[1 + e*x]) + 6*c*(5*c^2 + 1 
2*a*c*e^2 + 8*a^2*e^4)*ArcTanh[Sqrt[(-1 + e*x)/(1 + e*x)]])/(8*e^7)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.48, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {648, 315, 25, 27, 403, 299, 224, 219}

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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 299

\(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {c \left (\frac {\frac {3 \left (8 a^2 e^4+12 a c e^2+5 c^2\right ) \int \frac {1}{\sqrt {e^2 x^2-1}}dx}{2 e^2}+\frac {x \sqrt {e^2 x^2-1} \left (2 a e^2+5 c\right ) \left (4 a e^2+3 c\right )}{2 e^2}}{4 e^2}+\frac {1}{4} x \sqrt {e^2 x^2-1} \left (4 a+\frac {5 c}{e^2}\right ) \left (a+c x^2\right )\right )}{e^2}-\frac {x \left (a e^2+c\right ) \left (a+c x^2\right )^2}{e^2 \sqrt {e^2 x^2-1}}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {c \left (\frac {\frac {3 \left (8 a^2 e^4+12 a c e^2+5 c^2\right ) \int \frac {1}{1-\frac {e^2 x^2}{e^2 x^2-1}}d\frac {x}{\sqrt {e^2 x^2-1}}}{2 e^2}+\frac {x \sqrt {e^2 x^2-1} \left (2 a e^2+5 c\right ) \left (4 a e^2+3 c\right )}{2 e^2}}{4 e^2}+\frac {1}{4} x \sqrt {e^2 x^2-1} \left (4 a+\frac {5 c}{e^2}\right ) \left (a+c x^2\right )\right )}{e^2}-\frac {x \left (a e^2+c\right ) \left (a+c x^2\right )^2}{e^2 \sqrt {e^2 x^2-1}}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {e^2 x^2-1} \left (\frac {c \left (\frac {\frac {3 \left (8 a^2 e^4+12 a c e^2+5 c^2\right ) \text {arctanh}\left (\frac {e x}{\sqrt {e^2 x^2-1}}\right )}{2 e^3}+\frac {x \sqrt {e^2 x^2-1} \left (2 a e^2+5 c\right ) \left (4 a e^2+3 c\right )}{2 e^2}}{4 e^2}+\frac {1}{4} x \sqrt {e^2 x^2-1} \left (4 a+\frac {5 c}{e^2}\right ) \left (a+c x^2\right )\right )}{e^2}-\frac {x \left (a e^2+c\right ) \left (a+c x^2\right )^2}{e^2 \sqrt {e^2 x^2-1}}\right )}{\sqrt {e x-1} \sqrt {e x+1}}\)

Input:

Int[(a + c*x^2)^3/((-1 + e*x)^(3/2)*(1 + e*x)^(3/2)),x]
 

Output:

(Sqrt[-1 + e^2*x^2]*(-(((c + a*e^2)*x*(a + c*x^2)^2)/(e^2*Sqrt[-1 + e^2*x^ 
2])) + (c*(((4*a + (5*c)/e^2)*x*(a + c*x^2)*Sqrt[-1 + e^2*x^2])/4 + (((5*c 
 + 2*a*e^2)*(3*c + 4*a*e^2)*x*Sqrt[-1 + e^2*x^2])/(2*e^2) + (3*(5*c^2 + 12 
*a*c*e^2 + 8*a^2*e^4)*ArcTanh[(e*x)/Sqrt[-1 + e^2*x^2]])/(2*e^3))/(4*e^2)) 
)/e^2))/(Sqrt[-1 + e*x]*Sqrt[1 + e*x])
 

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 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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 299

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

rule 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(320\) vs. \(2(119)=238\).

Time = 0.78 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.34

method	result	size
risch	\(\frac {c^{2} x \left (2 x^{2} c \,e^{2}+12 a \,e^{2}+7 c \right ) \sqrt {e x -1}\, \sqrt {e x +1}}{8 e^{6}}+\frac {\left (\frac {\left (-4 e^{6} a^{3}-12 e^{4} a^{2} c -12 c^{2} a \,e^{2}-4 c^{3}\right ) \sqrt {e^{2} \left (x +\frac {1}{e}\right )^{2}-2 e \left (x +\frac {1}{e}\right )}}{e^{2} \left (x +\frac {1}{e}\right )}-\frac {\left (4 e^{6} a^{3}+12 e^{4} a^{2} c +12 c^{2} a \,e^{2}+4 c^{3}\right ) \sqrt {e^{2} \left (x -\frac {1}{e}\right )^{2}+2 e \left (x -\frac {1}{e}\right )}}{e^{2} \left (x -\frac {1}{e}\right )}+\frac {15 c^{3} \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right )}{\sqrt {e^{2}}}+\frac {36 c^{2} a \,e^{2} \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right )}{\sqrt {e^{2}}}+\frac {24 e^{4} a^{2} c \ln \left (\frac {e^{2} x}{\sqrt {e^{2}}}+\sqrt {e^{2} x^{2}-1}\right )}{\sqrt {e^{2}}}\right ) \sqrt {\left (e x -1\right ) \left (e x +1\right )}}{8 e^{6} \sqrt {e x -1}\, \sqrt {e x +1}}\)	\(321\)
default	\(-\frac {\left (-2 \,\operatorname {csgn}\left (e \right ) c^{3} e^{5} x^{5} \sqrt {e^{2} x^{2}-1}+8 \,\operatorname {csgn}\left (e \right ) e^{7} \sqrt {e^{2} x^{2}-1}\, a^{3} x -12 \,\operatorname {csgn}\left (e \right ) a \,c^{2} e^{5} x^{3} \sqrt {e^{2} x^{2}-1}-24 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a^{2} c \,e^{6} x^{2}+24 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{5} a^{2} c x -5 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{3} c^{3} x^{3}-36 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a \,c^{2} e^{4} x^{2}+36 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e^{3} a \,c^{2} x +24 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a^{2} c \,e^{4}-15 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) c^{3} e^{2} x^{2}+15 \sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right ) e \,c^{3} x +36 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) a \,c^{2} e^{2}+15 \ln \left (\left (\sqrt {e^{2} x^{2}-1}\, \operatorname {csgn}\left (e \right )+e x \right ) \operatorname {csgn}\left (e \right )\right ) c^{3}\right ) \operatorname {csgn}\left (e \right )}{8 \sqrt {e^{2} x^{2}-1}\, e^{7} \sqrt {e x +1}\, \sqrt {e x -1}}\)	\(385\)

Input:

int((c*x^2+a)^3/(e*x-1)^(3/2)/(e*x+1)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/8*c^2*x*(2*c*e^2*x^2+12*a*e^2+7*c)*(e*x-1)^(1/2)*(e*x+1)^(1/2)/e^6+1/8/e 
^6*((-4*a^3*e^6-12*a^2*c*e^4-12*a*c^2*e^2-4*c^3)/e^2/(x+1/e)*(e^2*(x+1/e)^ 
2-2*e*(x+1/e))^(1/2)-(4*a^3*e^6+12*a^2*c*e^4+12*a*c^2*e^2+4*c^3)/e^2/(x-1/ 
e)*(e^2*(x-1/e)^2+2*e*(x-1/e))^(1/2)+15*c^3*ln(e^2*x/(e^2)^(1/2)+(e^2*x^2- 
1)^(1/2))/(e^2)^(1/2)+36*c^2*a*e^2*ln(e^2*x/(e^2)^(1/2)+(e^2*x^2-1)^(1/2)) 
/(e^2)^(1/2)+24*e^4*a^2*c*ln(e^2*x/(e^2)^(1/2)+(e^2*x^2-1)^(1/2))/(e^2)^(1 
/2))*((e*x-1)*(e*x+1))^(1/2)/(e*x-1)^(1/2)/(e*x+1)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (119) = 238\).

Time = 0.10 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {8 \, a^{3} e^{6} + 24 \, a^{2} c e^{4} + 24 \, a c^{2} e^{2} + 8 \, c^{3} - 8 \, {\left (a^{3} e^{8} + 3 \, a^{2} c e^{6} + 3 \, a c^{2} e^{4} + c^{3} e^{2}\right )} x^{2} + {\left (2 \, c^{3} e^{5} x^{5} + {\left (12 \, a c^{2} e^{5} + 5 \, c^{3} e^{3}\right )} x^{3} - {\left (8 \, a^{3} e^{7} + 24 \, a^{2} c e^{5} + 36 \, a c^{2} e^{3} + 15 \, c^{3} e\right )} x\right )} \sqrt {e x + 1} \sqrt {e x - 1} + 3 \, {\left (8 \, a^{2} c e^{4} + 12 \, a c^{2} e^{2} + 5 \, c^{3} - {\left (8 \, a^{2} c e^{6} + 12 \, a c^{2} e^{4} + 5 \, c^{3} e^{2}\right )} x^{2}\right )} \log \left (-e x + \sqrt {e x + 1} \sqrt {e x - 1}\right )}{8 \, {\left (e^{9} x^{2} - e^{7}\right )}} \] Input:

integrate((c*x^2+a)^3/(e*x-1)^(3/2)/(e*x+1)^(3/2),x, algorithm="fricas")
 

Output:

1/8*(8*a^3*e^6 + 24*a^2*c*e^4 + 24*a*c^2*e^2 + 8*c^3 - 8*(a^3*e^8 + 3*a^2* 
c*e^6 + 3*a*c^2*e^4 + c^3*e^2)*x^2 + (2*c^3*e^5*x^5 + (12*a*c^2*e^5 + 5*c^ 
3*e^3)*x^3 - (8*a^3*e^7 + 24*a^2*c*e^5 + 36*a*c^2*e^3 + 15*c^3*e)*x)*sqrt( 
e*x + 1)*sqrt(e*x - 1) + 3*(8*a^2*c*e^4 + 12*a*c^2*e^2 + 5*c^3 - (8*a^2*c* 
e^6 + 12*a*c^2*e^4 + 5*c^3*e^2)*x^2)*log(-e*x + sqrt(e*x + 1)*sqrt(e*x - 1 
)))/(e^9*x^2 - e^7)
 

Sympy [F]

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

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

Output:

Integral((a + c*x**2)**3/((e*x - 1)**(3/2)*(e*x + 1)**(3/2)), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (119) = 238\).

Time = 0.04 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {c^{3} x^{5}}{4 \, \sqrt {e^{2} x^{2} - 1} e^{2}} - \frac {a^{3} x}{\sqrt {e^{2} x^{2} - 1}} + \frac {3 \, a c^{2} x^{3}}{2 \, \sqrt {e^{2} x^{2} - 1} e^{2}} - \frac {3 \, a^{2} c x}{\sqrt {e^{2} x^{2} - 1} e^{2}} + \frac {5 \, c^{3} x^{3}}{8 \, \sqrt {e^{2} x^{2} - 1} e^{4}} + \frac {3 \, a^{2} c \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{\sqrt {e^{2}} e^{2}} - \frac {9 \, a c^{2} x}{2 \, \sqrt {e^{2} x^{2} - 1} e^{4}} + \frac {9 \, a c^{2} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{2 \, \sqrt {e^{2}} e^{4}} - \frac {15 \, c^{3} x}{8 \, \sqrt {e^{2} x^{2} - 1} e^{6}} + \frac {15 \, c^{3} \log \left (2 \, e^{2} x + 2 \, \sqrt {e^{2} x^{2} - 1} \sqrt {e^{2}}\right )}{8 \, \sqrt {e^{2}} e^{6}} \] Input:

integrate((c*x^2+a)^3/(e*x-1)^(3/2)/(e*x+1)^(3/2),x, algorithm="maxima")
 

Output:

1/4*c^3*x^5/(sqrt(e^2*x^2 - 1)*e^2) - a^3*x/sqrt(e^2*x^2 - 1) + 3/2*a*c^2* 
x^3/(sqrt(e^2*x^2 - 1)*e^2) - 3*a^2*c*x/(sqrt(e^2*x^2 - 1)*e^2) + 5/8*c^3* 
x^3/(sqrt(e^2*x^2 - 1)*e^4) + 3*a^2*c*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sq 
rt(e^2))/(sqrt(e^2)*e^2) - 9/2*a*c^2*x/(sqrt(e^2*x^2 - 1)*e^4) + 9/2*a*c^2 
*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt(e^2))/(sqrt(e^2)*e^4) - 15/8*c^3*x 
/(sqrt(e^2*x^2 - 1)*e^6) + 15/8*c^3*log(2*e^2*x + 2*sqrt(e^2*x^2 - 1)*sqrt 
(e^2))/(sqrt(e^2)*e^6)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 251 vs. \(2 (119) = 238\).

Time = 0.15 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.83 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {{\left ({\left ({\left (e x + 1\right )} {\left (2 \, {\left (e x + 1\right )} {\left (\frac {{\left (e x + 1\right )} c^{3}}{e^{7}} - \frac {5 \, c^{3}}{e^{7}}\right )} + \frac {12 \, a c^{2} e^{37} + 25 \, c^{3} e^{35}}{e^{42}}\right )} - \frac {36 \, a c^{2} e^{37} + 35 \, c^{3} e^{35}}{e^{42}}\right )} {\left (e x + 1\right )} - \frac {2 \, {\left (2 \, a^{3} e^{41} + 6 \, a^{2} c e^{39} - 6 \, a c^{2} e^{37} - 7 \, c^{3} e^{35}\right )}}{e^{42}}\right )} \sqrt {e x + 1}}{8 \, \sqrt {e x - 1}} - \frac {3 \, {\left (8 \, a^{2} c e^{4} + 12 \, a c^{2} e^{2} + 5 \, c^{3}\right )} \log \left ({\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{2}\right )}{8 \, e^{7}} - \frac {2 \, {\left (a^{3} e^{6} + 3 \, a^{2} c e^{4} + 3 \, a c^{2} e^{2} + c^{3}\right )}}{{\left ({\left (\sqrt {e x + 1} - \sqrt {e x - 1}\right )}^{2} + 2\right )} e^{7}} \] Input:

integrate((c*x^2+a)^3/(e*x-1)^(3/2)/(e*x+1)^(3/2),x, algorithm="giac")
 

Output:

1/8*(((e*x + 1)*(2*(e*x + 1)*((e*x + 1)*c^3/e^7 - 5*c^3/e^7) + (12*a*c^2*e 
^37 + 25*c^3*e^35)/e^42) - (36*a*c^2*e^37 + 35*c^3*e^35)/e^42)*(e*x + 1) - 
 2*(2*a^3*e^41 + 6*a^2*c*e^39 - 6*a*c^2*e^37 - 7*c^3*e^35)/e^42)*sqrt(e*x 
+ 1)/sqrt(e*x - 1) - 3/8*(8*a^2*c*e^4 + 12*a*c^2*e^2 + 5*c^3)*log((sqrt(e* 
x + 1) - sqrt(e*x - 1))^2)/e^7 - 2*(a^3*e^6 + 3*a^2*c*e^4 + 3*a*c^2*e^2 + 
c^3)/(((sqrt(e*x + 1) - sqrt(e*x - 1))^2 + 2)*e^7)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.27 \[ \int \frac {\left (a+c x^2\right )^3}{(-1+e x)^{3/2} (1+e x)^{3/2}} \, dx=\frac {48 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a^{2} c \,e^{5} x +48 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a^{2} c \,e^{4}+72 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a \,c^{2} e^{3} x +72 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) a \,c^{2} e^{2}+30 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) c^{3} e x +30 \sqrt {e x -1}\, \mathrm {log}\left (\frac {\sqrt {e x -1}+\sqrt {e x +1}}{\sqrt {2}}\right ) c^{3}-8 \sqrt {e x -1}\, a^{3} e^{7} x -8 \sqrt {e x -1}\, a^{3} e^{6}-24 \sqrt {e x -1}\, a^{2} c \,e^{5} x -24 \sqrt {e x -1}\, a^{2} c \,e^{4}-27 \sqrt {e x -1}\, a \,c^{2} e^{3} x -27 \sqrt {e x -1}\, a \,c^{2} e^{2}-10 \sqrt {e x -1}\, c^{3} e x -10 \sqrt {e x -1}\, c^{3}-8 \sqrt {e x +1}\, a^{3} e^{7} x -24 \sqrt {e x +1}\, a^{2} c \,e^{5} x +12 \sqrt {e x +1}\, a \,c^{2} e^{5} x^{3}-36 \sqrt {e x +1}\, a \,c^{2} e^{3} x +2 \sqrt {e x +1}\, c^{3} e^{5} x^{5}+5 \sqrt {e x +1}\, c^{3} e^{3} x^{3}-15 \sqrt {e x +1}\, c^{3} e x}{8 \sqrt {e x -1}\, e^{7} \left (e x +1\right )} \] Input:

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

Output:

(48*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a**2*c*e**5 
*x + 48*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a**2*c* 
e**4 + 72*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a*c** 
2*e**3*x + 72*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))*a 
*c**2*e**2 + 30*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2)) 
*c**3*e*x + 30*sqrt(e*x - 1)*log((sqrt(e*x - 1) + sqrt(e*x + 1))/sqrt(2))* 
c**3 - 8*sqrt(e*x - 1)*a**3*e**7*x - 8*sqrt(e*x - 1)*a**3*e**6 - 24*sqrt(e 
*x - 1)*a**2*c*e**5*x - 24*sqrt(e*x - 1)*a**2*c*e**4 - 27*sqrt(e*x - 1)*a* 
c**2*e**3*x - 27*sqrt(e*x - 1)*a*c**2*e**2 - 10*sqrt(e*x - 1)*c**3*e*x - 1 
0*sqrt(e*x - 1)*c**3 - 8*sqrt(e*x + 1)*a**3*e**7*x - 24*sqrt(e*x + 1)*a**2 
*c*e**5*x + 12*sqrt(e*x + 1)*a*c**2*e**5*x**3 - 36*sqrt(e*x + 1)*a*c**2*e* 
*3*x + 2*sqrt(e*x + 1)*c**3*e**5*x**5 + 5*sqrt(e*x + 1)*c**3*e**3*x**3 - 1 
5*sqrt(e*x + 1)*c**3*e*x)/(8*sqrt(e*x - 1)*e**7*(e*x + 1))