\(\int \frac {e^{3 \text {arctanh}(a x)}}{(c-a^2 c x^2)^4} \, dx\) [1177]

Optimal result

Integrand size = 22, antiderivative size = 127 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {2 (1+a x)^2}{9 a c^4 \left (1-a^2 x^2\right )^{9/2}}+\frac {5 (1+a x)}{63 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {2 x}{21 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}} \] Output:

2/9*(a*x+1)^2/a/c^4/(-a^2*x^2+1)^(9/2)+5/63*(a*x+1)/a/c^4/(-a^2*x^2+1)^(7/ 
2)+2/21*x/c^4/(-a^2*x^2+1)^(5/2)+8/63*x/c^4/(-a^2*x^2+1)^(3/2)+16/63*x/c^4 
/(-a^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.59 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {19+6 a x-66 a^2 x^2+56 a^3 x^3+24 a^4 x^4-48 a^5 x^5+16 a^6 x^6}{63 a c^4 (1-a x)^{9/2} (1+a x)^{3/2}} \] Input:

Integrate[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^4,x]
 

Output:

(19 + 6*a*x - 66*a^2*x^2 + 56*a^3*x^3 + 24*a^4*x^4 - 48*a^5*x^5 + 16*a^6*x 
^6)/(63*a*c^4*(1 - a*x)^(9/2)*(1 + a*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6688, 464, 461, 461, 470, 209, 208}

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.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx\)

\(\Big \downarrow \) 6688

\(\displaystyle \frac {\int \frac {(a x+1)^3}{\left (1-a^2 x^2\right )^{11/2}}dx}{c^4}\)

\(\Big \downarrow \) 464

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

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {2}{3} \int \frac {1}{(1-a x)^2 \left (1-a^2 x^2\right )^{5/2}}dx+\frac {1}{9 a (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}}{c^4}\)

\(\Big \downarrow \) 461

\(\displaystyle \frac {\frac {2}{3} \left (\frac {5}{7} \int \frac {1}{(1-a x) \left (1-a^2 x^2\right )^{5/2}}dx+\frac {1}{7 a (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{9 a (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}}{c^4}\)

\(\Big \downarrow \) 470

\(\displaystyle \frac {\frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}}dx+\frac {1}{5 a (1-a x) \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{7 a (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{9 a (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}}{c^4}\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{5 a (1-a x) \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{7 a (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{9 a (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}}{c^4}\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {2}{3} \left (\frac {5}{7} \left (\frac {4}{5} \left (\frac {2 x}{3 \sqrt {1-a^2 x^2}}+\frac {x}{3 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{5 a (1-a x) \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{7 a (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}\right )+\frac {1}{9 a (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}}}{c^4}\)

Input:

Int[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^4,x]
 

Output:

(1/(9*a*(1 - a*x)^3*(1 - a^2*x^2)^(3/2)) + (2*(1/(7*a*(1 - a*x)^2*(1 - a^2 
*x^2)^(3/2)) + (5*(1/(5*a*(1 - a*x)*(1 - a^2*x^2)^(3/2)) + (4*(x/(3*(1 - a 
^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - a^2*x^2])))/5))/7))/3)/c^4
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

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

rule 461

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
(-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl 
ify[n + 2*p + 2]/(2*c*(n + p + 1))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x 
], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp 
lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
 

rule 464

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

rule 470

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

rule 6688

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

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.58

method	result	size
gosper	\(-\frac {16 x^{6} a^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19}{63 \left (a x -1\right )^{3} c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\)	\(74\)
trager	\(-\frac {\left (16 x^{6} a^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19\right ) \sqrt {-a^{2} x^{2}+1}}{63 c^{4} \left (a x -1\right )^{5} \left (a x +1\right )^{2} a}\)	\(81\)
orering	\(-\frac {\left (16 x^{6} a^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19\right ) \left (a x -1\right ) \left (a x +1\right )^{4}}{63 a \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (-a^{2} c \,x^{2}+c \right )^{4}}\)	\(89\)
default	\(\frac {\frac {\frac {1}{9 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {5 a \left (\frac {1}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {4 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}\right )}{9}}{2 a^{4}}-\frac {\frac {1}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {4 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{4 a^{3}}+\frac {\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{8 a^{2}}-\frac {\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 \left (x -\frac {1}{a}\right ) a^{2}-2 a}{3 a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}}{16 a}+\frac {-\frac {1}{3 a \left (x +\frac {1}{a}\right ) \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}-\frac {-2 a^{2} \left (x +\frac {1}{a}\right )+2 a}{3 a \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}}{16 a}}{c^{4}}\)	\(739\)

Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x,method=_RETURNVERBOSE)
 

Output:

-1/63*(16*a^6*x^6-48*a^5*x^5+24*a^4*x^4+56*a^3*x^3-66*a^2*x^2+6*a*x+19)/(a 
*x-1)^3/c^4/(-a^2*x^2+1)^(3/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.56 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {19 \, a^{7} x^{7} - 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} + 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} - 19 \, a^{2} x^{2} + 57 \, a x - {\left (16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} - 19}{63 \, {\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="fric 
as")
 

Output:

1/63*(19*a^7*x^7 - 57*a^6*x^6 + 19*a^5*x^5 + 95*a^4*x^4 - 95*a^3*x^3 - 19* 
a^2*x^2 + 57*a*x - (16*a^6*x^6 - 48*a^5*x^5 + 24*a^4*x^4 + 56*a^3*x^3 - 66 
*a^2*x^2 + 6*a*x + 19)*sqrt(-a^2*x^2 + 1) - 19)/(a^8*c^4*x^7 - 3*a^7*c^4*x 
^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4 
*x - a*c^4)
 

Sympy [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {\int \frac {3 a x}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \] Input:

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**4,x)
 

Output:

(Integral(3*a*x/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqrt(-a** 
2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt(-a**2* 
x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + 
 Integral(3*a**2*x**2/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqr 
t(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt( 
-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)) 
, x) + Integral(a**3*x**3/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8 
*sqrt(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*s 
qrt(-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 
 1)), x) + Integral(1/(-a**10*x**10*sqrt(-a**2*x**2 + 1) + 5*a**8*x**8*sqr 
t(-a**2*x**2 + 1) - 10*a**6*x**6*sqrt(-a**2*x**2 + 1) + 10*a**4*x**4*sqrt( 
-a**2*x**2 + 1) - 5*a**2*x**2*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)) 
, x))/c**4
 

Maxima [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="maxi 
ma")
 

Output:

integrate((a*x + 1)^3/((a^2*c*x^2 - c)^4*(-a^2*x^2 + 1)^(3/2)), x)
 

Giac [F]

\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x, algorithm="giac 
")
 

Output:

integrate((a*x + 1)^3/((a^2*c*x^2 - c)^4*(-a^2*x^2 + 1)^(3/2)), x)
 

Mupad [B] (verification not implemented)

Time = 27.94 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.23 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {13\,\sqrt {1-a^2\,x^2}}{252\,a\,c^4\,{\left (a\,x-1\right )}^4}-\frac {23\,\sqrt {1-a^2\,x^2}}{336\,a\,c^4\,{\left (a\,x-1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{36\,a\,c^4\,{\left (a\,x-1\right )}^5}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {197\,x}{1008\,c^4}+\frac {155}{1008\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{63\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \] Input:

int((a*x + 1)^3/((c - a^2*c*x^2)^4*(1 - a^2*x^2)^(3/2)),x)
 

Output:

(13*(1 - a^2*x^2)^(1/2))/(252*a*c^4*(a*x - 1)^4) - (23*(1 - a^2*x^2)^(1/2) 
)/(336*a*c^4*(a*x - 1)^3) - (1 - a^2*x^2)^(1/2)/(36*a*c^4*(a*x - 1)^5) + ( 
(1 - a^2*x^2)^(1/2)*((197*x)/(1008*c^4) + 155/(1008*a*c^4)))/((a*x - 1)^2* 
(a*x + 1)^2) - (16*x*(1 - a^2*x^2)^(1/2))/(63*c^4*(a*x - 1)*(a*x + 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.65 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx=\frac {-2 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+6 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-4 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-4 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+6 \sqrt {-a^{2} x^{2}+1}\, a x -2 \sqrt {-a^{2} x^{2}+1}+16 a^{6} x^{6}-48 a^{5} x^{5}+24 a^{4} x^{4}+56 a^{3} x^{3}-66 a^{2} x^{2}+6 a x +19}{63 \sqrt {-a^{2} x^{2}+1}\, a \,c^{4} \left (a^{5} x^{5}-3 a^{4} x^{4}+2 a^{3} x^{3}+2 a^{2} x^{2}-3 a x +1\right )} \] Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^4,x)
 

Output:

( - 2*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 6*sqrt( - a**2*x**2 + 1)*a**4*x** 
4 - 4*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 4*sqrt( - a**2*x**2 + 1)*a**2*x** 
2 + 6*sqrt( - a**2*x**2 + 1)*a*x - 2*sqrt( - a**2*x**2 + 1) + 16*a**6*x**6 
 - 48*a**5*x**5 + 24*a**4*x**4 + 56*a**3*x**3 - 66*a**2*x**2 + 6*a*x + 19) 
/(63*sqrt( - a**2*x**2 + 1)*a*c**4*(a**5*x**5 - 3*a**4*x**4 + 2*a**3*x**3 
+ 2*a**2*x**2 - 3*a*x + 1))