\(\int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx\) [390]

Optimal result

Integrand size = 19, antiderivative size = 197 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=-\frac {29 \sqrt {1-a^2 x^2}}{2 a^6 c^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^6 c^4 (1-a x)^5}-\frac {33 \left (1-a^2 x^2\right )^{3/2}}{35 a^6 c^4 (1-a x)^4}+\frac {317 \left (1-a^2 x^2\right )^{3/2}}{105 a^6 c^4 (1-a x)^3}-\frac {10 \left (1-a^2 x^2\right )^{3/2}}{a^6 c^4 (1-a x)^2}-\frac {\left (1-a^2 x^2\right )^{3/2}}{2 a^6 c^4 (1-a x)}+\frac {29 \arcsin (a x)}{2 a^6 c^4} \] Output:

-29/2*(-a^2*x^2+1)^(1/2)/a^6/c^4+1/7*(-a^2*x^2+1)^(3/2)/a^6/c^4/(-a*x+1)^5 
-33/35*(-a^2*x^2+1)^(3/2)/a^6/c^4/(-a*x+1)^4+317/105*(-a^2*x^2+1)^(3/2)/a^ 
6/c^4/(-a*x+1)^3-10*(-a^2*x^2+1)^(3/2)/a^6/c^4/(-a*x+1)^2-1/2*(-a^2*x^2+1) 
^(3/2)/a^6/c^4/(-a*x+1)+29/2*arcsin(a*x)/a^6/c^4
 

Mathematica [A] (warning: unable to verify)

Time = 0.38 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.64 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=-\frac {(1+a x) \left (\sqrt {1-a^2 x^2} \left (4784-16091 a x+18916 a^2 x^2-8404 a^3 x^3+630 a^4 x^4+105 a^5 x^5\right )-945 (-1+a x)^4 \arcsin (a x)+4200 (-1+a x)^4 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{210 a^6 c^4 (-1+a x)^3 \left (-1+a^2 x^2\right )} \] Input:

Integrate[(E^ArcTanh[a*x]*x^5)/(c - a*c*x)^4,x]
 

Output:

-1/210*((1 + a*x)*(Sqrt[1 - a^2*x^2]*(4784 - 16091*a*x + 18916*a^2*x^2 - 8 
404*a^3*x^3 + 630*a^4*x^4 + 105*a^5*x^5) - 945*(-1 + a*x)^4*ArcSin[a*x] + 
4200*(-1 + a*x)^4*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/(a^6*c^4*(-1 + a*x)^3*(- 
1 + a^2*x^2))
 

Rubi [A] (verified)

Time = 1.34 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {6678, 27, 570, 529, 2166, 2166, 27, 2166, 27, 676, 223}

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.

\(\displaystyle \int \frac {x^5 e^{\text {arctanh}(a x)}}{(c-a c x)^4} \, dx\)

\(\Big \downarrow \) 6678

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 570

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

\(\Big \downarrow \) 529

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

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \int \frac {(a x+1)^3 \left (\frac {35 x^3}{a^2}+\frac {70 x^2}{a^3}+\frac {105 x}{a^4}+\frac {107}{a^5}\right )}{\left (1-a^2 x^2\right )^{5/2}}dx-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-\frac {1}{3} \int \frac {105 (a x+1)^2 \left (\frac {x^2}{a^3}+\frac {3 x}{a^4}+\frac {6}{a^5}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-35 \int \frac {(a x+1)^2 \left (\frac {x^2}{a^3}+\frac {3 x}{a^4}+\frac {6}{a^5}\right )}{\left (1-a^2 x^2\right )^{3/2}}dx\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 2166

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-35 \left (\frac {10 (a x+1)^2}{a^6 \sqrt {1-a^2 x^2}}-\int \frac {(a x+1) (a x+14)}{a^5 \sqrt {1-a^2 x^2}}dx\right )\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-35 \left (\frac {10 (a x+1)^2}{a^6 \sqrt {1-a^2 x^2}}-\frac {\int \frac {(a x+1) (a x+14)}{\sqrt {1-a^2 x^2}}dx}{a^5}\right )\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 676

\(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-35 \left (\frac {10 (a x+1)^2}{a^6 \sqrt {1-a^2 x^2}}-\frac {\frac {29}{2} \int \frac {1}{\sqrt {1-a^2 x^2}}dx-\frac {1}{2} x \sqrt {1-a^2 x^2}-\frac {15 \sqrt {1-a^2 x^2}}{a}}{a^5}\right )\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )+\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}}{c^4}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {\frac {(a x+1)^5}{7 a^6 \left (1-a^2 x^2\right )^{7/2}}+\frac {1}{7} \left (\frac {1}{5} \left (\frac {317 (a x+1)^3}{3 a^6 \left (1-a^2 x^2\right )^{3/2}}-35 \left (\frac {10 (a x+1)^2}{a^6 \sqrt {1-a^2 x^2}}-\frac {-\frac {1}{2} x \sqrt {1-a^2 x^2}-\frac {15 \sqrt {1-a^2 x^2}}{a}+\frac {29 \arcsin (a x)}{2 a}}{a^5}\right )\right )-\frac {33 (a x+1)^4}{5 a^6 \left (1-a^2 x^2\right )^{5/2}}\right )}{c^4}\)

Input:

Int[(E^ArcTanh[a*x]*x^5)/(c - a*c*x)^4,x]
 

Output:

((1 + a*x)^5/(7*a^6*(1 - a^2*x^2)^(7/2)) + ((-33*(1 + a*x)^4)/(5*a^6*(1 - 
a^2*x^2)^(5/2)) + ((317*(1 + a*x)^3)/(3*a^6*(1 - a^2*x^2)^(3/2)) - 35*((10 
*(1 + a*x)^2)/(a^6*Sqrt[1 - a^2*x^2]) - ((-15*Sqrt[1 - a^2*x^2])/a - (x*Sq 
rt[1 - a^2*x^2])/2 + (29*ArcSin[a*x])/(2*a))/a^5))/5)/7)/c^4
 

Defintions of rubi rules used

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 223

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

rule 529

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem 
ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ 
(2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1))   Int[(c + d*x)^(n - 1)*(a + b* 
x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; 
 FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* 
c^2 + a*d^2, 0]
 

rule 570

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

rule 676

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x 
_Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim 
p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p 
+ 3))/(c*(2*p + 3))   Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g 
, p}, x] &&  !LeQ[p, -1]
 

rule 2166

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde 
r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e 
*(p + 1))), x] + Simp[d/(2*a*(p + 1))   Int[(d + e*x)^(m - 1)*(a + b*x^2)^( 
p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ 
[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 
 0] && GtQ[m, 0]
 

rule 6678

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

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.19

method	result	size
risch	\(\frac {\left (a x +10\right ) \left (a^{2} x^{2}-1\right )}{2 a^{6} \sqrt {-a^{2} x^{2}+1}\, c^{4}}+\frac {\frac {29 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{5} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a^{10} \left (x -\frac {1}{a}\right )^{4}}+\frac {71 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{35 a^{9} \left (x -\frac {1}{a}\right )^{3}}+\frac {733 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 a^{8} \left (x -\frac {1}{a}\right )^{2}}+\frac {2417 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{105 a^{7} \left (x -\frac {1}{a}\right )}}{c^{4}}\)	\(235\)
default	\(\frac {\frac {-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}}{a^{3}}+\frac {14 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{5} \sqrt {a^{2}}}-\frac {5 \sqrt {-a^{2} x^{2}+1}}{a^{6}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{9}}+\frac {\frac {11 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {22 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{8}}+\frac {\frac {25 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {25 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{7}}+\frac {30 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{7} \left (x -\frac {1}{a}\right )}}{c^{4}}\)	\(537\)

Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x,method=_RETURNVERBOSE)
 

Output:

1/2*(a*x+10)*(a^2*x^2-1)/a^6/(-a^2*x^2+1)^(1/2)/c^4+(29/2/a^5/(a^2)^(1/2)* 
arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))+2/7/a^10/(x-1/a)^4*(-(x-1/a)^2*a^ 
2-2*a*(x-1/a))^(1/2)+71/35/a^9/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2 
)+733/105/a^8/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+2417/105/a^7/(x 
-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))/c^4
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=-\frac {4784 \, a^{4} x^{4} - 19136 \, a^{3} x^{3} + 28704 \, a^{2} x^{2} - 19136 \, a x + 6090 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{5} x^{5} + 630 \, a^{4} x^{4} - 8404 \, a^{3} x^{3} + 18916 \, a^{2} x^{2} - 16091 \, a x + 4784\right )} \sqrt {-a^{2} x^{2} + 1} + 4784}{210 \, {\left (a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x, algorithm="fricas 
")
 

Output:

-1/210*(4784*a^4*x^4 - 19136*a^3*x^3 + 28704*a^2*x^2 - 19136*a*x + 6090*(a 
^4*x^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1 
)/(a*x)) + (105*a^5*x^5 + 630*a^4*x^4 - 8404*a^3*x^3 + 18916*a^2*x^2 - 160 
91*a*x + 4784)*sqrt(-a^2*x^2 + 1) + 4784)/(a^10*c^4*x^4 - 4*a^9*c^4*x^3 + 
6*a^8*c^4*x^2 - 4*a^7*c^4*x + a^6*c^4)
 

Sympy [F]

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

integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a*c*x+c)**4,x)
 

Output:

(Integral(x**5/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a**2*x* 
*2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + 
sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**6/(a**4*x**4*sqrt(-a**2*x**2 + 1 
) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 
4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
 

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.27 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{10} c^{4} x^{4} - 4 \, a^{9} c^{4} x^{3} + 6 \, a^{8} c^{4} x^{2} - 4 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} + \frac {71 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{9} c^{4} x^{3} - 3 \, a^{8} c^{4} x^{2} + 3 \, a^{7} c^{4} x - a^{6} c^{4}\right )}} + \frac {733 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{8} c^{4} x^{2} - 2 \, a^{7} c^{4} x + a^{6} c^{4}\right )}} + \frac {2417 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{7} c^{4} x - a^{6} c^{4}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{5} c^{4}} + \frac {29 \, \arcsin \left (a x\right )}{2 \, a^{6} c^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6} c^{4}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x, algorithm="maxima 
")
 

Output:

2/7*sqrt(-a^2*x^2 + 1)/(a^10*c^4*x^4 - 4*a^9*c^4*x^3 + 6*a^8*c^4*x^2 - 4*a 
^7*c^4*x + a^6*c^4) + 71/35*sqrt(-a^2*x^2 + 1)/(a^9*c^4*x^3 - 3*a^8*c^4*x^ 
2 + 3*a^7*c^4*x - a^6*c^4) + 733/105*sqrt(-a^2*x^2 + 1)/(a^8*c^4*x^2 - 2*a 
^7*c^4*x + a^6*c^4) + 2417/105*sqrt(-a^2*x^2 + 1)/(a^7*c^4*x - a^6*c^4) - 
1/2*sqrt(-a^2*x^2 + 1)*x/(a^5*c^4) + 29/2*arcsin(a*x)/(a^6*c^4) - 5*sqrt(- 
a^2*x^2 + 1)/(a^6*c^4)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.28 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=-\frac {1}{2} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {x}{a^{5} c^{4}} + \frac {10}{a^{6} c^{4}}\right )} + \frac {29 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{2 \, a^{5} c^{4} {\left | a \right |}} + \frac {2 \, {\left (\frac {11599 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {29442 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {38500 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {26845 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {9765 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {1470 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 1867\right )}}{105 \, a^{5} c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \] Input:

integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x, algorithm="giac")
 

Output:

-1/2*sqrt(-a^2*x^2 + 1)*(x/(a^5*c^4) + 10/(a^6*c^4)) + 29/2*arcsin(a*x)*sg 
n(a)/(a^5*c^4*abs(a)) + 2/105*(11599*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2* 
x) - 29442*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 38500*(sqrt(-a^2* 
x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 26845*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^4 
/(a^8*x^4) + 9765*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^5/(a^10*x^5) - 1470*(sqr 
t(-a^2*x^2 + 1)*abs(a) + a)^6/(a^12*x^6) - 1867)/(a^5*c^4*((sqrt(-a^2*x^2 
+ 1)*abs(a) + a)/(a^2*x) - 1)^7*abs(a))
 

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=\frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^{10}\,c^4\,x^4-4\,a^9\,c^4\,x^3+6\,a^8\,c^4\,x^2-4\,a^7\,c^4\,x+a^6\,c^4\right )}+\frac {733\,\sqrt {1-a^2\,x^2}}{105\,\left (a^8\,c^4\,x^2-2\,a^7\,c^4\,x+a^6\,c^4\right )}+\frac {71\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a^4\,c^4\,\sqrt {-a^2}+3\,a^6\,c^4\,x^2\,\sqrt {-a^2}-a^7\,c^4\,x^3\,\sqrt {-a^2}-3\,a^5\,c^4\,x\,\sqrt {-a^2}\right )}+\frac {2417\,\sqrt {1-a^2\,x^2}}{105\,\left (a^4\,c^4\,\sqrt {-a^2}-a^5\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {5\,\sqrt {1-a^2\,x^2}}{a^6\,c^4}-\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^5\,c^4}+\frac {29\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^5\,c^4\,\sqrt {-a^2}} \] Input:

int((x^5*(a*x + 1))/((1 - a^2*x^2)^(1/2)*(c - a*c*x)^4),x)
                                                                                    
                                                                                    
 

Output:

(2*(1 - a^2*x^2)^(1/2))/(7*(a^6*c^4 - 4*a^7*c^4*x + 6*a^8*c^4*x^2 - 4*a^9* 
c^4*x^3 + a^10*c^4*x^4)) + (733*(1 - a^2*x^2)^(1/2))/(105*(a^6*c^4 - 2*a^7 
*c^4*x + a^8*c^4*x^2)) + (71*(1 - a^2*x^2)^(1/2))/(35*(-a^2)^(1/2)*(a^4*c^ 
4*(-a^2)^(1/2) + 3*a^6*c^4*x^2*(-a^2)^(1/2) - a^7*c^4*x^3*(-a^2)^(1/2) - 3 
*a^5*c^4*x*(-a^2)^(1/2))) + (2417*(1 - a^2*x^2)^(1/2))/(105*(a^4*c^4*(-a^2 
)^(1/2) - a^5*c^4*x*(-a^2)^(1/2))*(-a^2)^(1/2)) - (5*(1 - a^2*x^2)^(1/2))/ 
(a^6*c^4) - (x*(1 - a^2*x^2)^(1/2))/(2*a^5*c^4) + (29*asinh(x*(-a^2)^(1/2) 
))/(2*a^5*c^4*(-a^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 424, normalized size of antiderivative = 2.15 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{(c-a c x)^4} \, dx=\frac {-3045 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{4} x^{4}+12180 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{3} x^{3}-18270 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a^{2} x^{2}+12180 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right ) a x -3045 \mathit {atan} \left (\frac {2 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}{2 a^{3} x^{3}-2 a x}\right )-210 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-1260 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+16808 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-37832 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+32182 \sqrt {-a^{2} x^{2}+1}\, a x -9568 \sqrt {-a^{2} x^{2}+1}}{420 a^{6} c^{4} \left (a^{4} x^{4}-4 a^{3} x^{3}+6 a^{2} x^{2}-4 a x +1\right )} \] Input:

int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a*c*x+c)^4,x)
 

Output:

( - 3045*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1) 
)/(2*a**3*x**3 - 2*a*x))*a**4*x**4 + 12180*atan((2*sqrt( - a**2*x**2 + 1)* 
a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a**3*x**3 - 182 
70*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a 
**3*x**3 - 2*a*x))*a**2*x**2 + 12180*atan((2*sqrt( - a**2*x**2 + 1)*a**2*x 
**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2*a*x))*a*x - 3045*atan((2*sq 
rt( - a**2*x**2 + 1)*a**2*x**2 - sqrt( - a**2*x**2 + 1))/(2*a**3*x**3 - 2* 
a*x)) - 210*sqrt( - a**2*x**2 + 1)*a**5*x**5 - 1260*sqrt( - a**2*x**2 + 1) 
*a**4*x**4 + 16808*sqrt( - a**2*x**2 + 1)*a**3*x**3 - 37832*sqrt( - a**2*x 
**2 + 1)*a**2*x**2 + 32182*sqrt( - a**2*x**2 + 1)*a*x - 9568*sqrt( - a**2* 
x**2 + 1))/(420*a**6*c**4*(a**4*x**4 - 4*a**3*x**3 + 6*a**2*x**2 - 4*a*x + 
 1))