\(\int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx\) [339]

Optimal result

Integrand size = 17, antiderivative size = 138 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=\frac {7 c^4 x \sqrt {1-a^2 x^2}}{16 a}-\frac {7}{8} a c^4 x^3 \sqrt {1-a^2 x^2}-\frac {4 c^4 \left (1-a^2 x^2\right )^{3/2}}{3 a^2}+\frac {1}{6} a c^4 x^3 \left (1-a^2 x^2\right )^{3/2}+\frac {3 c^4 \left (1-a^2 x^2\right )^{5/2}}{5 a^2}-\frac {7 c^4 \arcsin (a x)}{16 a^2} \] Output:

7/16*c^4*x*(-a^2*x^2+1)^(1/2)/a-7/8*a*c^4*x^3*(-a^2*x^2+1)^(1/2)-4/3*c^4*( 
-a^2*x^2+1)^(3/2)/a^2+1/6*a*c^4*x^3*(-a^2*x^2+1)^(3/2)+3/5*c^4*(-a^2*x^2+1 
)^(5/2)/a^2-7/16*c^4*arcsin(a*x)/a^2
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.60 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=-\frac {c^4 \left (\sqrt {1-a^2 x^2} \left (176-105 a x-32 a^2 x^2+170 a^3 x^3-144 a^4 x^4+40 a^5 x^5\right )-210 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )\right )}{240 a^2} \] Input:

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

Output:

-1/240*(c^4*(Sqrt[1 - a^2*x^2]*(176 - 105*a*x - 32*a^2*x^2 + 170*a^3*x^3 - 
 144*a^4*x^4 + 40*a^5*x^5) - 210*ArcSin[Sqrt[1 - a*x]/Sqrt[2]]))/a^2
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.09, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.765, Rules used = {6678, 27, 541, 27, 2340, 25, 27, 533, 25, 27, 455, 211, 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.

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

\(\Big \downarrow \) 6678

\(\displaystyle c \int c^3 x (1-a x)^3 \sqrt {1-a^2 x^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \int x (1-a x)^3 \sqrt {1-a^2 x^2}dx\)

\(\Big \downarrow \) 541

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2340

\(\displaystyle c^4 \left (\frac {-\frac {\int -a^4 x (22-35 a x) \sqrt {1-a^2 x^2}dx}{5 a^2}-\frac {6}{5} a^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^2}+\frac {1}{6} a x^3 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle c^4 \left (\frac {\frac {\int a^4 x (22-35 a x) \sqrt {1-a^2 x^2}dx}{5 a^2}-\frac {6}{5} a^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^2}+\frac {1}{6} a x^3 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle c^4 \left (\frac {\frac {1}{5} a^2 \int x (22-35 a x) \sqrt {1-a^2 x^2}dx-\frac {6}{5} a^2 x^2 \left (1-a^2 x^2\right )^{3/2}}{2 a^2}+\frac {1}{6} a x^3 \left (1-a^2 x^2\right )^{3/2}\right )\)

\(\Big \downarrow \) 533

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 455

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 223

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

Input:

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

Output:

c^4*((a*x^3*(1 - a^2*x^2)^(3/2))/6 + ((-6*a^2*x^2*(1 - a^2*x^2)^(3/2))/5 + 
 (a^2*((35*x*(1 - a^2*x^2)^(3/2))/(4*a) - ((88*(1 - a^2*x^2)^(3/2))/(3*a) 
+ 35*((x*Sqrt[1 - a^2*x^2])/2 + ArcSin[a*x]/(2*a)))/(4*a)))/5)/(2*a^2))
 

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 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 455

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

rule 533

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

rule 541

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Simp[d^n*x^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*(m + n + 2*p + 1))), x 
] + Simp[1/(b*(m + n + 2*p + 1))   Int[x^m*(a + b*x^2)^p*ExpandToSum[b*(m + 
 n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1)*x^n - a*d^n*(m + n - 1) 
*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, m, p}, x] && IGtQ[n, 1] && IGt 
Q[m, -2] && GtQ[p, -1] && IntegerQ[2*p]
 

rule 2340

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 
)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m 
+ q + 2*p + 1))   Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) 
*Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; 
GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ 
[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])
 

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 = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.74

method	result	size
risch	\(\frac {\left (40 a^{5} x^{5}-144 a^{4} x^{4}+170 a^{3} x^{3}-32 a^{2} x^{2}-105 a x +176\right ) \left (a^{2} x^{2}-1\right ) c^{4}}{240 a^{2} \sqrt {-a^{2} x^{2}+1}}-\frac {7 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c^{4}}{16 a \sqrt {a^{2}}}\)	\(102\)
meijerg	\(-\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {7}{2}} \left (56 a^{4} x^{4}+70 a^{2} x^{2}+105\right ) \sqrt {-a^{2} x^{2}+1}}{168 a^{6}}+\frac {5 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {7}{2}} \arcsin \left (a x \right )}{8 a^{7}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {3 c^{4} \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 a^{4} x^{4}+8 a^{2} x^{2}+16\right ) \sqrt {-a^{2} x^{2}+1}}{15}\right )}{2 a^{2} \sqrt {\pi }}+\frac {c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {5}{2}} \left (10 a^{2} x^{2}+15\right ) \sqrt {-a^{2} x^{2}+1}}{20 a^{4}}+\frac {3 \sqrt {\pi }\, \left (-a^{2}\right )^{\frac {5}{2}} \arcsin \left (a x \right )}{4 a^{5}}\right )}{a \sqrt {\pi }\, \sqrt {-a^{2}}}+\frac {c^{4} \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{6}\right )}{a^{2} \sqrt {\pi }}+\frac {3 c^{4} \left (-\frac {\sqrt {\pi }\, x \left (-a^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+\frac {\sqrt {\pi }\, \left (-a^{2}\right )^{\frac {3}{2}} \arcsin \left (a x \right )}{a^{3}}\right )}{2 a \sqrt {\pi }\, \sqrt {-a^{2}}}-\frac {c^{4} \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a^{2} x^{2}+1}\right )}{2 a^{2} \sqrt {\pi }}\)	\(355\)
default	\(c^{4} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{2}}+a^{5} \left (-\frac {x^{5} \sqrt {-a^{2} x^{2}+1}}{6 a^{2}}+\frac {-\frac {5 x^{3} \sqrt {-a^{2} x^{2}+1}}{24 a^{2}}+\frac {5 \left (-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}\right )}{6 a^{2}}}{a^{2}}\right )-3 a \left (-\frac {x \sqrt {-a^{2} x^{2}+1}}{2 a^{2}}+\frac {\arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 a^{2} \sqrt {a^{2}}}\right )+2 a^{2} \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )+2 a^{3} \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )-3 a^{4} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )\right )\)	\(372\)

Input:

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

Output:

1/240*(40*a^5*x^5-144*a^4*x^4+170*a^3*x^3-32*a^2*x^2-105*a*x+176)*(a^2*x^2 
-1)/a^2/(-a^2*x^2+1)^(1/2)*c^4-7/16/a/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a 
^2*x^2+1)^(1/2))*c^4
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.75 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=\frac {210 \, c^{4} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - {\left (40 \, a^{5} c^{4} x^{5} - 144 \, a^{4} c^{4} x^{4} + 170 \, a^{3} c^{4} x^{3} - 32 \, a^{2} c^{4} x^{2} - 105 \, a c^{4} x + 176 \, c^{4}\right )} \sqrt {-a^{2} x^{2} + 1}}{240 \, a^{2}} \] Input:

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

Output:

1/240*(210*c^4*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - (40*a^5*c^4*x^5 - 
144*a^4*c^4*x^4 + 170*a^3*c^4*x^3 - 32*a^2*c^4*x^2 - 105*a*c^4*x + 176*c^4 
)*sqrt(-a^2*x^2 + 1))/a^2
 

Sympy [A] (verification not implemented)

Time = 0.77 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.41 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=\begin {cases} \sqrt {- a^{2} x^{2} + 1} \left (- \frac {a^{3} c^{4} x^{5}}{6} + \frac {3 a^{2} c^{4} x^{4}}{5} - \frac {17 a c^{4} x^{3}}{24} + \frac {2 c^{4} x^{2}}{15} + \frac {7 c^{4} x}{16 a} - \frac {11 c^{4}}{15 a^{2}}\right ) - \frac {7 c^{4} \log {\left (- 2 a^{2} x + 2 \sqrt {- a^{2}} \sqrt {- a^{2} x^{2} + 1} \right )}}{16 a \sqrt {- a^{2}}} & \text {for}\: a^{2} \neq 0 \\\frac {a^{5} c^{4} x^{7}}{7} - \frac {a^{4} c^{4} x^{6}}{2} + \frac {2 a^{3} c^{4} x^{5}}{5} + \frac {a^{2} c^{4} x^{4}}{2} - a c^{4} x^{3} + \frac {c^{4} x^{2}}{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((sqrt(-a**2*x**2 + 1)*(-a**3*c**4*x**5/6 + 3*a**2*c**4*x**4/5 - 
17*a*c**4*x**3/24 + 2*c**4*x**2/15 + 7*c**4*x/(16*a) - 11*c**4/(15*a**2)) 
- 7*c**4*log(-2*a**2*x + 2*sqrt(-a**2)*sqrt(-a**2*x**2 + 1))/(16*a*sqrt(-a 
**2)), Ne(a**2, 0)), (a**5*c**4*x**7/7 - a**4*c**4*x**6/2 + 2*a**3*c**4*x* 
*5/5 + a**2*c**4*x**4/2 - a*c**4*x**3 + c**4*x**2/2, True))
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.02 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=-\frac {1}{6} \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{4} x^{5} + \frac {3}{5} \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{4} x^{4} - \frac {17}{24} \, \sqrt {-a^{2} x^{2} + 1} a c^{4} x^{3} + \frac {2}{15} \, \sqrt {-a^{2} x^{2} + 1} c^{4} x^{2} + \frac {7 \, \sqrt {-a^{2} x^{2} + 1} c^{4} x}{16 \, a} - \frac {7 \, c^{4} \arcsin \left (a x\right )}{16 \, a^{2}} - \frac {11 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{15 \, a^{2}} \] Input:

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

Output:

-1/6*sqrt(-a^2*x^2 + 1)*a^3*c^4*x^5 + 3/5*sqrt(-a^2*x^2 + 1)*a^2*c^4*x^4 - 
 17/24*sqrt(-a^2*x^2 + 1)*a*c^4*x^3 + 2/15*sqrt(-a^2*x^2 + 1)*c^4*x^2 + 7/ 
16*sqrt(-a^2*x^2 + 1)*c^4*x/a - 7/16*c^4*arcsin(a*x)/a^2 - 11/15*sqrt(-a^2 
*x^2 + 1)*c^4/a^2
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.68 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=-\frac {7 \, c^{4} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{16 \, a {\left | a \right |}} - \frac {1}{240} \, \sqrt {-a^{2} x^{2} + 1} {\left (\frac {176 \, c^{4}}{a^{2}} - {\left (\frac {105 \, c^{4}}{a} + 2 \, {\left (16 \, c^{4} - {\left (85 \, a c^{4} + 4 \, {\left (5 \, a^{3} c^{4} x - 18 \, a^{2} c^{4}\right )} x\right )} x\right )} x\right )} x\right )} \] Input:

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

Output:

-7/16*c^4*arcsin(a*x)*sgn(a)/(a*abs(a)) - 1/240*sqrt(-a^2*x^2 + 1)*(176*c^ 
4/a^2 - (105*c^4/a + 2*(16*c^4 - (85*a*c^4 + 4*(5*a^3*c^4*x - 18*a^2*c^4)* 
x)*x)*x)*x)
 

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.12 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=\frac {2\,c^4\,x^2\,\sqrt {1-a^2\,x^2}}{15}-\frac {11\,c^4\,\sqrt {1-a^2\,x^2}}{15\,a^2}+\frac {7\,c^4\,x\,\sqrt {1-a^2\,x^2}}{16\,a}-\frac {17\,a\,c^4\,x^3\,\sqrt {1-a^2\,x^2}}{24}-\frac {7\,c^4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{16\,a\,\sqrt {-a^2}}+\frac {3\,a^2\,c^4\,x^4\,\sqrt {1-a^2\,x^2}}{5}-\frac {a^3\,c^4\,x^5\,\sqrt {1-a^2\,x^2}}{6} \] Input:

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

Output:

(2*c^4*x^2*(1 - a^2*x^2)^(1/2))/15 - (11*c^4*(1 - a^2*x^2)^(1/2))/(15*a^2) 
 + (7*c^4*x*(1 - a^2*x^2)^(1/2))/(16*a) - (17*a*c^4*x^3*(1 - a^2*x^2)^(1/2 
))/24 - (7*c^4*asinh(x*(-a^2)^(1/2)))/(16*a*(-a^2)^(1/2)) + (3*a^2*c^4*x^4 
*(1 - a^2*x^2)^(1/2))/5 - (a^3*c^4*x^5*(1 - a^2*x^2)^(1/2))/6
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.87 \[ \int e^{\text {arctanh}(a x)} x (c-a c x)^4 \, dx=\frac {c^{4} \left (-105 \mathit {asin} \left (a x \right )-40 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+144 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}-170 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+32 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+105 \sqrt {-a^{2} x^{2}+1}\, a x -176 \sqrt {-a^{2} x^{2}+1}+176\right )}{240 a^{2}} \] Input:

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

Output:

(c**4*( - 105*asin(a*x) - 40*sqrt( - a**2*x**2 + 1)*a**5*x**5 + 144*sqrt( 
- a**2*x**2 + 1)*a**4*x**4 - 170*sqrt( - a**2*x**2 + 1)*a**3*x**3 + 32*sqr 
t( - a**2*x**2 + 1)*a**2*x**2 + 105*sqrt( - a**2*x**2 + 1)*a*x - 176*sqrt( 
 - a**2*x**2 + 1) + 176))/(240*a**2)