\(\int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx\) [333]

Optimal result

Integrand size = 23, antiderivative size = 310 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {1576 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{315 a^4 \sqrt {1-\frac {1}{a x}}}+\frac {472 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{315 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {92 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {38 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{63 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{a x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{a x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a^4 \sqrt {1-\frac {1}{a x}}} \] Output:

1576/315*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^4/(1-1/a/x)^(1/2)+472/315*(1+1 
/a/x)^(1/2)*x*(-a*c*x+c)^(1/2)/a^3/(1-1/a/x)^(1/2)+92/105*(1+1/a/x)^(1/2)* 
x^2*(-a*c*x+c)^(1/2)/a^2/(1-1/a/x)^(1/2)+38/63*(1+1/a/x)^(1/2)*x^3*(-a*c*x 
+c)^(1/2)/a/(1-1/a/x)^(1/2)+2/9*(1+1/a/x)^(1/2)*x^4*(-a*c*x+c)^(1/2)/(1-1/ 
a/x)^(1/2)-4*2^(1/2)*(1/a/x)^(1/2)*(-a*c*x+c)^(1/2)*arctanh(2^(1/2)*(1/a/x 
)^(1/2)/(1+1/a/x)^(1/2))/a^4/(1-1/a/x)^(1/2)
 

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.42 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} \left (788+236 a x+138 a^2 x^2+95 a^3 x^3+35 a^4 x^4\right )-630 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{315 a^{9/2} \sqrt {1-\frac {1}{a x}}} \] Input:

Integrate[E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x],x]
 

Output:

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(788 + 236*a*x + 138*a^2*x^2 
 + 95*a^3*x^3 + 35*a^4*x^4) - 630*Sqrt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sq 
rt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(315*a^(9/2)*Sqrt[1 - 1/(a*x)])
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {6730, 27, 109, 27, 169, 27, 169, 25, 169, 27, 169, 27, 104, 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.

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

\(\Big \downarrow \) 6730

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (-\frac {2 \int -\frac {19 a+\frac {17}{x}}{2 a \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{9 a}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\int \frac {19 a+\frac {17}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{9/2}}d\frac {1}{x}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {-\frac {2 \int -\frac {3 \left (23 a+\frac {19}{x}\right )}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \int \frac {23 a+\frac {19}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{7/2}}d\frac {1}{x}}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (-\frac {2 \int -\frac {59 a+\frac {46}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \int \frac {59 a+\frac {46}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{5/2}}d\frac {1}{x}}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \left (-\frac {2 \int -\frac {197 a+\frac {118}{x}}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \left (\frac {\int \frac {197 a+\frac {118}{x}}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \left (\frac {1}{x}\right )^{3/2}}d\frac {1}{x}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 169

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \left (\frac {-\frac {2 \int -\frac {315 a}{2 \left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}}{a}-\frac {394 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \left (\frac {315 \int \frac {1}{\left (a-\frac {1}{x}\right ) \sqrt {1+\frac {1}{a x}} \sqrt {\frac {1}{x}}}d\frac {1}{x}-\frac {394 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \sqrt {c-a c x} \left (\frac {\frac {6 \left (\frac {2 \left (\frac {630 \int \frac {1}{a-\frac {2}{x^2}}d\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}-\frac {394 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right )}{\sqrt {1-\frac {1}{a x}}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {a \sqrt {\frac {1}{x}} \left (\frac {\frac {6 \left (\frac {2 \left (\frac {\frac {315 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right )}{\sqrt {a}}-\frac {394 \sqrt {\frac {1}{a x}+1}}{\sqrt {\frac {1}{x}}}}{3 a}-\frac {118 \sqrt {\frac {1}{a x}+1}}{3 \left (\frac {1}{x}\right )^{3/2}}\right )}{5 a}-\frac {46 \sqrt {\frac {1}{a x}+1}}{5 \left (\frac {1}{x}\right )^{5/2}}\right )}{7 a}-\frac {38 \sqrt {\frac {1}{a x}+1}}{7 \left (\frac {1}{x}\right )^{7/2}}}{9 a^2}-\frac {2 \sqrt {\frac {1}{a x}+1}}{9 a \left (\frac {1}{x}\right )^{9/2}}\right ) \sqrt {c-a c x}}{\sqrt {1-\frac {1}{a x}}}\)

Input:

Int[E^(3*ArcCoth[a*x])*x^3*Sqrt[c - a*c*x],x]
 

Output:

-((a*Sqrt[x^(-1)]*Sqrt[c - a*c*x]*((-2*Sqrt[1 + 1/(a*x)])/(9*a*(x^(-1))^(9 
/2)) + ((-38*Sqrt[1 + 1/(a*x)])/(7*(x^(-1))^(7/2)) + (6*((-46*Sqrt[1 + 1/( 
a*x)])/(5*(x^(-1))^(5/2)) + (2*((-118*Sqrt[1 + 1/(a*x)])/(3*(x^(-1))^(3/2) 
) + ((-394*Sqrt[1 + 1/(a*x)])/Sqrt[x^(-1)] + (315*Sqrt[2]*ArcTanh[(Sqrt[2] 
*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])])/Sqrt[a])/(3*a)))/(5*a)))/(7*a 
))/(9*a^2)))/Sqrt[1 - 1/(a*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 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 109

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

rule 169

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 
2*m, 2*n, 2*p]
 

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 6730

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.47

method	result	size
risch	\(-\frac {2 \left (35 a^{4} x^{4}+95 a^{3} x^{3}+138 a^{2} x^{2}+236 a x +788\right ) c \left (a x -1\right )}{315 a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a^{4} \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \sqrt {-c \left (a x -1\right )}}\)	\(146\)
default	\(-\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-35 a^{4} x^{4} \sqrt {-c \left (a x +1\right )}-95 a^{3} x^{3} \sqrt {-c \left (a x +1\right )}-138 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}+630 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-236 a x \sqrt {-c \left (a x +1\right )}-788 \sqrt {-c \left (a x +1\right )}\right )}{315 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a^{4}}\)	\(161\)

Input:

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

Output:

-2/315*(35*a^4*x^4+95*a^3*x^3+138*a^2*x^2+236*a*x+788)/a^4*c/((a*x-1)/(a*x 
+1))^(1/2)/(-c*(a*x-1))^(1/2)*(a*x-1)-4/a^4*2^(1/2)*c^(1/2)*arctan(1/2*(-a 
*c*x-c)^(1/2)*2^(1/2)/c^(1/2))/((a*x-1)/(a*x+1))^(1/2)/(a*x+1)*(-c*(a*x+1) 
)^(1/2)/(-c*(a*x-1))^(1/2)*(a*x-1)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.00 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}, -\frac {2 \, {\left (630 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{2 \, {\left (a c x - c\right )}}\right ) - {\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}\right ] \] Input:

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

Output:

[2/315*(315*sqrt(2)*(a*x - 1)*sqrt(-c)*log(-(a^2*c*x^2 + 2*a*c*x + 2*sqrt( 
2)*sqrt(-a*c*x + c)*(a*x + 1)*sqrt(-c)*sqrt((a*x - 1)/(a*x + 1)) - 3*c)/(a 
^2*x^2 - 2*a*x + 1)) + (35*a^5*x^5 + 130*a^4*x^4 + 233*a^3*x^3 + 374*a^2*x 
^2 + 1024*a*x + 788)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*x - 
a^4), -2/315*(630*sqrt(2)*(a*x - 1)*sqrt(c)*arctan(1/2*sqrt(2)*sqrt(-a*c*x 
 + c)*(a*x + 1)*sqrt(c)*sqrt((a*x - 1)/(a*x + 1))/(a*c*x - c)) - (35*a^5*x 
^5 + 130*a^4*x^4 + 233*a^3*x^3 + 374*a^2*x^2 + 1024*a*x + 788)*sqrt(-a*c*x 
 + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^5*x - a^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.30 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c}\, i \left (-35 \sqrt {a x +1}\, a^{4} x^{4}-95 \sqrt {a x +1}\, a^{3} x^{3}-138 \sqrt {a x +1}\, a^{2} x^{2}-236 \sqrt {a x +1}\, a x -788 \sqrt {a x +1}-630 \sqrt {2}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-a x +1}}{\sqrt {2}}\right )}{2}\right )\right )+1292 \sqrt {2}\right )}{315 a^{4}} \] Input:

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

Output:

(2*sqrt(c)*i*( - 35*sqrt(a*x + 1)*a**4*x**4 - 95*sqrt(a*x + 1)*a**3*x**3 - 
 138*sqrt(a*x + 1)*a**2*x**2 - 236*sqrt(a*x + 1)*a*x - 788*sqrt(a*x + 1) - 
 630*sqrt(2)*log(tan(asin(sqrt( - a*x + 1)/sqrt(2))/2)) + 1292*sqrt(2)))/( 
315*a**4)