\(\int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx\) [310]

Optimal result

Integrand size = 23, antiderivative size = 309 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {1576 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{315 a^4 \sqrt {1-\frac {1}{a x}}}+\frac {472 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{315 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {92 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {38 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{63 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^4 \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{a^{9/2} \sqrt {1-\frac {1}{a x}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6311, 6316, 100, 157, 12, 95, 212} \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^{9/2} \sqrt {1-\frac {1}{a x}}}+\frac {1576 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{315 a^4 \sqrt {1-\frac {1}{a x}}}+\frac {472 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{315 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {92 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{105 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^4 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{9 \sqrt {1-\frac {1}{a x}}}+\frac {38 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{63 a \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[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] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> 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] + Dist[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 157

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> 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] + Dist[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 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 6311

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

Rule 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[Int[(1 + d*(x/c))^p*((1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, m,
n, p}, x] && EqQ[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) &&  !IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c-a c x} \int e^{3 \coth ^{-1}(a x)} \sqrt {1-\frac {1}{a x}} x^{7/2} \, dx}{\sqrt {1-\frac {1}{a x}} \sqrt {x}} \\ & = -\frac {\left (\sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^{11/2} \left (1-\frac {x}{a}\right )} \, dx,x,\frac {1}{x}\right )}{\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 {\left (2 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {19}{2 a}-\frac {17 x}{2 a^2}}{x^{9/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{9 \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 {\left (4 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\frac {69}{2 a^2}+\frac {57 x}{2 a^3}}{x^{7/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{63 \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 {\left (8 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {-\frac {177}{2 a^3}-\frac {69 x}{a^4}}{x^{5/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{315 \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 {\left (16 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {\frac {591}{4 a^4}+\frac {177 x}{2 a^5}}{x^{3/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{945 \sqrt {1-\frac {1}{a x}}} \\ & = \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 {\left (32 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int -\frac {945}{8 a^5 \sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{945 \sqrt {1-\frac {1}{a x}}} \\ & = \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 {\left (4 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a^5 \sqrt {1-\frac {1}{a x}}} \\ & = \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 {\left (8 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 x^2}{a}} \, dx,x,\frac {\sqrt {\frac {1}{x}}}{\sqrt {1+\frac {1}{a x}}}\right )}{a^5 \sqrt {1-\frac {1}{a x}}} \\ & = \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}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{a^{9/2} \sqrt {1-\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.42 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} \left (788+236 a x+138 a^2 x^2+95 a^3 x^3+35 a^4 x^4\right )-630 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{315 a^{9/2} \sqrt {1-\frac {1}{a x}}} \]

[In]

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

[Out]

(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*Sq
rt[2]*Sqrt[x^(-1)]*ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(315*a^(9/2)*Sqrt[1 - 1/(a*x)
])

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.47

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.98 \[ \int e^{3 \coth ^{-1}(a x)} x^3 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (315 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}, -\frac {2 \, {\left (630 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (35 \, a^{5} x^{5} + 130 \, a^{4} x^{4} + 233 \, a^{3} x^{3} + 374 \, a^{2} x^{2} + 1024 \, a x + 788\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{315 \, {\left (a^{5} x - a^{4}\right )}}\right ] \]

[In]

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

[Out]

[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(sqrt(2)*sqrt(-a*c*x + c)*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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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