\(\int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx\) [311]

Optimal result

Integrand size = 23, antiderivative size = 261 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \sqrt {c-a c x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )}{a^{7/2} \sqrt {1-\frac {1}{a x}}} \]

[Out]

104/21*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^3/(1-1/a/x)^(1/2)+32/21*x*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a^2/(1-1/
a/x)^(1/2)+6/7*x^2*(1+1/a/x)^(1/2)*(-a*c*x+c)^(1/2)/a/(1-1/a/x)^(1/2)+2/7*x^3*(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
^(7/2)/(1-1/a/x)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6311, 6316, 100, 157, 12, 95, 212} \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {4 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {\frac {1}{a x}+1}}\right ) \sqrt {c-a c x}}{a^{7/2} \sqrt {1-\frac {1}{a x}}}+\frac {104 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 x \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {2 x^3 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}+\frac {6 x^2 \sqrt {\frac {1}{a x}+1} \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}} \]

[In]

Int[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]

[Out]

(104*Sqrt[1 + 1/(a*x)]*Sqrt[c - a*c*x])/(21*a^3*Sqrt[1 - 1/(a*x)]) + (32*Sqrt[1 + 1/(a*x)]*x*Sqrt[c - a*c*x])/
(21*a^2*Sqrt[1 - 1/(a*x)]) + (6*Sqrt[1 + 1/(a*x)]*x^2*Sqrt[c - a*c*x])/(7*a*Sqrt[1 - 1/(a*x)]) + (2*Sqrt[1 + 1
/(a*x)]*x^3*Sqrt[c - a*c*x])/(7*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^(7/2)*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^{5/2} \, 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^{9/2} \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^3 \sqrt {c-a c x}}{7 \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 {15}{2 a}-\frac {13 x}{2 a^2}}{x^{7/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{7 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \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 {20}{a^2}+\frac {15 x}{a^3}}{x^{5/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{35 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \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 {65}{2 a^3}-\frac {20 x}{a^4}}{x^{3/2} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \sqrt {1-\frac {1}{a x}}}-\frac {\left (16 \sqrt {\frac {1}{x}} \sqrt {c-a c x}\right ) \text {Subst}\left (\int \frac {105}{4 a^4 \sqrt {x} \left (1-\frac {x}{a}\right ) \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \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^4 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \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^4 \sqrt {1-\frac {1}{a x}}} \\ & = \frac {104 \sqrt {1+\frac {1}{a x}} \sqrt {c-a c x}}{21 a^3 \sqrt {1-\frac {1}{a x}}}+\frac {32 \sqrt {1+\frac {1}{a x}} x \sqrt {c-a c x}}{21 a^2 \sqrt {1-\frac {1}{a x}}}+\frac {6 \sqrt {1+\frac {1}{a x}} x^2 \sqrt {c-a c x}}{7 a \sqrt {1-\frac {1}{a x}}}+\frac {2 \sqrt {1+\frac {1}{a x}} x^3 \sqrt {c-a c x}}{7 \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^{7/2} \sqrt {1-\frac {1}{a x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.47 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\frac {2 \sqrt {c-a c x} \left (\sqrt {a} \sqrt {1+\frac {1}{a x}} \left (52+16 a x+9 a^2 x^2+3 a^3 x^3\right )-42 \sqrt {2} \sqrt {\frac {1}{x}} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {\frac {1}{x}}}{\sqrt {a} \sqrt {1+\frac {1}{a x}}}\right )\right )}{21 a^{7/2} \sqrt {1-\frac {1}{a x}}} \]

[In]

Integrate[E^(3*ArcCoth[a*x])*x^2*Sqrt[c - a*c*x],x]

[Out]

(2*Sqrt[c - a*c*x]*(Sqrt[a]*Sqrt[1 + 1/(a*x)]*(52 + 16*a*x + 9*a^2*x^2 + 3*a^3*x^3) - 42*Sqrt[2]*Sqrt[x^(-1)]*
ArcTanh[(Sqrt[2]*Sqrt[x^(-1)])/(Sqrt[a]*Sqrt[1 + 1/(a*x)])]))/(21*a^(7/2)*Sqrt[1 - 1/(a*x)])

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.53

method	result	size
risch	\(-\frac {2 \left (3 a^{3} x^{3}+9 a^{2} x^{2}+16 a x +52\right ) c \left (a x -1\right )}{21 a^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}-\frac {4 \sqrt {2}\, \sqrt {c}\, \arctan \left (\frac {\sqrt {-a c x -c}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {-c \left (a x +1\right )}\, \left (a x -1\right )}{a^{3} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {-c \left (a x -1\right )}}\)	\(138\)
default	\(-\frac {2 \left (a x -1\right ) \sqrt {-c \left (a x -1\right )}\, \left (-3 a^{3} x^{3} \sqrt {-c \left (a x +1\right )}-9 a^{2} x^{2} \sqrt {-c \left (a x +1\right )}+42 \sqrt {c}\, \sqrt {2}\, \arctan \left (\frac {\sqrt {-c \left (a x +1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right )-16 a x \sqrt {-c \left (a x +1\right )}-52 \sqrt {-c \left (a x +1\right )}\right )}{21 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {-c \left (a x +1\right )}\, a^{3}}\)	\(143\)

[In]

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

[Out]

-2/21*(3*a^3*x^3+9*a^2*x^2+16*a*x+52)/a^3*c/((a*x-1)/(a*x+1))^(1/2)/(-c*(a*x-1))^(1/2)*(a*x-1)-4/a^3*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 = 0.27 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.10 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=\left [\frac {2 \, {\left (21 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {-c} \log \left (-\frac {a^{2} c x^{2} + 2 \, a c x + 2 \, \sqrt {2} \sqrt {-a c x + c} {\left (a x + 1\right )} \sqrt {-c} \sqrt {\frac {a x - 1}{a x + 1}} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + {\left (3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 68 \, a x + 52\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}, -\frac {2 \, {\left (42 \, \sqrt {2} {\left (a x - 1\right )} \sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x + c} \sqrt {c} \sqrt {\frac {a x - 1}{a x + 1}}}{a c x - c}\right ) - {\left (3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} + 25 \, a^{2} x^{2} + 68 \, a x + 52\right )} \sqrt {-a c x + c} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{21 \, {\left (a^{4} x - a^{3}\right )}}\right ] \]

[In]

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

[Out]

[2/21*(21*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)) + (3*a^4*x^4 + 12*a^3*x^3 + 25*a^2*x^2 + 68*a*x + 52)
*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^3), -2/21*(42*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)) - (3*a^4*x^4 + 12*a^3*x^3 + 25*a^2*x^2 + 68*a
*x + 52)*sqrt(-a*c*x + c)*sqrt((a*x - 1)/(a*x + 1)))/(a^4*x - a^3)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.56 \[ \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt {c-a c x} \, dx=-\frac {2 \, c^{2} {\left (\frac {2 \, \sqrt {2} {\left (21 \, \sqrt {c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {c}}\right ) - 40 \, \sqrt {-c}\right )}}{a^{2} c} - \frac {42 \, \sqrt {2} c^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {-a c x - c}}{2 \, \sqrt {c}}\right ) - 3 \, {\left (a c x + c\right )}^{3} \sqrt {-a c x - c} + 7 \, {\left (-a c x - c\right )}^{\frac {3}{2}} c^{2} - 42 \, \sqrt {-a c x - c} c^{3}}{a^{2} c^{4}}\right )}}{21 \, a {\left | c \right |} \mathrm {sgn}\left (a x + 1\right )} \]

[In]

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

[Out]

-2/21*c^2*(2*sqrt(2)*(21*sqrt(c)*arctan(sqrt(-c)/sqrt(c)) - 40*sqrt(-c))/(a^2*c) - (42*sqrt(2)*c^(7/2)*arctan(
1/2*sqrt(2)*sqrt(-a*c*x - c)/sqrt(c)) - 3*(a*c*x + c)^3*sqrt(-a*c*x - c) + 7*(-a*c*x - c)^(3/2)*c^2 - 42*sqrt(
-a*c*x - c)*c^3)/(a^2*c^4))/(a*abs(c)*sgn(a*x + 1))

Mupad [F(-1)]

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

[In]

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

[Out]

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