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

Optimal. Leaf size=261 \[ \frac{32 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{104 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{7/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{6 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 a \sqrt{1-\frac{1}{a x}}} \]

[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)])

________________________________________________________________________________________

Rubi [A]  time = 0.299959, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6176, 6181, 98, 152, 12, 93, 206} \[ \frac{32 x \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^2 \sqrt{1-\frac{1}{a x}}}+\frac{104 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{21 a^3 \sqrt{1-\frac{1}{a x}}}-\frac{4 \sqrt{2} \sqrt{\frac{1}{x}} \sqrt{c-a c x} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )}{a^{7/2} \sqrt{1-\frac{1}{a x}}}+\frac{2 x^3 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 \sqrt{1-\frac{1}{a x}}}+\frac{6 x^2 \sqrt{\frac{1}{a x}+1} \sqrt{c-a c x}}{7 a \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[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 6176

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 6181

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> -Dist[c^p*x^m*(1/x)^m, Sub
st[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]

Rule 98

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 152

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 12

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

Rule 93

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 206

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

Rubi steps

\begin{align*} \int e^{3 \coth ^{-1}(a x)} x^2 \sqrt{c-a c x} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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} \tanh ^{-1}\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]  time = 0.0844121, size = 122, normalized size = 0.47 \[ \frac{2 \sqrt{c-a c x} \left (\sqrt{a} \sqrt{\frac{1}{a x}+1} \left (3 a^3 x^3+9 a^2 x^2+16 a x+52\right )-42 \sqrt{2} \sqrt{\frac{1}{x}} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{1}{x}}}{\sqrt{a} \sqrt{\frac{1}{a x}+1}}\right )\right )}{21 a^{7/2} \sqrt{1-\frac{1}{a x}}} \]

Antiderivative was successfully verified.

[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]  time = 0.191, size = 143, normalized size = 0.6 \begin{align*} -{\frac{2\,ax-2}{ \left ( 21\,ax+21 \right ){a}^{3}}\sqrt{-c \left ( ax-1 \right ) } \left ( -3\,{x}^{3}{a}^{3}\sqrt{-c \left ( ax+1 \right ) }-9\,{x}^{2}{a}^{2}\sqrt{-c \left ( ax+1 \right ) }+42\,\sqrt{c}\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{-c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) -16\,xa\sqrt{-c \left ( ax+1 \right ) }-52\,\sqrt{-c \left ( ax+1 \right ) } \right ) \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-c \left ( ax+1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21/((a*x-1)/(a*x+1))^(3/2)*(a*x-1)/(a*x+1)*(-c*(a*x-1))^(1/2)*(-3*x^3*a^3*(-c*(a*x+1))^(1/2)-9*x^2*a^2*(-c*
(a*x+1))^(1/2)+42*c^(1/2)*2^(1/2)*arctan(1/2*(-c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))-16*x*a*(-c*(a*x+1))^(1/2)-52*
(-c*(a*x+1))^(1/2))/(-c*(a*x+1))^(1/2)/a^3

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c} x^{2}}{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 1.66304, size = 690, normalized size = 2.64 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [C]  time = 1.27086, size = 182, normalized size = 0.7 \begin{align*} -\frac{84 i \, \sqrt{2} \sqrt{-c} \arctan \left (-i\right ) - 160 \, \sqrt{2} \sqrt{-c}}{21 \, a^{3} \mathrm{sgn}\left (c\right )} - \frac{2 \,{\left (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}\right )}}{21 \, a^{3} c^{3} \mathrm{sgn}\left (-a c x - c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(84*I*sqrt(2)*sqrt(-c)*arctan(-I) - 160*sqrt(2)*sqrt(-c))/(a^3*sgn(c)) - 2/21*(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^3*c^3*sgn(-a*c*x - c))