3.410 \(\int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx\)

Optimal. Leaf size=259 \[ -\frac{107 a^2 \sqrt{a x+1} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{a x+1} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}-\frac{363 a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{64 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}-\frac{17 a \sqrt{a x+1} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{\sqrt{a x+1} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}} \]

[Out]

-(Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(4*c*x^4*(1 - a*x)^(3/2)) - (17*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(24*c*x^
3*(1 - a*x)^(3/2)) - (107*a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(96*c*x^2*(1 - a*x)^(3/2)) - (149*a^3*Sqrt[1 +
a*x]*(c - a*c*x)^(3/2))/(64*c*x*(1 - a*x)^(3/2)) - (363*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(64*c*(1
- a*x)^(3/2)) + (4*Sqrt[2]*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))

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Rubi [A]  time = 0.191846, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {6130, 23, 98, 151, 156, 63, 208} \[ -\frac{107 a^2 \sqrt{a x+1} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{a x+1} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}-\frac{363 a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{a x+1}\right )}{64 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}-\frac{17 a \sqrt{a x+1} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{\sqrt{a x+1} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^5,x]

[Out]

-(Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(4*c*x^4*(1 - a*x)^(3/2)) - (17*a*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(24*c*x^
3*(1 - a*x)^(3/2)) - (107*a^2*Sqrt[1 + a*x]*(c - a*c*x)^(3/2))/(96*c*x^2*(1 - a*x)^(3/2)) - (149*a^3*Sqrt[1 +
a*x]*(c - a*c*x)^(3/2))/(64*c*x*(1 - a*x)^(3/2)) - (363*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]])/(64*c*(1
- a*x)^(3/2)) + (4*Sqrt[2]*a^4*(c - a*c*x)^(3/2)*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]])/(c*(1 - a*x)^(3/2))

Rule 6130

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

Rule 23

Int[(u_.)*((a_) + (b_.)*(v_))^(m_)*((c_) + (d_.)*(v_))^(n_), x_Symbol] :> Dist[(a + b*v)^m/(c + d*v)^m, Int[u*
(c + d*v)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] &&  !(IntegerQ[m] || IntegerQ[n
] || GtQ[b/d, 0])

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 151

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] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{e^{3 \tanh ^{-1}(a x)} \sqrt{c-a c x}}{x^5} \, dx &=\int \frac{(1+a x)^{3/2} \sqrt{c-a c x}}{x^5 (1-a x)^{3/2}} \, dx\\ &=\frac{(c-a c x)^{3/2} \int \frac{(1+a x)^{3/2}}{x^5 (c-a c x)} \, dx}{(1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{(c-a c x)^{3/2} \int \frac{-\frac{17 a c}{2}-\frac{15}{2} a^2 c x}{x^4 \sqrt{1+a x} (c-a c x)} \, dx}{4 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{\frac{107 a^2 c^2}{4}+\frac{85}{4} a^3 c^2 x}{x^3 \sqrt{1+a x} (c-a c x)} \, dx}{12 c^2 (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{107 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{(c-a c x)^{3/2} \int \frac{-\frac{447}{8} a^3 c^3-\frac{321}{8} a^4 c^3 x}{x^2 \sqrt{1+a x} (c-a c x)} \, dx}{24 c^3 (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{107 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{1+a x} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}+\frac{(c-a c x)^{3/2} \int \frac{\frac{1089 a^4 c^4}{16}+\frac{447}{16} a^5 c^4 x}{x \sqrt{1+a x} (c-a c x)} \, dx}{24 c^4 (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{107 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{1+a x} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}+\frac{\left (4 a^5 (c-a c x)^{3/2}\right ) \int \frac{1}{\sqrt{1+a x} (c-a c x)} \, dx}{(1-a x)^{3/2}}+\frac{\left (363 a^4 (c-a c x)^{3/2}\right ) \int \frac{1}{x \sqrt{1+a x}} \, dx}{128 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{107 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{1+a x} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}+\frac{\left (8 a^4 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 c-c x^2} \, dx,x,\sqrt{1+a x}\right )}{(1-a x)^{3/2}}+\frac{\left (363 a^3 (c-a c x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a}+\frac{x^2}{a}} \, dx,x,\sqrt{1+a x}\right )}{64 c (1-a x)^{3/2}}\\ &=-\frac{\sqrt{1+a x} (c-a c x)^{3/2}}{4 c x^4 (1-a x)^{3/2}}-\frac{17 a \sqrt{1+a x} (c-a c x)^{3/2}}{24 c x^3 (1-a x)^{3/2}}-\frac{107 a^2 \sqrt{1+a x} (c-a c x)^{3/2}}{96 c x^2 (1-a x)^{3/2}}-\frac{149 a^3 \sqrt{1+a x} (c-a c x)^{3/2}}{64 c x (1-a x)^{3/2}}-\frac{363 a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\sqrt{1+a x}\right )}{64 c (1-a x)^{3/2}}+\frac{4 \sqrt{2} a^4 (c-a c x)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+a x}}{\sqrt{2}}\right )}{c (1-a x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0747908, size = 108, normalized size = 0.42 \[ -\frac{\sqrt{c-a c x} \left (\sqrt{a x+1} \left (447 a^3 x^3+214 a^2 x^2+136 a x+48\right )+1089 a^4 x^4 \tanh ^{-1}\left (\sqrt{a x+1}\right )-768 \sqrt{2} a^4 x^4 \tanh ^{-1}\left (\frac{\sqrt{a x+1}}{\sqrt{2}}\right )\right )}{192 x^4 \sqrt{1-a x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*ArcTanh[a*x])*Sqrt[c - a*c*x])/x^5,x]

[Out]

-(Sqrt[c - a*c*x]*(Sqrt[1 + a*x]*(48 + 136*a*x + 214*a^2*x^2 + 447*a^3*x^3) + 1089*a^4*x^4*ArcTanh[Sqrt[1 + a*
x]] - 768*Sqrt[2]*a^4*x^4*ArcTanh[Sqrt[1 + a*x]/Sqrt[2]]))/(192*x^4*Sqrt[1 - a*x])

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Maple [A]  time = 0.109, size = 171, normalized size = 0.7 \begin{align*} -{\frac{1}{ \left ( 192\,ax-192 \right ){x}^{4}}\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ( ax-1 \right ) } \left ( 768\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( ax+1 \right ) }\sqrt{2}}{\sqrt{c}}} \right ) c{a}^{4}{x}^{4}-1089\,c{\it Artanh} \left ({\frac{\sqrt{c \left ( ax+1 \right ) }}{\sqrt{c}}} \right ){x}^{4}{a}^{4}-447\,{x}^{3}{a}^{3}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-214\,{x}^{2}{a}^{2}\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-136\,xa\sqrt{c \left ( ax+1 \right ) }\sqrt{c}-48\,\sqrt{c \left ( ax+1 \right ) }\sqrt{c} \right ){\frac{1}{\sqrt{c \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^5,x)

[Out]

-1/192*(-a^2*x^2+1)^(1/2)*(-c*(a*x-1))^(1/2)*(768*2^(1/2)*arctanh(1/2*(c*(a*x+1))^(1/2)*2^(1/2)/c^(1/2))*c*a^4
*x^4-1089*c*arctanh((c*(a*x+1))^(1/2)/c^(1/2))*x^4*a^4-447*x^3*a^3*(c*(a*x+1))^(1/2)*c^(1/2)-214*x^2*a^2*(c*(a
*x+1))^(1/2)*c^(1/2)-136*x*a*(c*(a*x+1))^(1/2)*c^(1/2)-48*(c*(a*x+1))^(1/2)*c^(1/2))/c^(1/2)/(a*x-1)/(c*(a*x+1
))^(1/2)/x^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a c x + c}{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(-a*c*x + c)*(a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*x^5), x)

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Fricas [A]  time = 2.13876, size = 967, normalized size = 3.73 \begin{align*} \left [\frac{768 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + 2 \, a c x - 2 \, \sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 3 \, c}{a^{2} x^{2} - 2 \, a x + 1}\right ) + 1089 \,{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{c} \log \left (-\frac{a^{2} c x^{2} + a c x + 2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{c} - 2 \, c}{a x^{2} - x}\right ) + 2 \,{\left (447 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 136 \, a x + 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{384 \,{\left (a x^{5} - x^{4}\right )}}, \frac{768 \, \sqrt{2}{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{2} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) - 1089 \,{\left (a^{5} x^{5} - a^{4} x^{4}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c} \sqrt{-c}}{a^{2} c x^{2} - c}\right ) +{\left (447 \, a^{3} x^{3} + 214 \, a^{2} x^{2} + 136 \, a x + 48\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-a c x + c}}{192 \,{\left (a x^{5} - x^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^5,x, algorithm="fricas")

[Out]

[1/384*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + 2*a*c*x - 2*sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(
-a*c*x + c)*sqrt(c) - 3*c)/(a^2*x^2 - 2*a*x + 1)) + 1089*(a^5*x^5 - a^4*x^4)*sqrt(c)*log(-(a^2*c*x^2 + a*c*x +
2*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(c) - 2*c)/(a*x^2 - x)) + 2*(447*a^3*x^3 + 214*a^2*x^2 + 136*a*x +
48)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^5 - x^4), 1/192*(768*sqrt(2)*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arctan
(sqrt(2)*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) - 1089*(a^5*x^5 - a^4*x^4)*sqrt(-c)*arc
tan(sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c)*sqrt(-c)/(a^2*c*x^2 - c)) + (447*a^3*x^3 + 214*a^2*x^2 + 136*a*x + 48)
*sqrt(-a^2*x^2 + 1)*sqrt(-a*c*x + c))/(a*x^5 - x^4)]

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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((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a*c*x+c)**(1/2)/x**5,x)

[Out]

Timed out

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Giac [A]  time = 1.41393, size = 279, normalized size = 1.08 \begin{align*} -\frac{1}{192} \, a^{4} c^{5}{\left (\frac{768 \, \sqrt{2} \arctan \left (\frac{\sqrt{2} \sqrt{a c x + c}}{2 \, \sqrt{-c}}\right )}{\sqrt{-c} c^{3}{\left | c \right |}} - \frac{1089 \, \arctan \left (\frac{\sqrt{a c x + c}}{\sqrt{-c}}\right )}{\sqrt{-c} c^{3}{\left | c \right |}} + \frac{447 \,{\left (a c x + c\right )}^{\frac{7}{2}} - 1127 \,{\left (a c x + c\right )}^{\frac{5}{2}} c + 1049 \,{\left (a c x + c\right )}^{\frac{3}{2}} c^{2} - 321 \, \sqrt{a c x + c} c^{3}}{a^{4} c^{7} x^{4}{\left | c \right |}}\right )} - \frac{\sqrt{2}{\left (1089 \, \sqrt{2} a^{4} c^{2} \arctan \left (\frac{\sqrt{2} \sqrt{c}}{\sqrt{-c}}\right ) - 1536 \, a^{4} c^{2} \arctan \left (\frac{\sqrt{c}}{\sqrt{-c}}\right ) - 1690 \, a^{4} \sqrt{-c} c^{\frac{3}{2}}\right )}}{384 \, \sqrt{-c}{\left | c \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a*c*x+c)^(1/2)/x^5,x, algorithm="giac")

[Out]

-1/192*a^4*c^5*(768*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c^3*abs(c)) - 1089*arctan(s
qrt(a*c*x + c)/sqrt(-c))/(sqrt(-c)*c^3*abs(c)) + (447*(a*c*x + c)^(7/2) - 1127*(a*c*x + c)^(5/2)*c + 1049*(a*c
*x + c)^(3/2)*c^2 - 321*sqrt(a*c*x + c)*c^3)/(a^4*c^7*x^4*abs(c))) - 1/384*sqrt(2)*(1089*sqrt(2)*a^4*c^2*arcta
n(sqrt(2)*sqrt(c)/sqrt(-c)) - 1536*a^4*c^2*arctan(sqrt(c)/sqrt(-c)) - 1690*a^4*sqrt(-c)*c^(3/2))/(sqrt(-c)*abs
(c))