\(\int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx\) [476]

Optimal result

Integrand size = 20, antiderivative size = 164 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{3/2}} \]

[Out]

(a*d+4*b*c)*arctanh((d*x+c)^(1/2)/c^(1/2))/a^3/c^(3/2)-b^(3/2)*(-5*a*d+4*b*c)*arctanh(b^(1/2)*(d*x+c)^(1/2)/(-
a*d+b*c)^(1/2))/a^3/(-a*d+b*c)^(3/2)-b*(-a*d+2*b*c)*(d*x+c)^(1/2)/a^2/c/(-a*d+b*c)/(b*x+a)-(d*x+c)^(1/2)/a/c/x
/(b*x+a)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {105, 156, 162, 65, 214} \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=-\frac {b^{3/2} (4 b c-5 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{3/2}}+\frac {(a d+4 b c) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{3/2}}-\frac {b \sqrt {c+d x} (2 b c-a d)}{a^2 c (a+b x) (b c-a d)}-\frac {\sqrt {c+d x}}{a c x (a+b x)} \]

[In]

Int[1/(x^2*(a + b*x)^2*Sqrt[c + d*x]),x]

[Out]

-((b*(2*b*c - a*d)*Sqrt[c + d*x])/(a^2*c*(b*c - a*d)*(a + b*x))) - Sqrt[c + d*x]/(a*c*x*(a + b*x)) + ((4*b*c +
a*d)*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/(a^3*c^(3/2)) - (b^(3/2)*(4*b*c - 5*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x])
/Sqrt[b*c - a*d]])/(a^3*(b*c - a*d)^(3/2))

Rule 65

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 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 156

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] && ILtQ[m, -1]

Rule 162

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 214

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {c+d x}}{a c x (a+b x)}-\frac {\int \frac {\frac {1}{2} (4 b c+a d)+\frac {3 b d x}{2}}{x (a+b x)^2 \sqrt {c+d x}} \, dx}{a c} \\ & = -\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}-\frac {\int \frac {\frac {1}{2} (b c-a d) (4 b c+a d)+\frac {1}{2} b d (2 b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a^2 c (b c-a d)} \\ & = -\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}+\frac {\left (b^2 (4 b c-5 a d)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3 (b c-a d)}-\frac {(4 b c+a d) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3 c} \\ & = -\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}+\frac {\left (b^2 (4 b c-5 a d)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 d (b c-a d)}-\frac {(4 b c+a d) \text {Subst}\left (\int \frac {1}{-\frac {c}{d}+\frac {x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{a^3 c d} \\ & = -\frac {b (2 b c-a d) \sqrt {c+d x}}{a^2 c (b c-a d) (a+b x)}-\frac {\sqrt {c+d x}}{a c x (a+b x)}+\frac {(4 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 c^{3/2}}-\frac {b^{3/2} (4 b c-5 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 (b c-a d)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\frac {-\frac {a \sqrt {c+d x} \left (-a b c+a^2 d-2 b^2 c x+a b d x\right )}{c (-b c+a d) x (a+b x)}-\frac {b^{3/2} (4 b c-5 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}+\frac {(4 b c+a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{c^{3/2}}}{a^3} \]

[In]

Integrate[1/(x^2*(a + b*x)^2*Sqrt[c + d*x]),x]

[Out]

(-((a*Sqrt[c + d*x]*(-(a*b*c) + a^2*d - 2*b^2*c*x + a*b*d*x))/(c*(-(b*c) + a*d)*x*(a + b*x))) - (b^(3/2)*(4*b*
c - 5*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2) + ((4*b*c + a*d)*ArcTanh[S
qrt[c + d*x]/Sqrt[c]])/c^(3/2))/a^3

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{3}}+\frac {b^{2} \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\)	\(160\)
default	\(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 c x}+\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 c^{\frac {3}{2}}}}{a^{3} d^{3}}+\frac {b^{2} \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\)	\(160\)
risch	\(-\frac {\sqrt {d x +c}}{c \,a^{2} x}-\frac {d \left (-\frac {2 b^{2} c \left (\frac {a d \sqrt {d x +c}}{2 \left (a d -b c \right ) \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (5 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \left (a d -b c \right ) \sqrt {\left (a d -b c \right ) b}}\right )}{a d}-\frac {\left (a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{c \,a^{2}}\)	\(167\)
pseudoelliptic	\(-\frac {4 \left (x \,b^{2} c^{\frac {5}{2}} \left (b x +a \right ) \left (b c -\frac {5 a d}{4}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\frac {\left (c x \left (a d +4 b c \right ) \left (a d -b c \right ) \left (b x +a \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+c^{\frac {3}{2}} \left (2 b^{2} c x +a \left (-d x +c \right ) b -a^{2} d \right ) a \sqrt {d x +c}\right ) \sqrt {\left (a d -b c \right ) b}}{4}\right )}{\sqrt {\left (a d -b c \right ) b}\, a^{3} \left (a d -b c \right ) \left (b x +a \right ) c^{\frac {5}{2}} x}\)	\(171\)

[In]

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

[Out]

2*d^3*(1/a^3/d^3*(-1/2*a/c*(d*x+c)^(1/2)/x+1/2*(a*d+4*b*c)/c^(3/2)*arctanh((d*x+c)^(1/2)/c^(1/2)))+b^2/a^3/d^3
*(1/2*a*d/(a*d-b*c)*(d*x+c)^(1/2)/((d*x+c)*b+a*d-b*c)+1/2*(5*a*d-4*b*c)/(a*d-b*c)/((a*d-b*c)*b)^(1/2)*arctan(b
*(d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2))))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 1164, normalized size of antiderivative = 7.10 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/2*(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b
*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a)) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2
)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) - 2*(a^2
*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3 -
a^5*c^2*d)*x), -1/2*(2*((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x)*sqrt(-b/(b*c - a*d
))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) - ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d
^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(c)*log((d*x + 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*(a
^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3
- a^5*c^2*d)*x), -1/2*(2*((4*b^3*c^2 - 3*a*b^2*c*d - a^2*b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x
)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) - ((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)
*x)*sqrt(b/(b*c - a*d))*log((b*d*x + 2*b*c - a*d - 2*(b*c - a*d)*sqrt(d*x + c)*sqrt(b/(b*c - a*d)))/(b*x + a))
+ 2*(a^2*b*c^2 - a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^
4*b*c^3 - a^5*c^2*d)*x), -(((4*b^3*c^3 - 5*a*b^2*c^2*d)*x^2 + (4*a*b^2*c^3 - 5*a^2*b*c^2*d)*x)*sqrt(-b/(b*c -
a*d))*arctan(-(b*c - a*d)*sqrt(d*x + c)*sqrt(-b/(b*c - a*d))/(b*d*x + b*c)) + ((4*b^3*c^2 - 3*a*b^2*c*d - a^2*
b*d^2)*x^2 + (4*a*b^2*c^2 - 3*a^2*b*c*d - a^3*d^2)*x)*sqrt(-c)*arctan(sqrt(d*x + c)*sqrt(-c)/c) + (a^2*b*c^2 -
a^3*c*d + (2*a*b^2*c^2 - a^2*b*c*d)*x)*sqrt(d*x + c))/((a^3*b^2*c^3 - a^4*b*c^2*d)*x^2 + (a^4*b*c^3 - a^5*c^2
*d)*x)]

Sympy [F]

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

[In]

integrate(1/x**2/(b*x+a)**2/(d*x+c)**(1/2),x)

[Out]

Integral(1/(x**2*(a + b*x)**2*sqrt(c + d*x)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.44 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\frac {{\left (4 \, b^{3} c - 5 \, a b^{2} d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{{\left (a^{3} b c - a^{4} d\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{2} c d - 2 \, \sqrt {d x + c} b^{2} c^{2} d - {\left (d x + c\right )}^{\frac {3}{2}} a b d^{2} + 2 \, \sqrt {d x + c} a b c d^{2} - \sqrt {d x + c} a^{2} d^{3}}{{\left (a^{2} b c^{2} - a^{3} c d\right )} {\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )}} - \frac {{\left (4 \, b c + a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c} c} \]

[In]

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

[Out]

(4*b^3*c - 5*a*b^2*d)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/((a^3*b*c - a^4*d)*sqrt(-b^2*c + a*b*d)) -
(2*(d*x + c)^(3/2)*b^2*c*d - 2*sqrt(d*x + c)*b^2*c^2*d - (d*x + c)^(3/2)*a*b*d^2 + 2*sqrt(d*x + c)*a*b*c*d^2 -
sqrt(d*x + c)*a^2*d^3)/((a^2*b*c^2 - a^3*c*d)*((d*x + c)^2*b - 2*(d*x + c)*b*c + b*c^2 + (d*x + c)*a*d - a*c*
d)) - (4*b*c + a*d)*arctan(sqrt(d*x + c)/sqrt(-c))/(a^3*sqrt(-c)*c)

Mupad [B] (verification not implemented)

Time = 1.45 (sec) , antiderivative size = 3784, normalized size of antiderivative = 23.07 \[ \int \frac {1}{x^2 (a+b x)^2 \sqrt {c+d x}} \, dx=\text {Too large to display} \]

[In]

int(1/(x^2*(a + b*x)^2*(c + d*x)^(1/2)),x)

[Out]

(((c + d*x)^(1/2)*(a^2*d^3 + 2*b^2*c^2*d - 2*a*b*c*d^2))/(a^2*(b*c^2 - a*c*d)) + (b*(a*d^2 - 2*b*c*d)*(c + d*x
)^(3/2))/(a^2*(b*c^2 - a*c*d)))/((a*d - 2*b*c)*(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d) + (atan(((((2*(c + d
*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c
^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7
*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5
*a*d - 4*b*c)*(c + d*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5
))/((a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(5
*a*d - 4*b*c))/(2*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*
d - 4*b*c)*1i)/(2*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) + (((2*(c + d*x)^(1/2)*(a^4*b^3*d
^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2
*a^5*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8
*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(c + d
*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/((a^4*b^2*c^4 + a
^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(5*a*d - 4*b*c))/(2*(a
^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*1i)/(2*(a
^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))/((2*(5*a^3*b^4*d^6 + 32*b^7*c^3*d^3 - 48*a*b^6*c^2*d
^4 + 6*a^2*b^5*c*d^5))/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + (((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^
7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3
*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d
^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(c + d*x)^(1/2)*
(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/((a^4*b^2*c^4 + a^6*c^2*d^2
- 2*a^5*b*c^3*d)*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(5*a*d - 4*b*c))/(2*(a^6*d^3 - a
^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c))/(2*(a^6*d^3 - a^3*
b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)) - (((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^
3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) + ((-b^3*(a*d - b*c
)^3)^(1/2)*((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*
c^2*d^2 - 2*a^7*b*c^3*d) - ((-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*(c + d*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20
*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/((a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)*(a^6*
d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d - 3*a^5*b*c*d^2)))*(5*a*d - 4*b*c))/(2*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*
c^2*d - 3*a^5*b*c*d^2)))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c))/(2*(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2
*d - 3*a^5*b*c*d^2))))*(-b^3*(a*d - b*c)^3)^(1/2)*(5*a*d - 4*b*c)*1i)/(a^6*d^3 - a^3*b^3*c^3 + 3*a^4*b^2*c^2*d
- 3*a^5*b*c*d^2) + (atan((((a*d + 4*b*c)*((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3
+ 6*a^3*b^4*c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) - (((4*a^9*b^2*c*d^6 + 8
*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((a*d
+ 4*b*c)*(c + d*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/(
a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(2*a^3*(c^3)^(1/2)))*1i)/(2*a^3*(
c^3)^(1/2)) + ((a*d + 4*b*c)*((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*
c*d^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) + (((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*
d^3 - 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((a*d + 4*b*c)*(c
+ d*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/(a^3*(c^3)^(1/
2)*(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(2*a^3*(c^3)^(1/2)))*1i)/(2*a^3*(c^3)^(1/2)))/
((2*(5*a^3*b^4*d^6 + 32*b^7*c^3*d^3 - 48*a*b^6*c^2*d^4 + 6*a^2*b^5*c*d^5))/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*
b*c^3*d) + ((a*d + 4*b*c)*((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d
^5 + 26*a^2*b^5*c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) - (((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3
- 16*a^7*b^4*c^3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) + ((a*d + 4*b*c)*(c + d
*x)^(1/2)*(8*a^6*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/(a^3*(c^3)^(1/2)*
(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(2*a^3*(c^3)^(1/2))))/(2*a^3*(c^3)^(1/2)) - ((a*d
+ 4*b*c)*((2*(c + d*x)^(1/2)*(a^4*b^3*d^6 + 32*b^7*c^4*d^2 - 64*a*b^6*c^3*d^3 + 6*a^3*b^4*c*d^5 + 26*a^2*b^5*
c^2*d^4))/(a^4*b^2*c^4 + a^6*c^2*d^2 - 2*a^5*b*c^3*d) + (((4*a^9*b^2*c*d^6 + 8*a^6*b^5*c^4*d^3 - 16*a^7*b^4*c^
3*d^4 + 4*a^8*b^3*c^2*d^5)/(a^6*b^2*c^4 + a^8*c^2*d^2 - 2*a^7*b*c^3*d) - ((a*d + 4*b*c)*(c + d*x)^(1/2)*(8*a^6
*b^5*c^5*d^2 - 20*a^7*b^4*c^4*d^3 + 16*a^8*b^3*c^3*d^4 - 4*a^9*b^2*c^2*d^5))/(a^3*(c^3)^(1/2)*(a^4*b^2*c^4 + a
^6*c^2*d^2 - 2*a^5*b*c^3*d)))*(a*d + 4*b*c))/(2*a^3*(c^3)^(1/2))))/(2*a^3*(c^3)^(1/2))))*(a*d + 4*b*c)*1i)/(a^
3*(c^3)^(1/2))