\(\int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx\) [465]

Optimal result

Integrand size = 20, antiderivative size = 140 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 140, 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 = {101, 156, 162, 65, 214} \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=-\frac {\sqrt {b} (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)} \]

[In]

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

[Out]

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

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 101

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

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 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} \\ & = -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {\int \frac {-\frac {1}{2} (b c-a d) (4 b c-a d)-b d (b c-a d) x}{x (a+b x) \sqrt {c+d x}} \, dx}{a^2 (b c-a d)} \\ & = -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(b (4 b c-3 a d)) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 a^3}-\frac {(4 b c-a d) \int \frac {1}{x \sqrt {c+d x}} \, dx}{2 a^3} \\ & = -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(b (4 b c-3 a d)) \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}-\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 d} \\ & = -\frac {2 b \sqrt {c+d x}}{a^2 (a+b x)}-\frac {\sqrt {c+d x}}{a x (a+b x)}+\frac {(4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a^3 \sqrt {c}}-\frac {\sqrt {b} (4 b c-3 a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a^3 \sqrt {b c-a d}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\frac {-\frac {a (a+2 b x) \sqrt {c+d x}}{x (a+b x)}+\frac {\sqrt {b} (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{\sqrt {-b c+a d}}+\frac {(4 b c-a d) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{\sqrt {c}}}{a^3} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(\frac {4 x \left (b c -\frac {3 a d}{4}\right ) b \sqrt {c}\, \left (b x +a \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\sqrt {\left (a d -b c \right ) b}\, \left (x \left (b x +a \right ) \left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )+\sqrt {d x +c}\, \left (2 b x +a \right ) \sqrt {c}\, a \right )}{a^{3} \left (b x +a \right ) \sqrt {\left (a d -b c \right ) b}\, x \sqrt {c}}\)	\(133\)
derivativedivides	\(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 x}-\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}-\frac {b \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\)	\(136\)
default	\(2 d^{3} \left (\frac {-\frac {a \sqrt {d x +c}}{2 x}-\frac {\left (a d -4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{2 \sqrt {c}}}{a^{3} d^{3}}-\frac {b \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a^{3} d^{3}}\right )\)	\(136\)
risch	\(-\frac {\sqrt {d x +c}}{a^{2} x}-\frac {d \left (\frac {2 b \left (\frac {\sqrt {d x +c}\, a d}{2 \left (d x +c \right ) b +2 a d -2 b c}+\frac {\left (3 a d -4 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{a d}-\frac {\left (-a d +4 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a d \sqrt {c}}\right )}{a^{2}}\)	\(139\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 798, normalized size of antiderivative = 5.70 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\left [-\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {c} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {2 \, {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) + 2 \, {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{2 \, {\left (a^{3} b c x^{2} + a^{4} c x\right )}}, -\frac {{\left ({\left (4 \, b^{2} c^{2} - 3 \, a b c d\right )} x^{2} + {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) + {\left ({\left (4 \, b^{2} c - a b d\right )} x^{2} + {\left (4 \, a b c - a^{2} d\right )} x\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + {\left (2 \, a b c x + a^{2} c\right )} \sqrt {d x + c}}{a^{3} b c x^{2} + a^{4} c x}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((d*x+c)^(1/2)/x^2/(b*x+a)^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 = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\frac {{\left (4 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a^{3}} - \frac {{\left (4 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{a^{3} \sqrt {-c}} - \frac {2 \, {\left (d x + c\right )}^{\frac {3}{2}} b d - 2 \, \sqrt {d x + c} b c d + \sqrt {d x + c} a d^{2}}{{\left ({\left (d x + c\right )}^{2} b - 2 \, {\left (d x + c\right )} b c + b c^{2} + {\left (d x + c\right )} a d - a c d\right )} a^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 1175, normalized size of antiderivative = 8.39 \[ \int \frac {\sqrt {c+d x}}{x^2 (a+b x)^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan((((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*
c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b
^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2
*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a^3*b*c)) + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(
5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2
*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c
+ d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)))*1i)/(2*(a^4*d - a^3*b*c)))/((4*(3*a^2*b^3*d^5 +
16*b^5*c^2*d^3 - 16*a*b^4*c*d^4))/a^6 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^
3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 - ((-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4
*a^6*b^3*c*d^3))/a^6 - (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(
1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c)) + ((-b*(a*d - b*c))^(1/2)*(3*a*d
- 4*b*c)*((4*(c + d*x)^(1/2)*(5*a^2*b^3*d^4 + 16*b^5*c^2*d^2 - 16*a*b^4*c*d^3))/a^4 + ((-b*(a*d - b*c))^(1/2)
*(3*a*d - 4*b*c)*((2*(2*a^7*b^2*d^4 - 4*a^6*b^3*c*d^3))/a^6 + (2*(2*a^7*b^2*d^3 - 4*a^6*b^3*c*d^2)*(-b*(a*d -
b*c))^(1/2)*(3*a*d - 4*b*c)*(c + d*x)^(1/2))/(a^4*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3*b*c))))/(2*(a^4*d - a^3
*b*c))))*(-b*(a*d - b*c))^(1/2)*(3*a*d - 4*b*c)*1i)/(a^4*d - a^3*b*c) - (atanh((8*b^3*d^5*(c + d*x)^(1/2))/(c^
(1/2)*(8*b^3*d^5 - (2*a*b^2*d^6)/c)) - (2*b^2*d^6*(c + d*x)^(1/2))/(c^(3/2)*((8*b^3*d^5)/a - (2*b^2*d^6)/c)))*
(a*d - 4*b*c))/(a^3*c^(1/2)) - ((2*b*d*(c + d*x)^(3/2))/a^2 + (d*(a*d - 2*b*c)*(c + d*x)^(1/2))/a^2)/((a*d - 2
*b*c)*(c + d*x) + b*(c + d*x)^2 + b*c^2 - a*c*d)