3.5.10 \(\int \frac {1}{x^2 (\sqrt {a+b x}+\sqrt {c+b x})^2} \, dx\) [410]

Optimal. Leaf size=141 \[ -\frac {a+c}{(a-c)^2 x}+\frac {2 \sqrt {a+b x} \sqrt {c+b x}}{(a-c)^2 x}-\frac {4 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {c+b x}}\right )}{(a-c)^2}+\frac {2 b (a+c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+b x}}\right )}{\sqrt {a} (a-c)^2 \sqrt {c}}+\frac {2 b \log (x)}{(a-c)^2} \]

[Out]

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

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Rubi [A]
time = 0.16, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {6821, 99, 163, 65, 223, 212, 95, 214} \begin {gather*} \frac {2 \sqrt {a+b x} \sqrt {b x+c}}{x (a-c)^2}+\frac {2 b \log (x)}{(a-c)^2}+\frac {2 b (a+c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {b x+c}}\right )}{\sqrt {a} \sqrt {c} (a-c)^2}-\frac {4 b \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {b x+c}}\right )}{(a-c)^2}-\frac {a+c}{x (a-c)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + c)/((a - c)^2*x)) + (2*Sqrt[a + b*x]*Sqrt[c + b*x])/((a - c)^2*x) - (4*b*ArcTanh[Sqrt[a + b*x]/Sqrt[c +
b*x]])/(a - c)^2 + (2*b*(a + c)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + b*x])])/(Sqrt[a]*(a - c)^2*
Sqrt[c]) + (2*b*Log[x])/(a - c)^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 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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 163

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

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 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]

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 6821

Int[(u_.)*((e_.)*Sqrt[(a_.) + (b_.)*(x_)^(n_.)] + (f_.)*Sqrt[(c_.) + (d_.)*(x_)^(n_.)])^(m_), x_Symbol] :> Dis
t[(a*e^2 - c*f^2)^m, Int[ExpandIntegrand[u/(e*Sqrt[a + b*x^n] - f*Sqrt[c + d*x^n])^m, x], x], x] /; FreeQ[{a,
b, c, d, e, f, n}, x] && ILtQ[m, 0] && EqQ[b*e^2 - d*f^2, 0]

Rubi steps

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

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Mathematica [A]
time = 0.52, size = 128, normalized size = 0.91 \begin {gather*} \frac {\frac {2 b (a+c) \tanh ^{-1}\left (\frac {-b x+\sqrt {a+b x} \sqrt {c+b x}}{\sqrt {a} \sqrt {c}}\right )}{\sqrt {a} \sqrt {c}}-\frac {a+c-2 b x-2 \sqrt {a+b x} \sqrt {c+b x}-2 b x \log \left (b x \left (a+c+2 b x-2 \sqrt {a+b x} \sqrt {c+b x}\right )\right )}{x}}{(a-c)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2*b*(a + c)*ArcTanh[(-(b*x) + Sqrt[a + b*x]*Sqrt[c + b*x])/(Sqrt[a]*Sqrt[c])])/(Sqrt[a]*Sqrt[c]) - (a + c -
2*b*x - 2*Sqrt[a + b*x]*Sqrt[c + b*x] - 2*b*x*Log[b*x*(a + c + 2*b*x - 2*Sqrt[a + b*x]*Sqrt[c + b*x])])/x)/(a
- c)^2

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.02, size = 274, normalized size = 1.94

method	result	size
default	\(-\frac {a}{x \left (a -c \right )^{2}}-\frac {c}{x \left (a -c \right )^{2}}+\frac {2 b \ln \left (x \right )}{\left (a -c \right )^{2}}+\frac {\sqrt {b x +a}\, \sqrt {b x +c}\, \left (\mathrm {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x a b +\mathrm {csgn}\left (b \right ) \ln \left (\frac {a b x +b c x +2 \sqrt {a c}\, \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 a c}{x}\right ) x b c -2 \ln \left (\frac {\left (2 \,\mathrm {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}+2 b x +a +c \right ) \mathrm {csgn}\left (b \right )}{2}\right ) x b \sqrt {a c}+2 \,\mathrm {csgn}\left (b \right ) \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, \sqrt {a c}\right ) \mathrm {csgn}\left (b \right )}{\left (a -c \right )^{2} \sqrt {b^{2} x^{2}+a b x +b c x +a c}\, x \sqrt {a c}}\)	\(274\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x/(a-c)^2*a-1/x/(a-c)^2*c+2*b*ln(x)/(a-c)^2+1/(a-c)^2*(b*x+a)^(1/2)*(b*x+c)^(1/2)*(csgn(b)*ln((a*b*x+b*c*x+
2*(a*c)^(1/2)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x*a*b+csgn(b)*ln((a*b*x+b*c*x+2*(a*c)^(1/2)*(b^2*x^2+a
*b*x+b*c*x+a*c)^(1/2)+2*a*c)/x)*x*b*c-2*ln(1/2*(2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)+2*b*x+a+c)*csgn(b))*
x*b*(a*c)^(1/2)+2*csgn(b)*(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)*(a*c)^(1/2))*csgn(b)/(b^2*x^2+a*b*x+b*c*x+a*c)^(1/2)
/x/(a*c)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.38, size = 367, normalized size = 2.60 \begin {gather*} \left [\frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x + {\left (a b + b c\right )} \sqrt {a c} x \log \left (\frac {2 \, a^{2} c + 2 \, a c^{2} + 2 \, {\left (2 \, a c + \sqrt {a c} {\left (a + c\right )}\right )} \sqrt {b x + a} \sqrt {b x + c} + {\left (a^{2} b + 2 \, a b c + b c^{2}\right )} x + 2 \, {\left (2 \, a c + {\left (a b + b c\right )} x\right )} \sqrt {a c}}{x}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}, \frac {2 \, a b c x \log \left (-2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b x + c} - a - c\right ) + 2 \, a b c x \log \left (x\right ) + 2 \, a b c x - 2 \, {\left (a b + b c\right )} \sqrt {-a c} x \arctan \left (-\frac {\sqrt {-a c} b x - \sqrt {-a c} \sqrt {b x + a} \sqrt {b x + c}}{a c}\right ) + 2 \, \sqrt {b x + a} \sqrt {b x + c} a c - a^{2} c - a c^{2}}{{\left (a^{3} c - 2 \, a^{2} c^{2} + a c^{3}\right )} x}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(2*a*b*c*x*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c) + 2*a*b*c*x*log(x) + 2*a*b*c*x + (a*b + b*c)*s
qrt(a*c)*x*log((2*a^2*c + 2*a*c^2 + 2*(2*a*c + sqrt(a*c)*(a + c))*sqrt(b*x + a)*sqrt(b*x + c) + (a^2*b + 2*a*b
*c + b*c^2)*x + 2*(2*a*c + (a*b + b*c)*x)*sqrt(a*c))/x) + 2*sqrt(b*x + a)*sqrt(b*x + c)*a*c - a^2*c - a*c^2)/(
(a^3*c - 2*a^2*c^2 + a*c^3)*x), (2*a*b*c*x*log(-2*b*x + 2*sqrt(b*x + a)*sqrt(b*x + c) - a - c) + 2*a*b*c*x*log
(x) + 2*a*b*c*x - 2*(a*b + b*c)*sqrt(-a*c)*x*arctan(-(sqrt(-a*c)*b*x - sqrt(-a*c)*sqrt(b*x + a)*sqrt(b*x + c))
/(a*c)) + 2*sqrt(b*x + a)*sqrt(b*x + c)*a*c - a^2*c - a*c^2)/((a^3*c - 2*a^2*c^2 + a*c^3)*x)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{2} \left (\sqrt {a + b x} + \sqrt {b x + c}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (121) = 242\).
time = 5.04, size = 311, normalized size = 2.21 \begin {gather*} \frac {2 \, b \log \left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, b \log \left ({\left | b x \right |}\right )}{a^{2} - 2 \, a c + c^{2}} + \frac {2 \, {\left (a b + b c\right )} \arctan \left (\frac {{\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a - c}{2 \, \sqrt {-a c}}\right )}{{\left (a^{2} - 2 \, a c + c^{2}\right )} \sqrt {-a c}} - \frac {4 \, {\left (a b {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + b c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - a^{2} b + 2 \, a b c - b c^{2}\right )}}{{\left ({\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{4} - 2 \, a {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} - 2 \, c {\left (\sqrt {b x + a} - \sqrt {b x + c}\right )}^{2} + a^{2} - 2 \, a c + c^{2}\right )} {\left (a^{2} - 2 \, a c + c^{2}\right )}} - \frac {2 \, {\left (b x + a\right )} b - a b + b c}{{\left (a^{2} - 2 \, a c + c^{2}\right )} b x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*log((sqrt(b*x + a) - sqrt(b*x + c))^2)/(a^2 - 2*a*c + c^2) + 2*b*log(abs(b*x))/(a^2 - 2*a*c + c^2) + 2*(a*
b + b*c)*arctan(1/2*((sqrt(b*x + a) - sqrt(b*x + c))^2 - a - c)/sqrt(-a*c))/((a^2 - 2*a*c + c^2)*sqrt(-a*c)) -
4*(a*b*(sqrt(b*x + a) - sqrt(b*x + c))^2 + b*c*(sqrt(b*x + a) - sqrt(b*x + c))^2 - a^2*b + 2*a*b*c - b*c^2)/(
((sqrt(b*x + a) - sqrt(b*x + c))^4 - 2*a*(sqrt(b*x + a) - sqrt(b*x + c))^2 - 2*c*(sqrt(b*x + a) - sqrt(b*x + c
))^2 + a^2 - 2*a*c + c^2)*(a^2 - 2*a*c + c^2)) - (2*(b*x + a)*b - a*b + b*c)/((a^2 - 2*a*c + c^2)*b*x)

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Mupad [B]
time = 28.82, size = 2500, normalized size = 17.73 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*b*log(x))/(a^2 - 2*a*c + c^2) - ((((a + b*x)^(1/2) - a^(1/2))^2*((a^2*b)/2 + (b*c^2)/2 - (3*a*b*c)/2))/(((c
+ b*x)^(1/2) - c^(1/2))^2*(a*c^3 + a^3*c - 2*a^2*c^2)) - b/(2*(a^2 - 2*a*c + c^2)) + (a^(1/2)*c^(1/2)*((a*b)/
2 + (b*c)/2)*((a + b*x)^(1/2) - a^(1/2)))/(((c + b*x)^(1/2) - c^(1/2))*(a*c^3 + a^3*c - 2*a^2*c^2)))/(((a + b*
x)^(1/2) - a^(1/2))/((c + b*x)^(1/2) - c^(1/2)) + ((a + b*x)^(1/2) - a^(1/2))^3/((c + b*x)^(1/2) - c^(1/2))^3
- ((a + c)*((a + b*x)^(1/2) - a^(1/2))^2)/(a^(1/2)*c^(1/2)*((c + b*x)^(1/2) - c^(1/2))^2)) + (b*atan(((b*((4*(
4*a^4*b^3*c^12 + 8*a^5*b^3*c^11 - 32*a^6*b^3*c^10 - 8*a^7*b^3*c^9 + 56*a^8*b^3*c^8 - 8*a^9*b^3*c^7 - 32*a^10*b
^3*c^6 + 8*a^11*b^3*c^5 + 4*a^12*b^3*c^4))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11
- 56*a^12*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15*c^7) + (4*b*((4*b*((4*(16*a^6*b*c^14 - 4*a^5*b*c^15 + 12*a^7*
b*c^13 - 192*a^8*b*c^12 + 504*a^9*b*c^11 - 672*a^10*b*c^10 + 504*a^11*b*c^9 - 192*a^12*b*c^8 + 12*a^13*b*c^7 +
16*a^14*b*c^6 - 4*a^15*b*c^5))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c
^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15*c^7) + (4*b*((4*(a^(9/2)*c^(35/2) - 8*a^(11/2)*c^(33/2) + 27*a^(13/2)*c^
(31/2) - 49*a^(15/2)*c^(29/2) + 50*a^(17/2)*c^(27/2) - 27*a^(19/2)*c^(25/2) + 6*a^(21/2)*c^(23/2) + 6*a^(23/2)
*c^(21/2) - 27*a^(25/2)*c^(19/2) + 50*a^(27/2)*c^(17/2) - 49*a^(29/2)*c^(15/2) + 27*a^(31/2)*c^(13/2) - 8*a^(3
3/2)*c^(11/2) + a^(35/2)*c^(9/2)))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^1
2*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15*c^7) - (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^4*c^18 - 47*a^5*c^17 + 268
*a^6*c^16 - 982*a^7*c^15 + 2564*a^8*c^14 - 4993*a^9*c^13 + 7404*a^10*c^12 - 8436*a^11*c^11 + 7404*a^12*c^10 -
4993*a^13*c^9 + 2564*a^14*c^8 - 982*a^15*c^7 + 268*a^16*c^6 - 47*a^17*c^5 + 4*a^18*c^4))/(((c + b*x)^(1/2) - c
^(1/2))*(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 - 8*a^
14*c^8 + a^15*c^7))))/(a - c)^2 + (2*((a + b*x)^(1/2) - a^(1/2))*(4*a^(7/2)*b*c^(33/2) - 43*a^(9/2)*b*c^(31/2)
+ 231*a^(11/2)*b*c^(29/2) - 749*a^(13/2)*b*c^(27/2) + 1505*a^(15/2)*b*c^(25/2) - 1770*a^(17/2)*b*c^(23/2) + 8
22*a^(19/2)*b*c^(21/2) + 822*a^(21/2)*b*c^(19/2) - 1770*a^(23/2)*b*c^(17/2) + 1505*a^(25/2)*b*c^(15/2) - 749*a
^(27/2)*b*c^(13/2) + 231*a^(29/2)*b*c^(11/2) - 43*a^(31/2)*b*c^(9/2) + 4*a^(33/2)*b*c^(7/2)))/(((c + b*x)^(1/2
) - c^(1/2))*(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 -
8*a^14*c^8 + a^15*c^7))))/(a - c)^2 - (4*(a^(7/2)*b^2*c^(29/2) + 12*a^(9/2)*b^2*c^(27/2) - 100*a^(11/2)*b^2*c
^(25/2) + 285*a^(13/2)*b^2*c^(23/2) - 390*a^(15/2)*b^2*c^(21/2) + 192*a^(17/2)*b^2*c^(19/2) + 192*a^(19/2)*b^2
*c^(17/2) - 390*a^(21/2)*b^2*c^(15/2) + 285*a^(23/2)*b^2*c^(13/2) - 100*a^(25/2)*b^2*c^(11/2) + 12*a^(27/2)*b^
2*c^(9/2) + a^(29/2)*b^2*c^(7/2)))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^1
2*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15*c^7) + (2*((a + b*x)^(1/2) - a^(1/2))*(73*a^4*b^2*c^14 - 570*a^5*b^2*
c^13 + 2053*a^6*b^2*c^12 - 4568*a^7*b^2*c^11 + 7090*a^8*b^2*c^10 - 8156*a^9*b^2*c^9 + 7090*a^10*b^2*c^8 - 4568
*a^11*b^2*c^7 + 2053*a^12*b^2*c^6 - 570*a^13*b^2*c^5 + 73*a^14*b^2*c^4))/(((c + b*x)^(1/2) - c^(1/2))*(a^7*c^1
5 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15*c^
7))))/(a - c)^2 - (2*((a + b*x)^(1/2) - a^(1/2))*(65*a^(7/2)*b^3*c^(25/2) - 427*a^(9/2)*b^3*c^(23/2) + 1256*a^
(11/2)*b^3*c^(21/2) - 1856*a^(13/2)*b^3*c^(19/2) + 962*a^(15/2)*b^3*c^(17/2) + 962*a^(17/2)*b^3*c^(15/2) - 185
6*a^(19/2)*b^3*c^(13/2) + 1256*a^(21/2)*b^3*c^(11/2) - 427*a^(23/2)*b^3*c^(9/2) + 65*a^(25/2)*b^3*c^(7/2)))/((
(c + b*x)^(1/2) - c^(1/2))*(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 +
28*a^13*c^9 - 8*a^14*c^8 + a^15*c^7)))*4i)/(a - c)^2 + (b*((4*(4*a^4*b^3*c^12 + 8*a^5*b^3*c^11 - 32*a^6*b^3*c
^10 - 8*a^7*b^3*c^9 + 56*a^8*b^3*c^8 - 8*a^9*b^3*c^7 - 32*a^10*b^3*c^6 + 8*a^11*b^3*c^5 + 4*a^12*b^3*c^4))/(a^
7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^
15*c^7) + (4*b*((4*(a^(7/2)*b^2*c^(29/2) + 12*a^(9/2)*b^2*c^(27/2) - 100*a^(11/2)*b^2*c^(25/2) + 285*a^(13/2)*
b^2*c^(23/2) - 390*a^(15/2)*b^2*c^(21/2) + 192*a^(17/2)*b^2*c^(19/2) + 192*a^(19/2)*b^2*c^(17/2) - 390*a^(21/2
)*b^2*c^(15/2) + 285*a^(23/2)*b^2*c^(13/2) - 100*a^(25/2)*b^2*c^(11/2) + 12*a^(27/2)*b^2*c^(9/2) + a^(29/2)*b^
2*c^(7/2)))/(a^7*c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 -
8*a^14*c^8 + a^15*c^7) + (4*b*((4*(16*a^6*b*c^14 - 4*a^5*b*c^15 + 12*a^7*b*c^13 - 192*a^8*b*c^12 + 504*a^9*b*c
^11 - 672*a^10*b*c^10 + 504*a^11*b*c^9 - 192*a^12*b*c^8 + 12*a^13*b*c^7 + 16*a^14*b*c^6 - 4*a^15*b*c^5))/(a^7*
c^15 - 8*a^8*c^14 + 28*a^9*c^13 - 56*a^10*c^12 + 70*a^11*c^11 - 56*a^12*c^10 + 28*a^13*c^9 - 8*a^14*c^8 + a^15
*c^7) - (4*b*((4*(a^(9/2)*c^(35/2) - 8*a^(11/2)...

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