Integrand size = 22, antiderivative size = 198 \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}+\frac {5 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c^2 x^2}-\frac {\left (15 b^2 c^2+14 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^3 x}+\frac {(b c+a d) \left (5 b^2 c^2-2 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{7/2}} \]
1/8*(a*d+b*c)*(5*a^2*d^2-2*a*b*c*d+5*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2 )/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)-1/3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a /c/x^3+5/12*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2/x^2-1/24*(15*a^2 *d^2+14*a*b*c*d+15*b^2*c^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^3/x
Time = 0.49 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^2 c^2 x^2+2 a b c x (-5 c+7 d x)+a^2 \left (8 c^2-10 c d x+15 d^2 x^2\right )\right )}{24 a^3 c^3 x^3}+\frac {\left (5 b^3 c^3+3 a b^2 c^2 d+3 a^2 b c d^2+5 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{8 a^{7/2} c^{7/2}} \]
-1/24*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^2*c^2*x^2 + 2*a*b*c*x*(-5*c + 7*d *x) + a^2*(8*c^2 - 10*c*d*x + 15*d^2*x^2)))/(a^3*c^3*x^3) + ((5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x]) /(Sqrt[c]*Sqrt[a + b*x])])/(8*a^(7/2)*c^(7/2))
Time = 0.30 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {114, 27, 168, 27, 168, 27, 104, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {5 (b c+a d)+4 b d x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {5 (b c+a d)+4 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {15 b^2 c^2+14 a b d c+15 a^2 d^2+10 b d (b c+a d) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {15 b^2 c^2+14 a b d c+15 a^2 d^2+10 b d (b c+a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 (b c+a d) \left (5 b^2 c^2-2 a b d c+5 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {15 b^2 c}{a}+\frac {15 a d^2}{c}+14 b d\right )}{x}}{4 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {3 (a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {15 b^2 c}{a}+\frac {15 a d^2}{c}+14 b d\right )}{x}}{4 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {-\frac {3 (a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {15 b^2 c}{a}+\frac {15 a d^2}{c}+14 b d\right )}{x}}{4 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {3 (a d+b c) \left (5 a^2 d^2-2 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {15 b^2 c}{a}+\frac {15 a d^2}{c}+14 b d\right )}{x}}{4 a c}-\frac {5 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a c x^3}\) |
-1/3*(Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^3) - ((-5*(b*c + a*d)*Sqrt[a + b *x]*Sqrt[c + d*x])/(2*a*c*x^2) - (-((((15*b^2*c)/a + 14*b*d + (15*a*d^2)/c )*Sqrt[a + b*x]*Sqrt[c + d*x])/x) + (3*(b*c + a*d)*(5*b^2*c^2 - 2*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(a^ (3/2)*c^(3/2)))/(4*a*c))/(6*a*c)
3.8.40.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[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] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> 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] + Simp[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])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S imp[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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(407\) vs. \(2(166)=332\).
Time = 0.58 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.06
| method | result | size |
| default | \(\frac {\left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}-28 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}-30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}+20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x +20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x -16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{48 a^{3} c^{3} \sqrt {a c}\, x^{3} \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\) | \(408\) |
1/48/a^3/c^3*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a *c)/x)*a^3*d^3*x^3+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2) +2*a*c)/x)*a^2*b*c*d^2*x^3+9*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c ))^(1/2)+2*a*c)/x)*a*b^2*c^2*d*x^3+15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+ a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*x^3-30*(a*c)^(1/2)*((b*x+a)*(d*x+c))^( 1/2)*a^2*d^2*x^2-28*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d*x^2-30*(a* c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^2*c^2*x^2+20*(a*c)^(1/2)*((b*x+a)*(d*x+ c))^(1/2)*a^2*c*d*x+20*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*x-16*(( b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))*(d*x+c)^(1/2)*(b*x+a)^(1/2)/(a* c)^(1/2)/x^3/((b*x+a)*(d*x+c))^(1/2)
Time = 0.51 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.20 \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{4} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} + 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} + 14 \, a^{2} b c^{2} d + 15 \, a^{3} c d^{2}\right )} x^{2} - 10 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{4} x^{3}}\right ] \]
[1/96*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt(a*c) *x^3*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^2 + 4*(2*a*c + (b* c + a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)* x)/x^2) - 4*(8*a^3*c^3 + (15*a*b^2*c^3 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^ 2 - 10*(a^2*b*c^3 + a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^ 3), -1/48*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a^3*d^3)*sqrt( -a*c)*x^3*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt (d*x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) + 2*(8*a^3*c^3 + (15*a*b^2*c^3 + 14*a^2*b*c^2*d + 15*a^3*c*d^2)*x^2 - 10*(a^2*b*c^3 + a^3 *c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^4*x^3)]
\[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{x^{4} \sqrt {a + b x} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1932 vs. \(2 (166) = 332\).
Time = 2.18 (sec) , antiderivative size = 1932, normalized size of antiderivative = 9.76 \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Too large to display} \]
1/24*sqrt(b*d)*b^8*d^3*(3*(5*b^3*c^3 + 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 + 5*a ^3*d^3)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*a^3*b^7* c^3*d^3) - 2*(15*b^13*c^8 - 76*a*b^12*c^7*d + 156*a^2*b^11*c^6*d^2 - 180*a ^3*b^10*c^5*d^3 + 170*a^4*b^9*c^4*d^4 - 180*a^5*b^8*c^3*d^5 + 156*a^6*b^7* c^2*d^6 - 76*a^7*b^6*c*d^7 + 15*a^8*b^5*d^8 - 75*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^11*c^7 + 135*(sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^10*c^6*d + 45*(sqrt(b* d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^9*c^5*d^2 - 105*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^ 3*b^8*c^4*d^3 - 105*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^7*c^3*d^4 + 45*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ( b*x + a)*b*d - a*b*d))^2*a^5*b^6*c^2*d^5 + 135*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^5*c*d^6 - 75*(sqrt(b*d)*sqrt( b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^4*d^7 + 150*(sqrt( b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^9*c^6 + 60*( sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^8*c^5 *d + 42*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4* a^2*b^7*c^4*d^2 - 504*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^4*a^3*b^6*c^3*d^3 + 42*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*...
Time = 87.90 (sec) , antiderivative size = 1518, normalized size of antiderivative = 7.67 \[ \int \frac {1}{x^4 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Too large to display} \]
(((a + b*x)^(1/2) - a^(1/2))*((7*b*d^2)/(64*a^2*c^2) - (d*(a*d + b*c)^2)/( 4*a^3*c^3) + (d*(3*a^2*d^2 + 3*b^2*c^2 + 8*a*b*c*d))/(32*a^3*c^3)))/((c + d*x)^(1/2) - c^(1/2)) - (b^6/(192*a^2*c^2*d^3) + (((a + b*x)^(1/2) - a^(1/ 2))^2*((5*b^6*c^2)/64 + (5*a^2*b^4*d^2)/64))/(a^3*c^3*d^3*((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2) - a^(1/2))^4*((13*b^6*c^4)/64 + (13*a^4 *b^2*d^4)/64 + (55*a^3*b^3*c*d^3)/32 + (57*a^2*b^4*c^2*d^2)/32 + (55*a*b^5 *c^3*d)/32))/(a^4*c^4*d^3*((c + d*x)^(1/2) - c^(1/2))^4) - (((a + b*x)^(1/ 2) - a^(1/2))^5*((39*a^4*b^2*c*d^4)/32 - (17*a^5*b*d^5)/64 - (17*b^6*c^5)/ 64 + (25*a^2*b^4*c^3*d^2)/8 + (25*a^3*b^3*c^2*d^3)/8 + (39*a*b^5*c^4*d)/32 ))/(a^(9/2)*c^(9/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^5) - (((a + b*x)^(1/2) - a^(1/2))^7*((97*a^2*b^3*c^3*d^2)/64 - (27*b^5*c^5)/64 - (27*a^5*d^5)/64 + (97*a^3*b^2*c^2*d^3)/64 + (15*a*b^4*c^4*d)/64 + (15*a^4*b*c*d^4)/64))/( a^(9/2)*c^(9/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^7) + (((b^6*c)/64 + (a*b^5 *d)/64)*((a + b*x)^(1/2) - a^(1/2)))/(a^(5/2)*c^(5/2)*d^3*((c + d*x)^(1/2) - c^(1/2))) - (((a + b*x)^(1/2) - a^(1/2))^3*((73*b^6*c^3)/192 + (73*a^3* b^3*d^3)/192 + (9*a^2*b^4*c*d^2)/16 + (9*a*b^5*c^2*d)/16))/(a^(7/2)*c^(7/2 )*d^3*((c + d*x)^(1/2) - c^(1/2))^3) - (((a + b*x)^(1/2) - a^(1/2))^6*((37 *a^6*d^6)/192 + (37*b^6*c^6)/192 - (71*a^2*b^4*c^4*d^2)/32 - (49*a^3*b^3*c ^3*d^3)/16 - (71*a^4*b^2*c^2*d^4)/32 + (11*a*b^5*c^5*d)/32 + (11*a^5*b*c*d ^5)/32))/(a^5*c^5*d^3*((c + d*x)^(1/2) - c^(1/2))^6) + (((a + b*x)^(1/2...