Integrand size = 22, antiderivative size = 191 \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}+\frac {(5 b c-a d) \sqrt {a+b x} \sqrt {c+d x}}{12 a^2 c x^2}-\frac {(5 b c-3 a d) (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a^3 c^2 x}+\frac {(b c-a d) \left (5 b^2 c^2+2 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \]
1/8*(-a*d+b*c)*(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^(5/2)-1/3*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/ x^3+1/12*(-a*d+5*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c/x^2-1/24*(-3*a*d+5 *b*c)*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^2/x
Time = 10.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-15 b^2 c^2 x^2+2 a b c x (5 c+2 d x)+a^2 \left (-8 c^2-2 c d x+3 d^2 x^2\right )\right )}{24 a^3 c^2 x^3}-\frac {\left (-5 b^3 c^3+3 a b^2 c^2 d+a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 a^{7/2} c^{5/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-15*b^2*c^2*x^2 + 2*a*b*c*x*(5*c + 2*d*x) + a^2*(-8*c^2 - 2*c*d*x + 3*d^2*x^2)))/(24*a^3*c^2*x^3) - ((-5*b^3*c^3 + 3*a *b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[ a]*Sqrt[c + d*x])])/(8*a^(7/2)*c^(5/2))
Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {110, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {\int -\frac {5 b c-a d+4 b d x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {5 b c-a d+4 b d x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {(5 b c-3 a d) (3 b c+a d)+2 b d (5 b c-a d) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {(5 b c-3 a d) (3 b c+a d)+2 b d (5 b c-a d) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {3 (b c-a d) \left (5 b^2 c^2+2 a b d c+a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {-\frac {3 (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {-\frac {3 (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {-\frac {\frac {3 (b c-a d) \left (a^2 d^2+2 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-3 a d) (a d+3 b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (5 b c-a d)}{2 a c x^2}}{6 a}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{3 a x^3}\) |
-1/3*(Sqrt[a + b*x]*Sqrt[c + d*x])/(a*x^3) - (-1/2*((5*b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - (-(((5*b*c - 3*a*d)*(3*b*c + a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (3*(b*c - a*d)*(5*b^2*c^2 + 2*a*b*c*d + 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)
3.8.5.3.1 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[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[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] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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. \(2(159)=318\).
Time = 1.63 (sec) , antiderivative size = 408, normalized size of antiderivative = 2.14
| method | result | size |
| default | \(-\frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+9 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a \,b^{2} c^{2} d \,x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{3} c^{3} x^{3}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} d^{2} x^{2}-8 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c d \,x^{2}+30 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} x^{2}+4 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c d x -20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} x +16 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{48 a^{3} c^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x^{3} \sqrt {a c}}\) | \(408\) |
-1/48*(d*x+c)^(1/2)*(b*x+a)^(1/2)/a^3/c^2*(3*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+3*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^3-6*(a*c )^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^2*d^2*x^2-8*(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+4*( 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))/((b*x+ a)*(d*x+c))^(1/2)/x^3/(a*c)^(1/2)
Time = 0.41 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.30 \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\left [-\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {a c} x^{3} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, a^{4} c^{3} x^{3}}, -\frac {3 \, {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-a c} x^{3} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (8 \, a^{3} c^{3} + {\left (15 \, a b^{2} c^{3} - 4 \, a^{2} b c^{2} d - 3 \, a^{3} c d^{2}\right )} x^{2} - 2 \, {\left (5 \, a^{2} b c^{3} - a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, a^{4} c^{3} x^{3}}\right ] \]
[-1/96*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - 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 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2 *(5*a^2*b*c^3 - a^3*c^2*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^3), -1/48*(3*(5*b^3*c^3 - 3*a*b^2*c^2*d - a^2*b*c*d^2 - 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 - 4*a^2*b*c^2*d - 3*a^3*c*d^2)*x^2 - 2*(5*a^2*b*c^3 - a^3*c^2*d)*x )*sqrt(b*x + a)*sqrt(d*x + c))/(a^4*c^3*x^3)]
\[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\int \frac {\sqrt {c + d x}}{x^{4} \sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
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. 2140 vs. \(2 (159) = 318\).
Time = 2.18 (sec) , antiderivative size = 2140, normalized size of antiderivative = 11.20 \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\text {Too large to display} \]
1/24*(3*(5*sqrt(b*d)*b^6*c^3 - 3*sqrt(b*d)*a*b^5*c^2*d - sqrt(b*d)*a^2*b^4 *c*d^2 - sqrt(b*d)*a^3*b^3*d^3)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sq rt(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*c^2) - 2*(15*sqrt(b*d)*b^16*c^8 - 94*sqrt(b*d)*a*b^1 5*c^7*d + 246*sqrt(b*d)*a^2*b^14*c^6*d^2 - 342*sqrt(b*d)*a^3*b^13*c^5*d^3 + 260*sqrt(b*d)*a^4*b^12*c^4*d^4 - 90*sqrt(b*d)*a^5*b^11*c^3*d^5 - 6*sqrt( b*d)*a^6*b^10*c^2*d^6 + 14*sqrt(b*d)*a^7*b^9*c*d^7 - 3*sqrt(b*d)*a^8*b^8*d ^8 - 75*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^14*c^7 + 225*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^13*c^6*d - 111*sqrt(b*d)*(sqrt(b*d)*sqrt(b* x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^12*c^5*d^2 - 291*sqr t(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a ^3*b^11*c^4*d^3 + 399*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b *x + a)*b*d - a*b*d))^2*a^4*b^10*c^3*d^4 - 141*sqrt(b*d)*(sqrt(b*d)*sqrt(b *x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^5*b^9*c^2*d^5 - 21*sqrt (b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^ 6*b^8*c*d^6 + 15*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^7*b^7*d^7 + 150*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^12*c^6 - 120*sqrt(b*d)*(sqrt(b*d) *sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^11*c^5*d - ...
Time = 92.14 (sec) , antiderivative size = 1574, normalized size of antiderivative = 8.24 \[ \int \frac {\sqrt {c+d x}}{x^4 \sqrt {a+b x}} \, dx=\text {Too large to display} \]
((((a + b*x)^(1/2) - a^(1/2))^4*((5*a^4*b^2*d^4)/64 - (13*b^6*c^4)/64 + (a ^3*b^3*c*d^3)/2 + (7*a^2*b^4*c^2*d^2)/16 - (23*a*b^5*c^3*d)/16))/(a^4*c^3* d^3*((c + d*x)^(1/2) - c^(1/2))^4) - b^6/(192*a^2*c*d^3) - (((a + b*x)^(1/ 2) - a^(1/2))^5*((17*b^6*c^5)/64 - (a^5*b*d^5)/64 + (7*a^4*b^2*c*d^4)/16 - (47*a^2*b^4*c^3*d^2)/32 + (37*a^3*b^3*c^2*d^3)/32 - (3*a*b^5*c^4*d)/2))/( a^(9/2)*c^(7/2)*d^3*((c + d*x)^(1/2) - c^(1/2))^5) + (((a + b*x)^(1/2) - a ^(1/2))^2*((a^2*b^4*d^2)/64 - (5*b^6*c^2)/64 + (3*a*b^5*c*d)/32))/(a^3*c^2 *d^3*((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2) - a^(1/2))^7*((3*a ^5*d^5)/64 - (27*b^5*c^5)/64 + (69*a^2*b^3*c^3*d^2)/64 - (49*a^3*b^2*c^2*d ^3)/64 + (45*a*b^4*c^4*d)/64 - (13*a^4*b*c*d^4)/64))/(a^(9/2)*c^(7/2)*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^(3/2)*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 - (17*a^3*b^3*d^3)/192 - (9*a^ 2*b^4*c*d^2)/32 + (3*a*b^5*c^2*d)/32))/(a^(7/2)*c^(5/2)*d^3*((c + d*x)^(1/ 2) - c^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/2))^6*((37*b^6*c^6)/192 - (5*a ^6*d^6)/192 - (5*a^2*b^4*c^4*d^2)/2 + (5*a^3*b^3*c^3*d^3)/32 + a^4*b^2*c^2 *d^4 + (a*b^5*c^5*d)/8 + (a^5*b*c*d^5)/16))/(a^5*c^4*d^3*((c + d*x)^(1/2) - c^(1/2))^6) + (((a + b*x)^(1/2) - a^(1/2))^8*((15*b^4*c^4)/64 - (a^4*d^4 )/64 + (13*a^2*b^2*c^2*d^2)/64 - (19*a*b^3*c^3*d)/32 + (3*a^3*b*c*d^3)/32) )/(a^4*c^3*d*((c + d*x)^(1/2) - c^(1/2))^8))/(((a + b*x)^(1/2) - a^(1/2...