Integrand size = 22, antiderivative size = 115 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}-\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}+2 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right ) \]
-(a*d+b*c)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(1/2)/c^ (1/2)+2*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))*b^(1/2)*d^(1/ 2)-(b*x+a)^(1/2)*(d*x+c)^(1/2)/x
Time = 0.25 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}-\frac {(b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a} \sqrt {c}}+2 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right ) \]
-((Sqrt[a + b*x]*Sqrt[c + d*x])/x) - ((b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(Sqrt[a]*Sqrt[c]) + 2*Sqrt[b]*Sqrt[d]*Ar cTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])]
Time = 0.23 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {108, 27, 175, 66, 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.
| \(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \int \frac {b c+a d+2 b d x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {b c+a d+2 b d x}{x \sqrt {a+b x} \sqrt {c+d x}}dx-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
| \(\Big \downarrow \) 175 |
| \(\displaystyle \frac {1}{2} \left (2 b d \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx+(a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {1}{2} \left ((a d+b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx+4 b d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{2} \left (2 (a d+b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}+4 b d \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (4 \sqrt {b} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )-\frac {2 (a d+b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a} \sqrt {c}}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{x}\) |
-((Sqrt[a + b*x]*Sqrt[c + d*x])/x) + ((-2*(b*c + a*d)*ArcTanh[(Sqrt[c]*Sqr t[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(Sqrt[a]*Sqrt[c]) + 4*Sqrt[b]*Sqrt[d ]*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/2
3.6.52.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
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/(b*(m + 1))) , x] - Simp[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])
Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ )))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b Int[(c + d*x)^n*(e + f*x)^p, x] , x] + Simp[(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]
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. \(212\) vs. \(2(87)=174\).
Time = 1.50 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.85
| method | result | size |
| default | \(-\frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a d x \sqrt {b d}+\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b c x \sqrt {b d}-2 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b d x \sqrt {a c}+2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}\right )}{2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, x \sqrt {b d}\, \sqrt {a c}}\) | \(213\) |
-1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(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*d*x*(b*d)^(1/2)+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*c*x*(b*d)^(1/2)-2*ln(1/2*(2*b*d*x+2*((b* x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*b*d*x*(a*c)^(1/2)+2* ((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2)*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x /(b*d)^(1/2)/(a*c)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (87) = 174\).
Time = 0.32 (sec) , antiderivative size = 842, normalized size of antiderivative = 7.32 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=\left [\frac {2 \, \sqrt {b d} a c x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \sqrt {a c} {\left (b c + a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} a c}{4 \, a c x}, -\frac {4 \, \sqrt {-b d} a c x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - \sqrt {a c} {\left (b c + a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} a c}{4 \, a c x}, \frac {\sqrt {b d} a c x \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \sqrt {-a c} {\left (b c + a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} a c}{2 \, a c x}, -\frac {2 \, \sqrt {-b d} a c x \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - \sqrt {-a c} {\left (b c + a d\right )} x \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 \, \sqrt {b x + a} \sqrt {d x + c} a c}{2 \, a c x}\right ] \]
[1/4*(2*sqrt(b*d)*a*c*x*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c *d + a*b*d^2)*x) + sqrt(a*c)*(b*c + a*d)*x*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*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a*c*x), -1/4*(4*sqrt(-b*d)*a*c*x*arctan(1/2*(2*b*d*x + b*c + a*d )*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - sqrt(a*c)*(b*c + a*d)*x*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)*s qrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a*c*x), 1/2*(sqrt(b*d)*a*c*x*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b* c*d + a^2*d^2 + 4*(2*b*d*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + sqrt(-a*c)*(b*c + a*d)*x*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*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a*c *x), -1/2*(2*sqrt(-b*d)*a*c*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)* sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x )) - sqrt(-a*c)*(b*c + a*d)*x*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*sqrt(b*x + a)*sqrt(d*x + c)*a*c)/(a*c*x)]
\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=\text {Exception raised: ValueError} \]
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. 463 vs. \(2 (87) = 174\).
Time = 0.38 (sec) , antiderivative size = 463, normalized size of antiderivative = 4.03 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=-\frac {\sqrt {b d} {\left | b \right |} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right ) + \frac {{\left (\sqrt {b d} b^{2} c {\left | b \right |} + \sqrt {b d} a b d {\left | b \right |}\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b} + \frac {2 \, {\left (\sqrt {b d} b^{4} c^{2} {\left | b \right |} - 2 \, \sqrt {b d} a b^{3} c d {\left | b \right |} + \sqrt {b d} a^{2} b^{2} d^{2} {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c {\left | b \right |} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d {\left | b \right |}\right )}}{b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}}}{b} \]
-(sqrt(b*d)*abs(b)*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b *d - a*b*d))^2) + (sqrt(b*d)*b^2*c*abs(b) + sqrt(b*d)*a*b*d*abs(b))*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)*b) + 2*(sqrt(b*d)*b^4* c^2*abs(b) - 2*sqrt(b*d)*a*b^3*c*d*abs(b) + sqrt(b*d)*a^2*b^2*d^2*abs(b) - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) ^2*b^2*c*abs(b) - 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*d*abs(b))/(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2 *(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4))/b
Time = 20.65 (sec) , antiderivative size = 4568, normalized size of antiderivative = 39.72 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^2} \, dx=\text {Too large to display} \]
((((b^2*c)/4 + (a*b*d)/4)*((a + b*x)^(1/2) - a^(1/2)))/(a^(1/2)*c^(1/2)*d* ((c + d*x)^(1/2) - c^(1/2))) - b^2/(4*d) + (((a + b*x)^(1/2) - a^(1/2))^2* ((a^2*d^2)/4 + (b^2*c^2)/4 - (3*a*b*c*d)/4))/(a*c*d*((c + d*x)^(1/2) - c^( 1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^3/((c + d*x)^(1/2) - c^(1/2))^3 + ( b*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) - ((a*d + b *c)*((a + b*x)^(1/2) - a^(1/2))^2)/(a^(1/2)*c^(1/2)*d*((c + d*x)^(1/2) - c ^(1/2))^2)) - atan(((b*d)^(1/2)*(2*(b*d)^(1/2)*(2*(b*d)^(1/2)*(2*((2*(4*a^ (9/2)*b^9*c^(19/2) - 4*a^(13/2)*b^7*c^(15/2)*d^2 - 4*a^(15/2)*b^6*c^(13/2) *d^3 + 4*a^(19/2)*b^4*c^(9/2)*d^5))/(a^7*c^7*d^9) - (((a + b*x)^(1/2) - a^ (1/2))*(32*a^4*b^9*c^10 - 120*a^5*b^8*c^9*d + 288*a^6*b^7*c^8*d^2 - 400*a^ 7*b^6*c^7*d^3 + 288*a^8*b^5*c^6*d^4 - 120*a^9*b^4*c^5*d^5 + 32*a^10*b^3*c^ 4*d^6))/(2*a^7*c^7*d^9*((c + d*x)^(1/2) - c^(1/2))))*(b*d)^(1/2) - (2*(8*a ^5*b^9*c^9*d + 16*a^6*b^8*c^8*d^2 - 48*a^7*b^7*c^7*d^3 + 16*a^8*b^6*c^6*d^ 4 + 8*a^9*b^5*c^5*d^5))/(a^7*c^7*d^9) + (((a + b*x)^(1/2) - a^(1/2))*(16*a ^(7/2)*b^10*c^(21/2) - 76*a^(9/2)*b^9*c^(19/2)*d + 228*a^(11/2)*b^8*c^(17/ 2)*d^2 - 168*a^(13/2)*b^7*c^(15/2)*d^3 - 168*a^(15/2)*b^6*c^(13/2)*d^4 + 2 28*a^(17/2)*b^5*c^(11/2)*d^5 - 76*a^(19/2)*b^4*c^(9/2)*d^6 + 16*a^(21/2)*b ^3*c^(7/2)*d^7))/(2*a^7*c^7*d^9*((c + d*x)^(1/2) - c^(1/2)))) - (2*(a^(7/2 )*b^11*c^(21/2) + 16*a^(9/2)*b^10*c^(19/2)*d - 42*a^(11/2)*b^9*c^(17/2)*d^ 2 + 25*a^(13/2)*b^8*c^(15/2)*d^3 + 25*a^(15/2)*b^7*c^(13/2)*d^4 - 42*a^...