Integrand size = 22, antiderivative size = 137 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}+\frac {3 (b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{4 a^2 c^2 x}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{4 a^{5/2} c^{5/2}} \]
-1/4*(3*a^2*d^2+2*a*b*c*d+3*b^2*c^2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2) /(d*x+c)^(1/2))/a^(5/2)/c^(5/2)-1/2*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a/c/x^2+3/ 4*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^2/c^2/x
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.83 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\sqrt {a+b x} \sqrt {c+d x} (-2 a c+3 b c x+3 a d x)}{4 a^2 c^2 x^2}-\frac {\left (3 b^2 c^2+2 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{4 a^{5/2} c^{5/2}} \]
(Sqrt[a + b*x]*Sqrt[c + d*x]*(-2*a*c + 3*b*c*x + 3*a*d*x))/(4*a^2*c^2*x^2) - ((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(S qrt[c]*Sqrt[a + b*x])])/(4*a^(5/2)*c^(5/2))
Time = 0.24 (sec) , antiderivative size = 143, 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 = {114, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx\) |
| \(\Big \downarrow \) 114 |
| \(\displaystyle -\frac {\int \frac {3 (b c+a d)+2 b d x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {3 (b c+a d)+2 b d x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle -\frac {-\frac {\int \frac {3 b^2 c^2+2 a b d c+3 a^2 d^2}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {-\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {\left (3 a^2 d^2+2 a b c d+3 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {3 \sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x}}{2 a c x^2}\) |
-1/2*(Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - ((-3*(b*c + a*d)*Sqrt[a + b *x]*Sqrt[c + d*x])/(a*c*x) + ((3*b^2*c^2 + 2*a*b*c*d + 3*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)
3.8.39.3.1 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. \(257\) vs. \(2(111)=222\).
Time = 1.66 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.88
| method | result | size |
| default | \(-\frac {\left (3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x^{2}+2 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) a b c d \,x^{2}+3 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}+2 a c}{x}\right ) b^{2} c^{2} x^{2}-6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a d x -6 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b c x +4 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a c \sqrt {a c}\right ) \sqrt {d x +c}\, \sqrt {b x +a}}{8 a^{2} c^{2} \sqrt {a c}\, x^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}}\) | \(258\) |
-1/8/a^2/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^2*d^2*x^2+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*b*c*d*x^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)*b^2*c^2*x^2-6*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d*x-6*(a *c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*c*x+4*((b*x+a)*(d*x+c))^(1/2)*a*c*(a*c )^(1/2))*(d*x+c)^(1/2)*(b*x+a)^(1/2)/(a*c)^(1/2)/x^2/((b*x+a)*(d*x+c))^(1/ 2)
Time = 0.34 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.42 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\left [\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {a c} x^{2} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{16 \, a^{3} c^{3} x^{2}}, \frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-a c} x^{2} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (2 \, a^{2} c^{2} - 3 \, {\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{8 \, a^{3} c^{3} x^{2}}\right ] \]
[1/16*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*sqrt(a*c)*x^2*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*(2*a^2*c^2 - 3*(a*b*c^2 + a^2*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^3*c^3*x^2), 1/ 8*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*sqrt(-a*c)*x^2*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*(2*a^2*c^2 - 3*(a*b*c^2 + a^2*c*d)*x)*sqr t(b*x + a)*sqrt(d*x + c))/(a^3*c^3*x^2)]
\[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\int \frac {1}{x^{3} \sqrt {a + b x} \sqrt {c + d x}}\, dx \]
Exception generated. \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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. 951 vs. \(2 (111) = 222\).
Time = 0.59 (sec) , antiderivative size = 951, normalized size of antiderivative = 6.94 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=-\frac {\sqrt {b d} b^{6} d^{2} {\left (\frac {{\left (3 \, b^{2} c^{2} + 2 \, a b c d + 3 \, a^{2} d^{2}\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} a^{2} b^{5} c^{2} d^{2}} - \frac {2 \, {\left (3 \, b^{8} c^{5} - 9 \, a b^{7} c^{4} d + 6 \, a^{2} b^{6} c^{3} d^{2} + 6 \, a^{3} b^{5} c^{2} d^{3} - 9 \, a^{4} b^{4} c d^{4} + 3 \, a^{5} b^{3} d^{5} - 9 \, {\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^{6} c^{4} - 4 \, {\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^{5} c^{3} d + 26 \, {\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^{2} b^{4} c^{2} d^{2} - 4 \, {\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^{3} b^{3} c d^{3} - 9 \, {\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^{4} b^{2} d^{4} + 9 \, {\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^{4} c^{3} + 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{2} d + 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c d^{2} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b d^{3} - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c^{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 )}^{6} a b c d - 3 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a^{2} d^{2}\right )}}{{\left (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}\right )}^{2} a^{2} b^{4} c^{2} d^{2}}\right )}}{4 \, {\left | b \right |}} \]
-1/4*sqrt(b*d)*b^6*d^2*((3*b^2*c^2 + 2*a*b*c*d + 3*a^2*d^2)*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^2*b^5*c^2*d^2) - 2*(3*b^8*c^ 5 - 9*a*b^7*c^4*d + 6*a^2*b^6*c^3*d^2 + 6*a^3*b^5*c^2*d^3 - 9*a^4*b^4*c*d^ 4 + 3*a^5*b^3*d^5 - 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b* d - a*b*d))^2*b^6*c^4 - 4*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a )*b*d - a*b*d))^2*a*b^5*c^3*d + 26*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*b^4*c^2*d^2 - 4*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^3*c*d^3 - 9*(sqrt(b*d)*sqrt(b *x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^4*b^2*d^4 + 9*(sqrt(b*d )*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*b^4*c^3 + 15*(sqr t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a*b^3*c^2*d + 15*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4*a^2 *b^2*c*d^2 + 9*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b *d))^4*a^3*b*d^3 - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*b^2*c^2 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a) *b*d - a*b*d))^6*a*b*c*d - 3*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^6*a^2*d^2)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 2*(s qrt(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...
Time = 20.84 (sec) , antiderivative size = 900, normalized size of antiderivative = 6.57 \[ \int \frac {1}{x^3 \sqrt {a+b x} \sqrt {c+d x}} \, dx=\frac {\ln \left (\frac {\left (\sqrt {c}\,\sqrt {a+b\,x}-\sqrt {a}\,\sqrt {c+d\,x}\right )\,\left (b\,\sqrt {c}-\frac {\sqrt {a}\,d\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^3\,c^3}-\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2\,\left (\frac {11\,a^2\,b^2\,d^2}{32}+\frac {5\,a\,b^3\,c\,d}{8}+\frac {11\,b^4\,c^2}{32}\right )}{a^{5/2}\,c^{5/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}-\frac {b^4}{32\,a^{3/2}\,c^{3/2}\,d^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {a^3\,b\,d^3}{16}-\frac {9\,a^2\,b^2\,c\,d^2}{8}-\frac {9\,a\,b^3\,c^2\,d}{8}+\frac {b^4\,c^3}{16}\right )}{a^3\,c^3\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {\left (\frac {c\,b^4}{8}+\frac {a\,d\,b^3}{8}\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{a^2\,c^2\,d^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {a^3\,d^3}{4}-\frac {7\,a^2\,b\,c\,d^2}{16}-\frac {7\,a\,b^2\,c^2\,d}{16}+\frac {b^3\,c^3}{4}\right )}{a^3\,c^3\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (-\frac {7\,a^4\,d^4}{32}+\frac {a^3\,b\,c\,d^3}{4}+\frac {45\,a^2\,b^2\,c^2\,d^2}{32}+\frac {a\,b^3\,c^3\,d}{4}-\frac {7\,b^4\,c^4}{32}\right )}{a^{7/2}\,c^{7/2}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^6}+\frac {b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4\,\left (a^2\,d^2+4\,a\,b\,c\,d+b^2\,c^2\right )}{a\,c\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^4}-\frac {\left (2\,c\,b^2+2\,a\,d\,b\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{\sqrt {a}\,\sqrt {c}\,d^2\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^3}-\frac {\left (2\,a\,d+2\,b\,c\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5}{\sqrt {a}\,\sqrt {c}\,d\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^5}}-\frac {\ln \left (\frac {\sqrt {a+b\,x}-\sqrt {a}}{\sqrt {c+d\,x}-\sqrt {c}}\right )\,\left (3\,\sqrt {a}\,b^2\,c^{5/2}+3\,a^{5/2}\,\sqrt {c}\,d^2+2\,a^{3/2}\,b\,c^{3/2}\,d\right )}{8\,a^3\,c^3}+\frac {d^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{32\,a^{3/2}\,c^{3/2}\,{\left (\sqrt {c+d\,x}-\sqrt {c}\right )}^2}+\frac {3\,d\,\left (a\,d+b\,c\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{16\,a^2\,c^2\,\left (\sqrt {c+d\,x}-\sqrt {c}\right )} \]
(log(((c^(1/2)*(a + b*x)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/2) - (a^ (1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))))/((c + d *x)^(1/2) - c^(1/2)))*(3*a^(1/2)*b^2*c^(5/2) + 3*a^(5/2)*c^(1/2)*d^2 + 2*a ^(3/2)*b*c^(3/2)*d))/(8*a^3*c^3) - ((((a + b*x)^(1/2) - a^(1/2))^2*((11*b^ 4*c^2)/32 + (11*a^2*b^2*d^2)/32 + (5*a*b^3*c*d)/8))/(a^(5/2)*c^(5/2)*d^2*( (c + d*x)^(1/2) - c^(1/2))^2) - b^4/(32*a^(3/2)*c^(3/2)*d^2) + (((a + b*x) ^(1/2) - a^(1/2))^3*((b^4*c^3)/16 + (a^3*b*d^3)/16 - (9*a^2*b^2*c*d^2)/8 - (9*a*b^3*c^2*d)/8))/(a^3*c^3*d^2*((c + d*x)^(1/2) - c^(1/2))^3) - (((b^4* c)/8 + (a*b^3*d)/8)*((a + b*x)^(1/2) - a^(1/2)))/(a^2*c^2*d^2*((c + d*x)^( 1/2) - c^(1/2))) + (((a + b*x)^(1/2) - a^(1/2))^5*((a^3*d^3)/4 + (b^3*c^3) /4 - (7*a*b^2*c^2*d)/16 - (7*a^2*b*c*d^2)/16))/(a^3*c^3*d*((c + d*x)^(1/2) - c^(1/2))^5) + (((a + b*x)^(1/2) - a^(1/2))^4*((45*a^2*b^2*c^2*d^2)/32 - (7*b^4*c^4)/32 - (7*a^4*d^4)/32 + (a*b^3*c^3*d)/4 + (a^3*b*c*d^3)/4))/(a^ (7/2)*c^(7/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^4))/(((a + b*x)^(1/2) - a^(1 /2))^6/((c + d*x)^(1/2) - c^(1/2))^6 + (b^2*((a + b*x)^(1/2) - a^(1/2))^2) /(d^2*((c + d*x)^(1/2) - c^(1/2))^2) + (((a + b*x)^(1/2) - a^(1/2))^4*(a^2 *d^2 + b^2*c^2 + 4*a*b*c*d))/(a*c*d^2*((c + d*x)^(1/2) - c^(1/2))^4) - ((2 *b^2*c + 2*a*b*d)*((a + b*x)^(1/2) - a^(1/2))^3)/(a^(1/2)*c^(1/2)*d^2*((c + d*x)^(1/2) - c^(1/2))^3) - ((2*a*d + 2*b*c)*((a + b*x)^(1/2) - a^(1/2))^ 5)/(a^(1/2)*c^(1/2)*d*((c + d*x)^(1/2) - c^(1/2))^5)) - (log(((a + b*x)...