Integrand size = 24, antiderivative size = 140 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=-\frac {(b B d-4 A b e+3 a B e) \sqrt {a+b x} \sqrt {d+e x}}{4 b^2 e}+\frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(b d-a e) (b B d-4 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{4 b^{5/2} e^{3/2}} \]
-1/4*(-a*e+b*d)*(-4*A*b*e+3*B*a*e+B*b*d)*arctanh(e^(1/2)*(b*x+a)^(1/2)/b^( 1/2)/(e*x+d)^(1/2))/b^(5/2)/e^(3/2)+1/2*B*(e*x+d)^(3/2)*(b*x+a)^(1/2)/b/e- 1/4*(-4*A*b*e+3*B*a*e+B*b*d)*(b*x+a)^(1/2)*(e*x+d)^(1/2)/b^2/e
Time = 0.71 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=\frac {\sqrt {a+b x} \sqrt {d+e x} (4 A b e-3 a B e+b B (d+2 e x))}{4 b^2 e}+\frac {(b d-a e) (b B d-4 A b e+3 a B e) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {e} \left (\sqrt {a-\frac {b d}{e}}-\sqrt {a+b x}\right )}\right )}{2 b^{5/2} e^{3/2}} \]
(Sqrt[a + b*x]*Sqrt[d + e*x]*(4*A*b*e - 3*a*B*e + b*B*(d + 2*e*x)))/(4*b^2 *e) + ((b*d - a*e)*(b*B*d - 4*A*b*e + 3*a*B*e)*ArcTanh[(Sqrt[b]*Sqrt[d + e *x])/(Sqrt[e]*(Sqrt[a - (b*d)/e] - Sqrt[a + b*x]))])/(2*b^(5/2)*e^(3/2))
Time = 0.22 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 60, 66, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(3 a B e-4 A b e+b B d) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x}}dx}{4 b e}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(3 a B e-4 A b e+b B d) \left (\frac {(b d-a e) \int \frac {1}{\sqrt {a+b x} \sqrt {d+e x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b e}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(3 a B e-4 A b e+b B d) \left (\frac {(b d-a e) \int \frac {1}{b-\frac {e (a+b x)}{d+e x}}d\frac {\sqrt {a+b x}}{\sqrt {d+e x}}}{b}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {B \sqrt {a+b x} (d+e x)^{3/2}}{2 b e}-\frac {(3 a B e-4 A b e+b B d) \left (\frac {(b d-a e) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a+b x}}{\sqrt {b} \sqrt {d+e x}}\right )}{b^{3/2} \sqrt {e}}+\frac {\sqrt {a+b x} \sqrt {d+e x}}{b}\right )}{4 b e}\) |
(B*Sqrt[a + b*x]*(d + e*x)^(3/2))/(2*b*e) - ((b*B*d - 4*A*b*e + 3*a*B*e)*( (Sqrt[a + b*x]*Sqrt[d + e*x])/b + ((b*d - a*e)*ArcTanh[(Sqrt[e]*Sqrt[a + b *x])/(Sqrt[b]*Sqrt[d + e*x])])/(b^(3/2)*Sqrt[e])))/(4*b*e)
3.23.38.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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. \(374\) vs. \(2(114)=228\).
Time = 1.10 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.68
| method | result | size |
| default | \(-\frac {\sqrt {e x +d}\, \sqrt {b x +a}\, \left (4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b \,e^{2}-4 A \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d e -3 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a^{2} e^{2}+2 B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) a b d e +B \ln \left (\frac {2 b e x +2 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}+a e +b d}{2 \sqrt {b e}}\right ) b^{2} d^{2}-4 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, b e x -8 A \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b e +6 B \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, \sqrt {b e}\, a e -2 B \sqrt {b e}\, \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b d \right )}{8 \sqrt {\left (b x +a \right ) \left (e x +d \right )}\, b^{2} e \sqrt {b e}}\) | \(375\) |
-1/8*(e*x+d)^(1/2)*(b*x+a)^(1/2)*(4*A*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^ (1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a*b*e^2-4*A*ln(1/2*(2*b*e*x+2*((b* x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*b^2*d*e-3*B*ln(1/2*( 2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e)^(1/2))*a^2*e^ 2+2*B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a*e+b*d)/(b*e) ^(1/2))*a*b*d*e+B*ln(1/2*(2*b*e*x+2*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)+a* e+b*d)/(b*e)^(1/2))*b^2*d^2-4*B*((b*x+a)*(e*x+d))^(1/2)*(b*e)^(1/2)*b*e*x- 8*A*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b*e+6*B*((b*x+a)*(e*x+d))^(1/2)*(b *e)^(1/2)*a*e-2*B*(b*e)^(1/2)*((b*x+a)*(e*x+d))^(1/2)*b*d)/((b*x+a)*(e*x+d ))^(1/2)/b^2/e/(b*e)^(1/2)
Time = 0.25 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.61 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=\left [\frac {{\left (B b^{2} d^{2} + 2 \, {\left (B a b - 2 \, A b^{2}\right )} d e - {\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt {b e} \log \left (8 \, b^{2} e^{2} x^{2} + b^{2} d^{2} + 6 \, a b d e + a^{2} e^{2} - 4 \, {\left (2 \, b e x + b d + a e\right )} \sqrt {b e} \sqrt {b x + a} \sqrt {e x + d} + 8 \, {\left (b^{2} d e + a b e^{2}\right )} x\right ) + 4 \, {\left (2 \, B b^{2} e^{2} x + B b^{2} d e - {\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{16 \, b^{3} e^{2}}, \frac {{\left (B b^{2} d^{2} + 2 \, {\left (B a b - 2 \, A b^{2}\right )} d e - {\left (3 \, B a^{2} - 4 \, A a b\right )} e^{2}\right )} \sqrt {-b e} \arctan \left (\frac {{\left (2 \, b e x + b d + a e\right )} \sqrt {-b e} \sqrt {b x + a} \sqrt {e x + d}}{2 \, {\left (b^{2} e^{2} x^{2} + a b d e + {\left (b^{2} d e + a b e^{2}\right )} x\right )}}\right ) + 2 \, {\left (2 \, B b^{2} e^{2} x + B b^{2} d e - {\left (3 \, B a b - 4 \, A b^{2}\right )} e^{2}\right )} \sqrt {b x + a} \sqrt {e x + d}}{8 \, b^{3} e^{2}}\right ] \]
[1/16*((B*b^2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a^2 - 4*A*a*b)*e^2)*sqr t(b*e)*log(8*b^2*e^2*x^2 + b^2*d^2 + 6*a*b*d*e + a^2*e^2 - 4*(2*b*e*x + b* d + a*e)*sqrt(b*e)*sqrt(b*x + a)*sqrt(e*x + d) + 8*(b^2*d*e + a*b*e^2)*x) + 4*(2*B*b^2*e^2*x + B*b^2*d*e - (3*B*a*b - 4*A*b^2)*e^2)*sqrt(b*x + a)*sq rt(e*x + d))/(b^3*e^2), 1/8*((B*b^2*d^2 + 2*(B*a*b - 2*A*b^2)*d*e - (3*B*a ^2 - 4*A*a*b)*e^2)*sqrt(-b*e)*arctan(1/2*(2*b*e*x + b*d + a*e)*sqrt(-b*e)* sqrt(b*x + a)*sqrt(e*x + d)/(b^2*e^2*x^2 + a*b*d*e + (b^2*d*e + a*b*e^2)*x )) + 2*(2*B*b^2*e^2*x + B*b^2*d*e - (3*B*a*b - 4*A*b^2)*e^2)*sqrt(b*x + a) *sqrt(e*x + d))/(b^3*e^2)]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=\int \frac {\left (A + B x\right ) \sqrt {d + e x}}{\sqrt {a + b x}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (114) = 228\).
Time = 0.33 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.66 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=-\frac {\frac {4 \, {\left (\frac {{\left (b^{2} d - a b e\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e}} - \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \sqrt {b x + a}\right )} A {\left | b \right |}}{b^{2}} - \frac {{\left (\sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} {\left (2 \, b x + 2 \, a + \frac {b d e - 5 \, a e^{2}}{e^{2}}\right )} \sqrt {b x + a} + \frac {{\left (b^{3} d^{2} + 2 \, a b^{2} d e - 3 \, a^{2} b e^{2}\right )} \log \left ({\left | -\sqrt {b e} \sqrt {b x + a} + \sqrt {b^{2} d + {\left (b x + a\right )} b e - a b e} \right |}\right )}{\sqrt {b e} e}\right )} B {\left | b \right |}}{b^{3}}}{4 \, b} \]
-1/4*(4*((b^2*d - a*b*e)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + ( b*x + a)*b*e - a*b*e)))/sqrt(b*e) - sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*sq rt(b*x + a))*A*abs(b)/b^2 - (sqrt(b^2*d + (b*x + a)*b*e - a*b*e)*(2*b*x + 2*a + (b*d*e - 5*a*e^2)/e^2)*sqrt(b*x + a) + (b^3*d^2 + 2*a*b^2*d*e - 3*a^ 2*b*e^2)*log(abs(-sqrt(b*e)*sqrt(b*x + a) + sqrt(b^2*d + (b*x + a)*b*e - a *b*e)))/(sqrt(b*e)*e))*B*abs(b)/b^3)/b
Time = 24.48 (sec) , antiderivative size = 866, normalized size of antiderivative = 6.19 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a+b x}} \, dx=\frac {\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3\,\left (\frac {11\,B\,a^2\,e^2}{2}+23\,B\,a\,b\,d\,e+\frac {7\,B\,b^2\,d^2}{2}\right )}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}+\frac {\left (\sqrt {a+b\,x}-\sqrt {a}\right )\,\left (-\frac {3\,B\,a^2\,b\,e^2}{2}+B\,a\,b^2\,d\,e+\frac {B\,b^3\,d^2}{2}\right )}{e^5\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^7\,\left (-\frac {3\,B\,a^2\,e^2}{2}+B\,a\,b\,d\,e+\frac {B\,b^2\,d^2}{2}\right )}{b^2\,e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^7}+\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^5\,\left (\frac {11\,B\,a^2\,e^2}{2}+23\,B\,a\,b\,d\,e+\frac {7\,B\,b^2\,d^2}{2}\right )}{b\,e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^5}-\frac {\sqrt {a}\,\sqrt {d}\,\left (32\,B\,a\,e+16\,B\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {8\,B\,\sqrt {a}\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}-\frac {8\,B\,\sqrt {a}\,b^2\,d^{3/2}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^4\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^8}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^8}+\frac {b^4}{e^4}-\frac {4\,b^3\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e^3\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}+\frac {6\,b^2\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{e^2\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}-\frac {4\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^6}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^6}}+\frac {\frac {\left (2\,A\,a\,e+2\,A\,b\,d\right )\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{e^2\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}+\frac {\left (2\,A\,a\,e+2\,A\,b\,d\right )\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^3}{b\,e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^3}-\frac {8\,A\,\sqrt {a}\,\sqrt {d}\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}{\frac {{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^4}{{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^4}+\frac {b^2}{e^2}-\frac {2\,b\,{\left (\sqrt {a+b\,x}-\sqrt {a}\right )}^2}{e\,{\left (\sqrt {d+e\,x}-\sqrt {d}\right )}^2}}-\frac {2\,A\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )}{b^{3/2}\,\sqrt {e}}+\frac {B\,\mathrm {atanh}\left (\frac {\sqrt {e}\,\left (\sqrt {a+b\,x}-\sqrt {a}\right )}{\sqrt {b}\,\left (\sqrt {d+e\,x}-\sqrt {d}\right )}\right )\,\left (a\,e-b\,d\right )\,\left (3\,a\,e+b\,d\right )}{2\,b^{5/2}\,e^{3/2}} \]
((((a + b*x)^(1/2) - a^(1/2))^3*((11*B*a^2*e^2)/2 + (7*B*b^2*d^2)/2 + 23*B *a*b*d*e))/(e^4*((d + e*x)^(1/2) - d^(1/2))^3) + (((a + b*x)^(1/2) - a^(1/ 2))*((B*b^3*d^2)/2 - (3*B*a^2*b*e^2)/2 + B*a*b^2*d*e))/(e^5*((d + e*x)^(1/ 2) - d^(1/2))) + (((a + b*x)^(1/2) - a^(1/2))^7*((B*b^2*d^2)/2 - (3*B*a^2* e^2)/2 + B*a*b*d*e))/(b^2*e^2*((d + e*x)^(1/2) - d^(1/2))^7) + (((a + b*x) ^(1/2) - a^(1/2))^5*((11*B*a^2*e^2)/2 + (7*B*b^2*d^2)/2 + 23*B*a*b*d*e))/( b*e^3*((d + e*x)^(1/2) - d^(1/2))^5) - (a^(1/2)*d^(1/2)*(32*B*a*e + 16*B*b *d)*((a + b*x)^(1/2) - a^(1/2))^4)/(e^3*((d + e*x)^(1/2) - d^(1/2))^4) - ( 8*B*a^(1/2)*d^(3/2)*((a + b*x)^(1/2) - a^(1/2))^6)/(e^2*((d + e*x)^(1/2) - d^(1/2))^6) - (8*B*a^(1/2)*b^2*d^(3/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(e^ 4*((d + e*x)^(1/2) - d^(1/2))^2))/(((a + b*x)^(1/2) - a^(1/2))^8/((d + e*x )^(1/2) - d^(1/2))^8 + b^4/e^4 - (4*b^3*((a + b*x)^(1/2) - a^(1/2))^2)/(e^ 3*((d + e*x)^(1/2) - d^(1/2))^2) + (6*b^2*((a + b*x)^(1/2) - a^(1/2))^4)/( e^2*((d + e*x)^(1/2) - d^(1/2))^4) - (4*b*((a + b*x)^(1/2) - a^(1/2))^6)/( e*((d + e*x)^(1/2) - d^(1/2))^6)) + (((2*A*a*e + 2*A*b*d)*((a + b*x)^(1/2) - a^(1/2)))/(e^2*((d + e*x)^(1/2) - d^(1/2))) + ((2*A*a*e + 2*A*b*d)*((a + b*x)^(1/2) - a^(1/2))^3)/(b*e*((d + e*x)^(1/2) - d^(1/2))^3) - (8*A*a^(1 /2)*d^(1/2)*((a + b*x)^(1/2) - a^(1/2))^2)/(e*((d + e*x)^(1/2) - d^(1/2))^ 2))/(((a + b*x)^(1/2) - a^(1/2))^4/((d + e*x)^(1/2) - d^(1/2))^4 + b^2/e^2 - (2*b*((a + b*x)^(1/2) - a^(1/2))^2)/(e*((d + e*x)^(1/2) - d^(1/2))^2...