Integrand size = 22, antiderivative size = 153 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=\frac {(b d-2 a e+(2 c d-b e) x) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{3/2}} \] Output:
1/4*(b*d-2*a*e+(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e* x+d)^2-1/8*(-4*a*c+b^2)*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e^2-b*d* e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(3/2)
Time = 10.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=\frac {\sqrt {a+x (b+c x)} (-2 a e+2 c d x+b (d-e x))}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}} \] Input:
Integrate[Sqrt[a + b*x + c*x^2]/(d + e*x)^3,x]
Output:
(Sqrt[a + x*(b + c*x)]*(-2*a*e + 2*c*d*x + b*(d - e*x)))/(4*(c*d^2 + e*(-( b*d) + a*e))*(d + e*x)^2) + ((b^2 - 4*a*c)*ArcTanh[(-(b*d) + 2*a*e - 2*c*d *x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])])/(8* (c*d^2 + e*(-(b*d) + a*e))^(3/2))
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1152, 1154, 219}
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.
\(\displaystyle \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx\) |
\(\Big \downarrow \) 1152 |
\(\displaystyle \frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\left (b^2-4 a c\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\sqrt {a+b x+c x^2} (-2 a e+x (2 c d-b e)+b d)}{4 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (b^2-4 a c\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{3/2}}\) |
Input:
Int[Sqrt[a + b*x + c*x^2]/(d + e*x)^3,x]
Output:
((b*d - 2*a*e + (2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])/(4*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - ((b^2 - 4*a*c)*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e )*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/(8*(c*d^2 - b *d*e + a*e^2)^(3/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1137\) vs. \(2(139)=278\).
Time = 1.18 (sec) , antiderivative size = 1138, normalized size of antiderivative = 7.44
Input:
int((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
1/e^3*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c*d)/e*( x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)-1/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2 )*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+( a*e^2-b*d*e+c*d^2)/e^2)^(3/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*((c*(x +d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c* d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c*d)/e*( x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a *e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*( x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e )+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+2*c/(a*e^2-b*d*e+c*d^2)*e^2*(1 /4*(2*c*(x+d/e)+(b*e-2*c*d)/e)/c*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2 -b*d*e+c*d^2)/e^2)^(1/2)+1/8*(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^ 2)/c^(3/2)*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2*c* d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))))+1/2*c/(a*e^2-b*d*e+c*d^2)*e ^2*((c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2* (b*e-2*c*d)/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(1/2)+(c*(x+d/e)^2+(b*e-2 *c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2 )/e^2/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2 *c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(x+d/e)^2+(b*e-2*c*d) /e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e))))
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (139) = 278\).
Time = 0.47 (sec) , antiderivative size = 918, normalized size of antiderivative = 6.00 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x, algorithm="fricas")
Output:
[-1/16*(((b^2 - 4*a*c)*e^2*x^2 + 2*(b^2 - 4*a*c)*d*e*x + (b^2 - 4*a*c)*d^2 )*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d ^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 4*sqrt(c*d^2 - b*d* e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b* c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - 4 *(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3 *b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(c^2 *d^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2* e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x), -1/8*(((b^2 - 4*a*c)*e^2*x^2 + 2*(b^2 - 4*a*c)*d*e* x + (b^2 - 4*a*c)*d^2)*sqrt(-c*d^2 + b*d*e - a*e^2)*arctan(-1/2*sqrt(-c*d^ 2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x)/( a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a*c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) - 2*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x )*sqrt(c*x^2 + b*x + a))/(c^2*d^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2* e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a *b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x)]
\[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a + b x + c x^{2}}}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate((c*x**2+b*x+a)**(1/2)/(e*x+d)**3,x)
Output:
Integral(sqrt(a + b*x + c*x**2)/(d + e*x)**3, x)
Exception generated. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x, algorithm="maxima")
Output:
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*e^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (139) = 278\).
Time = 0.38 (sec) , antiderivative size = 693, normalized size of antiderivative = 4.53 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=-\frac {{\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{4 \, {\left (c d^{2} - b d e + a e^{2}\right )} \sqrt {-c d^{2} + b d e - a e^{2}}} + \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} c^{2} d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b c d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c^{\frac {5}{2}} d^{3} - 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{2} \sqrt {c} d e^{2} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a c^{\frac {3}{2}} d e^{2} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b \sqrt {c} e^{3} + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b c^{2} d^{3} - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{2} c d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a c^{2} d^{2} e - {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{3} d e^{2} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b c d e^{2} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} e^{3} + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c e^{3} + 2 \, b^{2} c^{\frac {3}{2}} d^{3} - b^{3} \sqrt {c} d^{2} e - 4 \, a b c^{\frac {3}{2}} d^{2} e + a b^{2} \sqrt {c} d e^{2} + 4 \, a^{2} c^{\frac {3}{2}} d e^{2}}{4 \, {\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} d + b d - a e\right )}^{2}} \] Input:
integrate((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x, algorithm="giac")
Output:
-1/4*(b^2 - 4*a*c)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a))*e + sqrt(c )*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c*d^2 - b*d*e + a*e^2)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*c^2*d^2*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b*c*d*e^2 + (sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^3*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c^(5/2)*d^3 - 5*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^2*b^2*sqrt(c)*d*e^2 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*c^(3/2)*d*e^2 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*a*b*sqr t(c)*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*c^2*d^3 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^2*c*d^2*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a ))*a*c^2*d^2*e - (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^3*d*e^2 + 4*(sqrt(c )*x - sqrt(c*x^2 + b*x + a))*a*b*c*d*e^2 + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*e^3 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*e^3 + 2*b^2*c ^(3/2)*d^3 - b^3*sqrt(c)*d^2*e - 4*a*b*c^(3/2)*d^2*e + a*b^2*sqrt(c)*d*e^2 + 4*a^2*c^(3/2)*d*e^2)/((c*d^2*e^2 - b*d*e^3 + a*e^4)*((sqrt(c)*x - sqrt( c*x^2 + b*x + a))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c)*d + b*d - a*e)^2)
Timed out. \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx=\int \frac {\sqrt {c\,x^2+b\,x+a}}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int((a + b*x + c*x^2)^(1/2)/(d + e*x)^3,x)
Output:
int((a + b*x + c*x^2)^(1/2)/(d + e*x)^3, x)
Time = 0.21 (sec) , antiderivative size = 1033, normalized size of antiderivative = 6.75 \[ \int \frac {\sqrt {a+b x+c x^2}}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
int((c*x^2+b*x+a)^(1/2)/(e*x+d)^3,x)
Output:
(4*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c*d**2 + 8*sqrt(a*e** 2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d **2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c*d*e*x + 4*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*c*e**2*x**2 - sqrt(a*e**2 - b*d*e + c*d**2)*lo g(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b *e*x + 2*c*d*x)*b**2*d**2 - 2*sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d *x)*b**2*d*e*x - sqrt(a*e**2 - b*d*e + c*d**2)*log(2*sqrt(a + b*x + c*x**2 )*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*b**2*e**2 *x**2 - 4*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a*c*d**2 - 8*sqrt(a*e **2 - b*d*e + c*d**2)*log(d + e*x)*a*c*d*e*x - 4*sqrt(a*e**2 - b*d*e + c*d **2)*log(d + e*x)*a*c*e**2*x**2 + sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e* x)*b**2*d**2 + 2*sqrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b**2*d*e*x + s qrt(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*b**2*e**2*x**2 - 4*sqrt(a + b*x + c*x**2)*a**2*e**3 + 6*sqrt(a + b*x + c*x**2)*a*b*d*e**2 - 2*sqrt(a + b*x + c*x**2)*a*b*e**3*x - 4*sqrt(a + b*x + c*x**2)*a*c*d**2*e + 4*sqrt(a + b *x + c*x**2)*a*c*d*e**2*x - 2*sqrt(a + b*x + c*x**2)*b**2*d**2*e + 2*sqrt( a + b*x + c*x**2)*b**2*d*e**2*x + 2*sqrt(a + b*x + c*x**2)*b*c*d**3 - 6...