Integrand size = 26, antiderivative size = 115 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\frac {3 \sqrt {a+b x+c x^2}}{16 c^2 d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}-\frac {3 \sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{32 c^{5/2} d^3} \] Output:
3/16*(c*x^2+b*x+a)^(1/2)/c^2/d^3-1/4*(c*x^2+b*x+a)^(3/2)/c/d^3/(2*c*x+b)^2 -3/32*(-4*a*c+b^2)^(1/2)*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2) ^(1/2))/c^(5/2)/d^3
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\frac {2 (a+x (b+c x))^{5/2} \operatorname {Hypergeometric2F1}\left (2,\frac {5}{2},\frac {7}{2},\frac {4 c (a+x (b+c x))}{-b^2+4 a c}\right )}{5 \left (b^2-4 a c\right )^2 d^3} \] Input:
Integrate[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^3,x]
Output:
(2*(a + x*(b + c*x))^(5/2)*Hypergeometric2F1[2, 5/2, 7/2, (4*c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)])/(5*(b^2 - 4*a*c)^2*d^3)
Time = 0.28 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1108, 27, 1109, 1112, 218}
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.
| \(\displaystyle \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx\) |
| \(\Big \downarrow \) 1108 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {c x^2+b x+a}}{d (b+2 c x)}dx}{8 c d^2}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {c x^2+b x+a}}{b+2 c x}dx}{8 c d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}\) |
| \(\Big \downarrow \) 1109 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{(b+2 c x) \sqrt {c x^2+b x+a}}dx}{4 c}\right )}{8 c d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}\) |
| \(\Big \downarrow \) 1112 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\left (b^2-4 a c\right ) \int \frac {1}{8 \left (c x^2+b x+a\right ) c^2+2 \left (b^2-4 a c\right ) c}d\sqrt {c x^2+b x+a}\right )}{8 c d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {3 \left (\frac {\sqrt {a+b x+c x^2}}{2 c}-\frac {\sqrt {b^2-4 a c} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{4 c^{3/2}}\right )}{8 c d^3}-\frac {\left (a+b x+c x^2\right )^{3/2}}{4 c d^3 (b+2 c x)^2}\) |
Input:
Int[(a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^3,x]
Output:
-1/4*(a + b*x + c*x^2)^(3/2)/(c*d^3*(b + 2*c*x)^2) + (3*(Sqrt[a + b*x + c* x^2]/(2*c) - (Sqrt[b^2 - 4*a*c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/S qrt[b^2 - 4*a*c]])/(4*c^(3/2))))/(8*c*d^3)
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_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si mp[b*(p/(d*e*(m + 1))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x ], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && LtQ[m, -1] && !(IntegerQ[m/2] && LtQ[m + 2*p + 3, 0] ) && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x ] - Simp[d*p*((b^2 - 4*a*c)/(b*e*(m + 2*p + 1))) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b* e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[p, 0] && !LtQ[m, -1] && !(IGtQ[(m - 1 )/2, 0] && ( !IntegerQ[p] || LtQ[m, 2*p])) && RationalQ[m] && IntegerQ[2*p]
Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symb ol] :> Simp[4*c Subst[Int[1/(b^2*e - 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Time = 1.06 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.99
| method | result | size |
| pseudoelliptic | \(\frac {\sqrt {c \,x^{2}+b x +a}+\frac {\left (4 a c -b^{2}\right ) \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{4 c^{2} x^{2}+4 c b x +b^{2}}-\frac {3 \,\operatorname {arctanh}\left (\frac {2 \sqrt {c \,x^{2}+b x +a}\, c}{\sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{2 \sqrt {\left (4 a c -b^{2}\right ) c}}\right )}{2}}{8 c^{2} d^{3}}\) | \(114\) |
| default | \(\frac {-\frac {2 c \left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {5}{2}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {6 c^{2} \left (\frac {\left (c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}{3}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {\sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}-\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{2 c \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 c}\right )}{4 a c -b^{2}}}{8 d^{3} c^{3}}\) | \(270\) |
| risch | \(\frac {\sqrt {c \,x^{2}+b x +a}}{8 c^{2} d^{3}}+\frac {-\frac {\left (8 a c -2 b^{2}\right ) \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{c \sqrt {\frac {4 a c -b^{2}}{c}}}+\frac {\left (16 a^{2} c^{2}-8 c a \,b^{2}+b^{4}\right ) \left (-\frac {2 c \sqrt {c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{4 c}}}{\left (4 a c -b^{2}\right ) \left (x +\frac {b}{2 c}\right )^{2}}+\frac {4 c^{2} \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 c \left (x +\frac {b}{2 c}\right )^{2}+\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{8 c^{3}}}{16 c^{2} d^{3}}\) | \(327\) |
Input:
int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^3,x,method=_RETURNVERBOSE)
Output:
1/8/c^2*((c*x^2+b*x+a)^(1/2)+1/2*(4*a*c-b^2)*(-(c*x^2+b*x+a)^(1/2)/(4*c^2* x^2+4*b*c*x+b^2)-3/2/((4*a*c-b^2)*c)^(1/2)*arctanh(2*(c*x^2+b*x+a)^(1/2)*c /((4*a*c-b^2)*c)^(1/2))))/d^3
Time = 0.19 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.88 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\left [\frac {3 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {-\frac {b^{2} - 4 \, a c}{c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c - 4 \, \sqrt {c x^{2} + b x + a} c \sqrt {-\frac {b^{2} - 4 \, a c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + 4 \, {\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt {c x^{2} + b x + a}}{64 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}}, \frac {3 \, {\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c}} \arctan \left (-\frac {2 \, \sqrt {c x^{2} + b x + a} c \sqrt {\frac {b^{2} - 4 \, a c}{c}}}{b^{2} - 4 \, a c}\right ) + 2 \, {\left (8 \, c^{2} x^{2} + 8 \, b c x + 3 \, b^{2} - 4 \, a c\right )} \sqrt {c x^{2} + b x + a}}{32 \, {\left (4 \, c^{4} d^{3} x^{2} + 4 \, b c^{3} d^{3} x + b^{2} c^{2} d^{3}\right )}}\right ] \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^3,x, algorithm="fricas")
Output:
[1/64*(3*(4*c^2*x^2 + 4*b*c*x + b^2)*sqrt(-(b^2 - 4*a*c)/c)*log(-(4*c^2*x^ 2 + 4*b*c*x - b^2 + 8*a*c - 4*sqrt(c*x^2 + b*x + a)*c*sqrt(-(b^2 - 4*a*c)/ c))/(4*c^2*x^2 + 4*b*c*x + b^2)) + 4*(8*c^2*x^2 + 8*b*c*x + 3*b^2 - 4*a*c) *sqrt(c*x^2 + b*x + a))/(4*c^4*d^3*x^2 + 4*b*c^3*d^3*x + b^2*c^2*d^3), 1/3 2*(3*(4*c^2*x^2 + 4*b*c*x + b^2)*sqrt((b^2 - 4*a*c)/c)*arctan(-2*sqrt(c*x^ 2 + b*x + a)*c*sqrt((b^2 - 4*a*c)/c)/(b^2 - 4*a*c)) + 2*(8*c^2*x^2 + 8*b*c *x + 3*b^2 - 4*a*c)*sqrt(c*x^2 + b*x + a))/(4*c^4*d^3*x^2 + 4*b*c^3*d^3*x + b^2*c^2*d^3)]
\[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\frac {\int \frac {a \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {b x \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx + \int \frac {c x^{2} \sqrt {a + b x + c x^{2}}}{b^{3} + 6 b^{2} c x + 12 b c^{2} x^{2} + 8 c^{3} x^{3}}\, dx}{d^{3}} \] Input:
integrate((c*x**2+b*x+a)**(3/2)/(2*c*d*x+b*d)**3,x)
Output:
(Integral(a*sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8 *c**3*x**3), x) + Integral(b*x*sqrt(a + b*x + c*x**2)/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c**3*x**3), x) + Integral(c*x**2*sqrt(a + b*x + c*x**2 )/(b**3 + 6*b**2*c*x + 12*b*c**2*x**2 + 8*c**3*x**3), x))/d**3
Exception generated. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*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(4*a*c-b^2>0)', see `assume?` for more deta
Leaf count of result is larger than twice the leaf count of optimal. 378 vs. \(2 (95) = 190\).
Time = 0.38 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.29 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=-\frac {3 \, {\left (b^{2} - 4 \, a c\right )} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{16 \, \sqrt {b^{2} c - 4 \, a c^{2}} c^{2} d^{3}} + \frac {\sqrt {c x^{2} + b x + a}}{8 \, c^{2} d^{3}} - \frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} c - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} b^{3} \sqrt {c} - 12 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b c^{\frac {3}{2}} + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b^{4} - 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} c - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c^{2} + a b^{3} \sqrt {c} - 4 \, a^{2} b c^{\frac {3}{2}}}{16 \, {\left (2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} c + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b \sqrt {c} + b^{2} - 2 \, a c\right )}^{2} c^{2} d^{3}} \] Input:
integrate((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^3,x, algorithm="giac")
Output:
-3/16*(b^2 - 4*a*c)*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*s qrt(c))/sqrt(b^2*c - 4*a*c^2))/(sqrt(b^2*c - 4*a*c^2)*c^2*d^3) + 1/8*sqrt( c*x^2 + b*x + a)/(c^2*d^3) - 1/16*(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3 *b^2*c - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^2 + 3*(sqrt(c)*x - sq rt(c*x^2 + b*x + a))^2*b^3*sqrt(c) - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) )^2*a*b*c^(3/2) + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b^4 - 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*c - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a^ 2*c^2 + a*b^3*sqrt(c) - 4*a^2*b*c^(3/2))/((2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*c + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b*sqrt(c) + b^2 - 2*a*c )^2*c^2*d^3)
Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{3/2}}{{\left (b\,d+2\,c\,d\,x\right )}^3} \,d x \] Input:
int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^3,x)
Output:
int((a + b*x + c*x^2)^(3/2)/(b*d + 2*c*d*x)^3, x)
Time = 0.30 (sec) , antiderivative size = 501, normalized size of antiderivative = 4.36 \[ \int \frac {\left (a+b x+c x^2\right )^{3/2}}{(b d+2 c d x)^3} \, dx=\frac {3 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}+12 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x +12 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {-\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}-3 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b^{2}-12 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) b c x -12 \sqrt {c}\, \sqrt {4 a c -b^{2}}\, \mathrm {log}\left (\frac {\sqrt {4 a c -b^{2}}+2 \sqrt {c}\, \sqrt {c \,x^{2}+b x +a}+b +2 c x}{\sqrt {4 a c -b^{2}}}\right ) c^{2} x^{2}-8 \sqrt {c \,x^{2}+b x +a}\, a \,c^{2}+6 \sqrt {c \,x^{2}+b x +a}\, b^{2} c +16 \sqrt {c \,x^{2}+b x +a}\, b \,c^{2} x +16 \sqrt {c \,x^{2}+b x +a}\, c^{3} x^{2}}{32 c^{3} d^{3} \left (4 c^{2} x^{2}+4 b c x +b^{2}\right )} \] Input:
int((c*x^2+b*x+a)^(3/2)/(2*c*d*x+b*d)^3,x)
Output:
(3*sqrt(c)*sqrt(4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt( a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b**2 + 12*sqrt(c)*sqrt( 4*a*c - b**2)*log(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2 ) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*x + 12*sqrt(c)*sqrt(4*a*c - b**2)*l og(( - sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/ sqrt(4*a*c - b**2))*c**2*x**2 - 3*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a *c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b* *2))*b**2 - 12*sqrt(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt (c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*x - 12*sqr t(c)*sqrt(4*a*c - b**2)*log((sqrt(4*a*c - b**2) + 2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*x**2 - 8*sqrt(a + b*x + c*x **2)*a*c**2 + 6*sqrt(a + b*x + c*x**2)*b**2*c + 16*sqrt(a + b*x + c*x**2)* b*c**2*x + 16*sqrt(a + b*x + c*x**2)*c**3*x**2)/(32*c**3*d**3*(b**2 + 4*b* c*x + 4*c**2*x**2))