\(\int \frac {A+B x}{x (a+b x+c x^2)^{5/2}} \, dx\) [167]

Optimal result

Integrand size = 23, antiderivative size = 184 \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (A b^2-a b B-2 a A c+(A b-2 a B) c x\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (8 a^2 b B c+A \left (3 b^4-22 a b^2 c+24 a^2 c^2\right )+c \left (3 A b^3-20 a A b c+16 a^2 B c\right ) x\right )}{3 a^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {A \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{5/2}} \] Output:

2/3*(A*b^2-a*b*B-2*A*a*c+(A*b-2*B*a)*c*x)/a/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/ 
2)+2/3*(8*a^2*b*B*c+A*(24*a^2*c^2-22*a*b^2*c+3*b^4)+c*(-20*A*a*b*c+3*A*b^3 
+16*B*a^2*c)*x)/a^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)-A*arctanh(1/2*(b*x+ 
2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 1.61 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {6 A b^3 x (b+c x)^2+8 a^3 c (3 b B+8 A c+6 B c x)+4 a A b \left (2 b^3-9 b^2 c x-21 b c^2 x^2-10 c^3 x^3\right )+2 a^2 \left (-b^3 B+24 b B c^2 x^2+8 c^3 x^2 (3 A+2 B x)+b^2 (-28 A c+6 B c x)\right )}{3 a^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}+\frac {2 A \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{a^{5/2}} \] Input:

Integrate[(A + B*x)/(x*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(6*A*b^3*x*(b + c*x)^2 + 8*a^3*c*(3*b*B + 8*A*c + 6*B*c*x) + 4*a*A*b*(2*b^ 
3 - 9*b^2*c*x - 21*b*c^2*x^2 - 10*c^3*x^3) + 2*a^2*(-(b^3*B) + 24*b*B*c^2* 
x^2 + 8*c^3*x^2*(3*A + 2*B*x) + b^2*(-28*A*c + 6*B*c*x)))/(3*a^2*(b^2 - 4* 
a*c)^2*(a + x*(b + c*x))^(3/2)) + (2*A*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b 
+ c*x)])/Sqrt[a]])/a^(5/2)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1235, 27, 1235, 27, 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.

\(\displaystyle \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \int -\frac {3 A \left (b^2-4 a c\right )+4 (A b-2 a B) c x}{2 x \left (c x^2+b x+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {3 A \left (b^2-4 a c\right )+4 (A b-2 a B) c x}{x \left (c x^2+b x+a\right )^{3/2}}dx}{3 a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {\frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \int -\frac {3 A \left (b^2-4 a c\right )^2}{2 x \sqrt {c x^2+b x+a}}dx}{a \left (b^2-4 a c\right )}}{3 a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 A \left (b^2-4 a c\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{a}+\frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}}{3 a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {6 A \left (b^2-4 a c\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}}{3 a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {2 \left (c x \left (16 a^2 B c-20 a A b c+3 A b^3\right )+A \left (24 a^2 c^2-22 a b^2 c+3 b^4\right )+8 a^2 b B c\right )}{a \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {3 A \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{a^{3/2}}}{3 a \left (b^2-4 a c\right )}+\frac {2 \left (c x (A b-2 a B)-2 a A c-a b B+A b^2\right )}{3 a \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\)

Input:

Int[(A + B*x)/(x*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(2*(A*b^2 - a*b*B - 2*a*A*c + (A*b - 2*a*B)*c*x))/(3*a*(b^2 - 4*a*c)*(a + 
b*x + c*x^2)^(3/2)) + ((2*(8*a^2*b*B*c + A*(3*b^4 - 22*a*b^2*c + 24*a^2*c^ 
2) + c*(3*A*b^3 - 20*a*A*b*c + 16*a^2*B*c)*x))/(a*(b^2 - 4*a*c)*Sqrt[a + b 
*x + c*x^2]) - (3*A*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + 
b*x + c*x^2])])/a^(3/2))/(3*a*(b^2 - 4*a*c))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

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])
 

rule 1154

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]
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.38

method	result	size
default	\(B \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+A \left (\frac {1}{3 a \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 a}+\frac {\frac {1}{a \sqrt {c \,x^{2}+b x +a}}-\frac {b \left (2 c x +b \right )}{a \left (4 a c -b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}-\frac {\ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{a^{\frac {3}{2}}}}{a}\right )\)	\(253\)

Input:

int((B*x+A)/x/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

B*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c 
*x+b)/(c*x^2+b*x+a)^(1/2))+A*(1/3/a/(c*x^2+b*x+a)^(3/2)-1/2*b/a*(2/3*(2*c* 
x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2 
+b*x+a)^(1/2))+1/a*(1/a/(c*x^2+b*x+a)^(1/2)-b/a*(2*c*x+b)/(4*a*c-b^2)/(c*x 
^2+b*x+a)^(1/2)-1/a^(3/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (169) = 338\).

Time = 1.15 (sec) , antiderivative size = 1077, normalized size of antiderivative = 5.85 \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/x/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

[1/6*(3*(A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*c^2 + (A*b^4*c^2 - 8*A*a*b^2 
*c^3 + 16*A*a^2*c^4)*x^4 + 2*(A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^ 
3 + (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*x^2 + 2*(A*a*b^5 - 8*A*a^2*b^3*c 
+ 16*A*a^3*b*c^2)*x)*sqrt(a)*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c* 
x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) - 4*(B*a^3*b^3 - 4*A*a^2* 
b^4 - 32*A*a^4*c^2 - (3*A*a*b^3*c^2 + 4*(4*B*a^3 - 5*A*a^2*b)*c^3)*x^3 - 6 
*(A*a*b^4*c + 4*A*a^3*c^3 + (4*B*a^3*b - 7*A*a^2*b^2)*c^2)*x^2 - 4*(3*B*a^ 
4*b - 7*A*a^3*b^2)*c - 3*(A*a*b^5 + 8*B*a^4*c^2 + 2*(B*a^3*b^2 - 3*A*a^2*b 
^3)*c)*x)*sqrt(c*x^2 + b*x + a))/(a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2 + (a^ 
3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*x^4 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 
 + 16*a^5*b*c^3)*x^3 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*x^2 + 2*(a^4*b 
^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*x), 1/3*(3*(A*a^2*b^4 - 8*A*a^3*b^2*c + 1 
6*A*a^4*c^2 + (A*b^4*c^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*x^4 + 2*(A*b^5*c 
- 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*x^3 + (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^ 
3)*x^2 + 2*(A*a*b^5 - 8*A*a^2*b^3*c + 16*A*a^3*b*c^2)*x)*sqrt(-a)*arctan(1 
/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 2 
*(B*a^3*b^3 - 4*A*a^2*b^4 - 32*A*a^4*c^2 - (3*A*a*b^3*c^2 + 4*(4*B*a^3 - 5 
*A*a^2*b)*c^3)*x^3 - 6*(A*a*b^4*c + 4*A*a^3*c^3 + (4*B*a^3*b - 7*A*a^2*b^2 
)*c^2)*x^2 - 4*(3*B*a^4*b - 7*A*a^3*b^2)*c - 3*(A*a*b^5 + 8*B*a^4*c^2 + 2* 
(B*a^3*b^2 - 3*A*a^2*b^3)*c)*x)*sqrt(c*x^2 + b*x + a))/(a^5*b^4 - 8*a^6...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)/x/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/x/(c*x^2+b*x+a)^(5/2),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
 

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.81 \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (3 \, A a^{5} b^{3} c^{2} + 16 \, B a^{7} c^{3} - 20 \, A a^{6} b c^{3}\right )} x}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}} + \frac {6 \, {\left (A a^{5} b^{4} c + 4 \, B a^{7} b c^{2} - 7 \, A a^{6} b^{2} c^{2} + 4 \, A a^{7} c^{3}\right )}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )} x + \frac {3 \, {\left (A a^{5} b^{5} + 2 \, B a^{7} b^{2} c - 6 \, A a^{6} b^{3} c + 8 \, B a^{8} c^{2}\right )}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )} x - \frac {B a^{7} b^{3} - 4 \, A a^{6} b^{4} - 12 \, B a^{8} b c + 28 \, A a^{7} b^{2} c - 32 \, A a^{8} c^{2}}{a^{7} b^{4} - 8 \, a^{8} b^{2} c + 16 \, a^{9} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} + \frac {2 \, A \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} \] Input:

integrate((B*x+A)/x/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

2/3*((((3*A*a^5*b^3*c^2 + 16*B*a^7*c^3 - 20*A*a^6*b*c^3)*x/(a^7*b^4 - 8*a^ 
8*b^2*c + 16*a^9*c^2) + 6*(A*a^5*b^4*c + 4*B*a^7*b*c^2 - 7*A*a^6*b^2*c^2 + 
 4*A*a^7*c^3)/(a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2))*x + 3*(A*a^5*b^5 + 2*B 
*a^7*b^2*c - 6*A*a^6*b^3*c + 8*B*a^8*c^2)/(a^7*b^4 - 8*a^8*b^2*c + 16*a^9* 
c^2))*x - (B*a^7*b^3 - 4*A*a^6*b^4 - 12*B*a^8*b*c + 28*A*a^7*b^2*c - 32*A* 
a^8*c^2)/(a^7*b^4 - 8*a^8*b^2*c + 16*a^9*c^2))/(c*x^2 + b*x + a)^(3/2) + 2 
*A*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{x\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int((A + B*x)/(x*(a + b*x + c*x^2)^(5/2)),x)
 

Output:

int((A + B*x)/(x*(a + b*x + c*x^2)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 17.97 (sec) , antiderivative size = 7582, normalized size of antiderivative = 41.21 \[ \int \frac {A+B x}{x \left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/x/(c*x^2+b*x+a)^(5/2),x)
 

Output:

( - 24*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sq 
rt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2) 
)*a**3*b*c + 6*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sq 
rt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c 
 - b**2))*a**2*b**3 - 48*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)* 
atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a) 
*b - 4*a*c - b**2))*a**2*b**2*c*x - 48*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 
4*a*c - b**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*s 
qrt(c)*sqrt(a)*b - 4*a*c - b**2))*a**2*b*c**2*x**2 + 12*sqrt(a)*sqrt(4*sqr 
t(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b 
+ 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2))*a*b**4*x - 12*sqrt(a)*s 
qrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x 
**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2))*a*b**3*c*x**2 
- 48*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*atan((2*sqrt(c)*sqrt 
(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2))* 
a*b**2*c**2*x**3 - 24*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a*c - b**2)*ata 
n((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt(c)*sqrt(a)*b 
- 4*a*c - b**2))*a*b*c**3*x**4 + 6*sqrt(a)*sqrt(4*sqrt(c)*sqrt(a)*b - 4*a* 
c - b**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2) + b + 2*c*x)/sqrt(4*sqrt( 
c)*sqrt(a)*b - 4*a*c - b**2))*b**5*x**2 + 12*sqrt(a)*sqrt(4*sqrt(c)*sqr...