\(\int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx\) [14]

Optimal result

Integrand size = 23, antiderivative size = 191 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=-\frac {(4 c d-b e-2 c e x) \sqrt {a+b x+c x^2}}{4 c e^2}-\frac {\left (\left (b^2-4 a c\right ) e^2-4 c d (2 c d-b e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{8 c^{3/2} e^3}-\frac {d \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^3} \] Output:

-1/4*(-2*c*e*x-b*e+4*c*d)*(c*x^2+b*x+a)^(1/2)/c/e^2-1/8*((-4*a*c+b^2)*e^2- 
4*c*d*(-b*e+2*c*d))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^( 
3/2)/e^3-d*(a*e^2-b*d*e+c*d^2)^(1/2)*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))/e^3
 

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {2 \sqrt {c} \left (e (-4 c d+b e+2 c e x) \sqrt {a+x (b+c x)}-8 c d \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )+\left (8 c^2 d^2-b^2 e^2+4 c e (-b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+x (b+c x)}}\right )}{8 c^{3/2} e^3} \] Input:

Integrate[(x*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
 

Output:

(2*Sqrt[c]*(e*(-4*c*d + b*e + 2*c*e*x)*Sqrt[a + x*(b + c*x)] - 8*c*d*Sqrt[ 
-(c*d^2) + b*d*e - a*e^2]*ArcTan[(Sqrt[c]*(d + e*x) - e*Sqrt[a + x*(b + c* 
x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]]) + (8*c^2*d^2 - b^2*e^2 + 4*c*e*(-(b* 
d) + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/(8*c^(3 
/2)*e^3)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {1231, 27, 1269, 1092, 219, 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.

\(\displaystyle \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx\)

\(\Big \downarrow \) 1231

\(\displaystyle -\frac {\int -\frac {d \left (-e b^2+4 c d b-4 a c e\right )+\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {d \left (-e b^2+4 c d b-4 a c e\right )+\left (8 c^2 d^2-b^2 e^2-4 c e (b d-a e)\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {2 \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {8 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {16 c d \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right )}{\sqrt {c} e}-\frac {8 c d \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}-\frac {\sqrt {a+b x+c x^2} (-b e+4 c d-2 c e x)}{4 c e^2}\)

Input:

Int[(x*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
 

Output:

-1/4*((4*c*d - b*e - 2*c*e*x)*Sqrt[a + b*x + c*x^2])/(c*e^2) + (((8*c^2*d^ 
2 - b^2*e^2 - 4*c*e*(b*d - a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b 
*x + c*x^2])])/(Sqrt[c]*e) - (8*c*d*Sqrt[c*d^2 - b*d*e + a*e^2]*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])])/e)/(8*c*e^2)
 

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

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 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 1.37 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {\left (2 c e x +b e -4 c d \right ) \sqrt {c \,x^{2}+b x +a}}{4 c \,e^{2}}+\frac {\frac {\left (4 a c \,e^{2}-b^{2} e^{2}-4 b c d e +8 c^{2} d^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}+\frac {8 d \left (a \,e^{2}-b d e +c \,d^{2}\right ) c \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{8 e^{2} c}\)	\(280\)
default	\(\frac {\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}}{e}-\frac {d \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{2}}\)	\(400\)

Input:

int(x*(c*x^2+b*x+a)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/4*(2*c*e*x+b*e-4*c*d)/c*(c*x^2+b*x+a)^(1/2)/e^2+1/8/e^2/c*((4*a*c*e^2-b^ 
2*e^2-4*b*c*d*e+8*c^2*d^2)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c 
^(1/2)+8*d*(a*e^2-b*d*e+c*d^2)*c/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)))
 

Fricas [A] (verification not implemented)

Time = 9.43 (sec) , antiderivative size = 1165, normalized size of antiderivative = 6.10 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Too large to display} \] Input:

integrate(x*(c*x^2+b*x+a)^(1/2)/(e*x+d),x, algorithm="fricas")
 

Output:

[1/16*(8*sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d*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)) + (8*c^2*d^2 - 4*b*c*d*e - (b^2 - 4*a*c)*e^2)*sqrt(c)*log(-8*c^2 
*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c 
) + 4*(2*c^2*e^2*x - 4*c^2*d*e + b*c*e^2)*sqrt(c*x^2 + b*x + a))/(c^2*e^3) 
, -1/16*(16*sqrt(-c*d^2 + b*d*e - a*e^2)*c^2*d*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)) - (8*c^2*d^2 - 4*b*c*d*e - (b^2 - 4*a*c)*e^2)*sqrt(c) 
*log(-8*c^2*x^2 - 8*b*c*x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt 
(c) - 4*a*c) - 4*(2*c^2*e^2*x - 4*c^2*d*e + b*c*e^2)*sqrt(c*x^2 + b*x + a) 
)/(c^2*e^3), 1/8*(4*sqrt(c*d^2 - b*d*e + a*e^2)*c^2*d*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)) - (8*c^2*d^2 - 4*b*c*d*e - (b^2 - 4*a*c)*e^2)*sqrt(-c 
)*arctan(1/2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c)/(c^2*x^2 + b*c*x + 
 a*c)) + 2*(2*c^2*e^2*x - 4*c^2*d*e + b*c*e^2)*sqrt(c*x^2 + b*x + a))/(...
 

Sympy [F]

\[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {x \sqrt {a + b x + c x^{2}}}{d + e x}\, dx \] Input:

integrate(x*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)
 

Output:

Integral(x*sqrt(a + b*x + c*x**2)/(d + e*x), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x*(c*x^2+b*x+a)^(1/2)/(e*x+d),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(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x*(c*x^2+b*x+a)^(1/2)/(e*x+d),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx=\int \frac {x\,\sqrt {c\,x^2+b\,x+a}}{d+e\,x} \,d x \] Input:

int((x*(a + b*x + c*x^2)^(1/2))/(d + e*x),x)
 

Output:

int((x*(a + b*x + c*x^2)^(1/2))/(d + e*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.52 (sec) , antiderivative size = 4268, normalized size of antiderivative = 22.35 \[ \int \frac {x \sqrt {a+b x+c x^2}}{d+e x} \, dx =\text {Too large to display} \] Input:

int(x*(c*x^2+b*x+a)^(1/2)/(e*x+d),x)
 

Output:

(8*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e** 
2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2 
)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + 
 b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt( 
c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e 
- 8*c**2*d**2))*b*c**2*d*e - 16*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d** 
2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e 
**2 + 8*b*c*d*e - 8*c**2*d**2)*sqrt(a*e**2 - b*d*e + c*d**2)*atan((2*sqrt( 
c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e**2 - 
b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 4*a*c* 
e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*c**3*d**2 + 16*sqrt(c)*sqrt(4 
*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e 
 + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2)*atan((2 
*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c)*sqrt(a*e 
**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*c*d - 
4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d**2))*a*c**2*d*e**2 - 16*sqrt 
(c)*sqrt(4*sqrt(c)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e* 
*2 - b*d*e + c*d**2)*c*d - 4*a*c*e**2 - b**2*e**2 + 8*b*c*d*e - 8*c**2*d** 
2)*atan((2*sqrt(c)*sqrt(a + b*x + c*x**2)*e + b*e + 2*c*e*x)/sqrt(4*sqrt(c 
)*sqrt(a*e**2 - b*d*e + c*d**2)*b*e - 8*sqrt(c)*sqrt(a*e**2 - b*d*e + c...