\(\int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx\) [225]

Optimal result

Integrand size = 28, antiderivative size = 137 \[ \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}} \] Output:

1/3*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/d-1/3*(-4*a*c+b^2)^(5/4)*(-c 
*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF((2*c*d*x+b*d)^(1/2)/(-4*a*c+b 
^2)^(1/4)/d^(1/2),I)/c^2/d^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {d (b+2 c x)} \sqrt {a+x (b+c x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{2 c d \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}}} \] Input:

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

Output:

(Sqrt[d*(b + 2*c*x)]*Sqrt[a + x*(b + c*x)]*Hypergeometric2F1[-1/2, 1/4, 5/ 
4, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(2*c*d*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 
 4*a*c)])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1109, 1115, 1113, 762}

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.

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

\(\Big \downarrow \) 1109

\(\displaystyle \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx}{6 c}\)

\(\Big \downarrow \) 1115

\(\displaystyle \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{6 c \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 1113

\(\displaystyle \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c^2 d \sqrt {a+b x+c x^2}}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}}{3 c d}-\frac {\left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c^2 \sqrt {d} \sqrt {a+b x+c x^2}}\)

Input:

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

Output:

(Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/(3*c*d) - ((b^2 - 4*a*c)^(5/4) 
*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[b*d + 
2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c^2*Sqrt[d]*Sqrt[a + b*x 
+ c*x^2])
 

Defintions of rubi rules used

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 1109

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]
 

rule 1113

Int[1/(Sqrt[(d_) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_ 
Symbol] :> Simp[(4/e)*Sqrt[-c/(b^2 - 4*a*c)]   Subst[Int[1/Sqrt[Simp[1 - b^ 
2*(x^4/(d^2*(b^2 - 4*a*c))), x]], x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, 
 c, d, e}, x] && EqQ[2*c*d - b*e, 0] && LtQ[c/(b^2 - 4*a*c), 0]
 

rule 1115

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sym 
bol] :> Simp[Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/Sqrt[a + b*x + c* 
x^2]   Int[(d + e*x)^m/Sqrt[(-a)*(c/(b^2 - 4*a*c)) - b*c*(x/(b^2 - 4*a*c)) 
- c^2*(x^2/(b^2 - 4*a*c))], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c* 
d - b*e, 0] && EqQ[m^2, 1/4]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs. \(2(115)=230\).

Time = 1.18 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.66

method	result	size
default	\(\frac {\sqrt {c \,x^{2}+b x +a}\, \sqrt {d \left (2 c x +b \right )}\, \left (4 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a c -\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{2}+4 c^{3} x^{3}+6 b \,c^{2} x^{2}+4 a \,c^{2} x +2 b^{2} c x +2 a b c \right )}{6 d \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right ) c^{2}}\)	\(364\)
risch	\(\frac {\sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right )}{3 c \sqrt {d \left (2 c x +b \right )}}+\frac {\left (4 a c -b^{2}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{3 c \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(454\)
elliptic	\(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}{3 c d}+\frac {2 \left (a -\frac {a c d +\frac {1}{2} b^{2} d}{3 c d}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right )}{\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\)	\(474\)

Input:

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

Output:

1/6*(c*x^2+b*x+a)^(1/2)*(d*(2*c*x+b))^(1/2)/d*(4*(-4*a*c+b^2)^(1/2)*(1/(-4 
*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^ 
2)^(1/2))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*E 
llipticF(1/2*2^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^( 
1/2),2^(1/2))*a*c-(-4*a*c+b^2)^(1/2)*(1/(-4*a*c+b^2)^(1/2)*(2*c*x+(-4*a*c+ 
b^2)^(1/2)+b))^(1/2)*(-(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*((-2*c*x+(-4*a* 
c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*EllipticF(1/2*2^(1/2)*(1/(-4*a*c 
+b^2)^(1/2)*(2*c*x+(-4*a*c+b^2)^(1/2)+b))^(1/2),2^(1/2))*b^2+4*c^3*x^3+6*b 
*c^2*x^2+4*a*c^2*x+2*b^2*c*x+2*a*b*c)/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2*x+a 
*b)/c^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx=\frac {2 \, \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} c^{2} - \sqrt {2} \sqrt {c^{2} d} {\left (b^{2} - 4 \, a c\right )} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )}{6 \, c^{3} d} \] Input:

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

Output:

1/6*(2*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)*c^2 - sqrt(2)*sqrt(c^2*d) 
*(b^2 - 4*a*c)*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c 
))/(c^3*d)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(sqrt(c*x^2 + b*x + a)/sqrt(2*c*d*x + b*d), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(c*x^2 + b*x + a)/sqrt(2*c*d*x + b*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b x+c x^2}}{\sqrt {b d+2 c d x}} \, dx=\frac {\sqrt {d}\, \left (2 \sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, a -8 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{2} c^{3}-2 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{2} c^{2}+\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{4} c -8 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{2} b \,c^{2}-2 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{3} c +\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{5}\right )}{d \left (2 a c +b^{2}\right )} \] Input:

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

Output:

(sqrt(d)*(2*sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*a - 8*int((sqrt(b + 2*c 
*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4* 
a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2* 
b**2*c**2*x**3),x)*a**2*c**3 - 2*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x** 
2)*x**2)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2* 
x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a*b** 
2*c**2 + int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**2*b*c + 4 
*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b 
**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*b**4*c - 8*int((sqrt(b + 2*c* 
x)*sqrt(a + b*x + c*x**2)*x)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b* 
*2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2 
*c**2*x**3),x)*a**2*b*c**2 - 2*int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2) 
*x)/(2*a**2*b*c + 4*a**2*c**2*x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 
+ 4*a*c**3*x**3 + b**4*x + 3*b**3*c*x**2 + 2*b**2*c**2*x**3),x)*a*b**3*c + 
 int((sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*x)/(2*a**2*b*c + 4*a**2*c**2* 
x + a*b**3 + 4*a*b**2*c*x + 6*a*b*c**2*x**2 + 4*a*c**3*x**3 + b**4*x + 3*b 
**3*c*x**2 + 2*b**2*c**2*x**3),x)*b**5))/(d*(2*a*c + b**2))