Integrand size = 28, antiderivative size = 132 \[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {4}{3} d \sqrt {b d+2 c d x} \sqrt {a+b x+c x^2}+\frac {2 \left (b^2-4 a c\right )^{5/4} d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}} \] Output:
4/3*d*(2*c*d*x+b*d)^(1/2)*(c*x^2+b*x+a)^(1/2)+2/3*(-4*a*c+b^2)^(5/4)*d^(3/ 2)*(-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/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.08 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.84 \[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 d \sqrt {d (b+2 c x)} \left (2 c (a+x (b+c x))+\left (b^2-4 a c\right ) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 c \sqrt {a+x (b+c x)}} \] Input:
Integrate[(b*d + 2*c*d*x)^(3/2)/Sqrt[a + b*x + c*x^2],x]
Output:
(2*d*Sqrt[d*(b + 2*c*x)]*(2*c*(a + x*(b + c*x)) + (b^2 - 4*a*c)*Sqrt[(c*(a + x*(b + c*x)))/(-b^2 + 4*a*c)]*Hypergeometric2F1[1/4, 1/2, 5/4, (b + 2*c *x)^2/(b^2 - 4*a*c)]))/(3*c*Sqrt[a + x*(b + c*x)])
Time = 0.32 (sec) , antiderivative size = 132, 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 = {1116, 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.
\(\displaystyle \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx\) |
\(\Big \downarrow \) 1116 |
\(\displaystyle \frac {1}{3} d^2 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 1115 |
\(\displaystyle \frac {d^2 \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{3 \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 1113 |
\(\displaystyle \frac {2 d \left (b^2-4 a c\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {2 d^{3/2} \left (b^2-4 a c\right )^{5/4} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 c \sqrt {a+b x+c x^2}}+\frac {4}{3} d \sqrt {a+b x+c x^2} \sqrt {b d+2 c d x}\) |
Input:
Int[(b*d + 2*c*d*x)^(3/2)/Sqrt[a + b*x + c*x^2],x]
Output:
(4*d*Sqrt[b*d + 2*c*d*x]*Sqrt[a + b*x + c*x^2])/3 + (2*(b^2 - 4*a*c)^(5/4) *d^(3/2)*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqr t[b*d + 2*c*d*x]/((b^2 - 4*a*c)^(1/4)*Sqrt[d])], -1])/(3*c*Sqrt[a + b*x + c*x^2])
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]
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]
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]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] + Simp[d^2*(m - 1)*((b^2 - 4*a*c)/(b^2*(m + 2*p + 1))) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && (IntegerQ[2*p] || (IntegerQ[m] && RationalQ[p]) || OddQ[m])
Leaf count of result is larger than twice the leaf count of optimal. \(361\) vs. \(2(110)=220\).
Time = 1.39 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.74
method | result | size |
default | \(-\frac {\sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}\, d \left (4 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a c -\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b^{2}-8 c^{3} x^{3}-12 b \,c^{2} x^{2}-8 a \,c^{2} x -4 b^{2} c x -4 a b c \right )}{3 c \left (2 x^{3} c^{2}+3 b c \,x^{2}+2 a c x +b^{2} x +a b \right )}\) | \(362\) |
risch | \(\frac {4 \sqrt {c \,x^{2}+b x +a}\, \left (2 c x +b \right ) d^{2}}{3 \sqrt {d \left (2 c x +b \right )}}+\frac {2 \left (\frac {b^{2}}{3}-\frac {4 a c}{3}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right ) d^{2} \sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}\, \sqrt {d \left (2 c x +b \right )}\, \sqrt {c \,x^{2}+b x +a}}\) | \(454\) |
elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (\frac {4 d \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}{3}+\frac {2 \left (b^{2} d^{2}-\frac {4 d \left (a c d +\frac {1}{2} b^{2} d \right )}{3}\right ) \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{\sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(481\) |
Input:
int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/3*(d*(2*c*x+b))^(1/2)*(c*x^2+b*x+a)^(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)* 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))*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^2-8*c^3*x^3-12 *b*c^2*x^2-8*a*c^2*x-4*b^2*c*x-4*a*b*c)/c/(2*c^2*x^3+3*b*c*x^2+2*a*c*x+b^2 *x+a*b)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61 \[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {4 \, \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a} c^{2} d + \sqrt {2} \sqrt {c^{2} d} {\left (b^{2} - 4 \, a c\right )} d {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right )}{3 \, c^{2}} \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
1/3*(4*sqrt(2*c*d*x + b*d)*sqrt(c*x^2 + b*x + a)*c^2*d + sqrt(2)*sqrt(c^2* d)*(b^2 - 4*a*c)*d*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c))/c^2
\[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (d \left (b + 2 c x\right )\right )^{\frac {3}{2}}}{\sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((d*(b + 2*c*x))**(3/2)/sqrt(a + b*x + c*x**2), x)
\[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate((2*c*d*x + b*d)^(3/2)/sqrt(c*x^2 + b*x + a), x)
\[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x + a}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate((2*c*d*x + b*d)^(3/2)/sqrt(c*x^2 + b*x + a), x)
Timed out. \[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(1/2),x)
Output:
int((b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(1/2), x)
\[ \int \frac {(b d+2 c d x)^{3/2}}{\sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {d}\, d \left (\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, b^{2}+8 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{2} c^{4}+2 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{2} c^{3}-\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x^{2}}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{4} c^{2}+8 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a^{2} b \,c^{3}+2 \left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) a \,b^{3} c^{2}-\left (\int \frac {\sqrt {2 c x +b}\, \sqrt {c \,x^{2}+b x +a}\, x}{4 a \,c^{3} x^{3}+2 b^{2} c^{2} x^{3}+6 a b \,c^{2} x^{2}+3 b^{3} c \,x^{2}+4 a^{2} c^{2} x +4 a \,b^{2} c x +b^{4} x +2 a^{2} b c +a \,b^{3}}d x \right ) b^{5} c \right )}{2 a c +b^{2}} \] Input:
int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
(2*sqrt(d)*d*(sqrt(b + 2*c*x)*sqrt(a + b*x + c*x**2)*b**2 + 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**4 + 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**3 - 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**2 + 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**3 + 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**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)*b**5*c))/(2*a*c + b**2)