Integrand size = 28, antiderivative size = 173 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}-\frac {4 c d \sqrt {b d+2 c d x}}{3 \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {8 c d^{3/2} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c d x}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{3 \left (b^2-4 a c\right )^{3/4} \sqrt {a+b x+c x^2}} \] Output:
-2/3*d*(2*c*d*x+b*d)^(1/2)/(c*x^2+b*x+a)^(3/2)-4/3*c*d*(2*c*d*x+b*d)^(1/2) /(-4*a*c+b^2)/(c*x^2+b*x+a)^(1/2)-8/3*c*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)/(- 4*a*c+b^2)^(3/4)/(c*x^2+b*x+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.11 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.75 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d \sqrt {d (b+2 c x)} \left (b^2+2 b c x+2 c \left (-a+c x^2\right )+4 c (a+x (b+c x)) \sqrt {\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {(b+2 c x)^2}{b^2-4 a c}\right )\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \] Input:
Integrate[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*d*Sqrt[d*(b + 2*c*x)]*(b^2 + 2*b*c*x + 2*c*(-a + c*x^2) + 4*c*(a + x*( b + c*x))*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*(b^2 - 4*a*c)*(a + x*(b + c* x))^(3/2))
Time = 0.39 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1110, 1111, 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.
| \(\displaystyle \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1110 |
| \(\displaystyle \frac {2}{3} c d^2 \int \frac {1}{\sqrt {b d+2 c x d} \left (c x^2+b x+a\right )^{3/2}}dx-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1111 |
| \(\displaystyle \frac {2}{3} c d^2 \left (-\frac {2 c \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {c x^2+b x+a}}dx}{b^2-4 a c}-\frac {2 \sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1115 |
| \(\displaystyle \frac {2}{3} c d^2 \left (-\frac {2 c \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {b d+2 c x d} \sqrt {-\frac {c^2 x^2}{b^2-4 a c}-\frac {b c x}{b^2-4 a c}-\frac {a c}{b^2-4 a c}}}dx}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1113 |
| \(\displaystyle \frac {2}{3} c d^2 \left (-\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {(b d+2 c x d)^2}{\left (b^2-4 a c\right ) d^2}}}d\sqrt {b d+2 c x d}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 762 |
| \(\displaystyle \frac {2}{3} c d^2 \left (-\frac {4 \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b d+2 c x d}}{\sqrt [4]{b^2-4 a c} \sqrt {d}}\right ),-1\right )}{\sqrt {d} \left (b^2-4 a c\right )^{3/4} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {b d+2 c d x}}{d \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}\right )-\frac {2 d \sqrt {b d+2 c d x}}{3 \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[(b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*d*Sqrt[b*d + 2*c*d*x])/(3*(a + b*x + c*x^2)^(3/2)) + (2*c*d^2*((-2*Sqr t[b*d + 2*c*d*x])/((b^2 - 4*a*c)*d*Sqrt[a + b*x + c*x^2]) - (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])/((b^2 - 4*a*c)^(3/4)*Sqrt[d]*Sqrt[a + b* x + c*x^2])))/3
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[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy mbol] :> Simp[d*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d*e*((m - 1)/(b*(p + 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] && N eQ[m + 2*p + 3, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[2*p]
Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_S ymbol] :> Simp[2*c*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)* (b^2 - 4*a*c))), x] - Simp[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*c))) 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] && LtQ[p, -1] && !G tQ[m, 1] && RationalQ[m] && IntegerQ[2*p]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(486\) vs. \(2(145)=290\).
Time = 2.85 (sec) , antiderivative size = 487, normalized size of antiderivative = 2.82
| method | result | size |
| default | \(\frac {2 \left (2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) c^{2} x^{2}+2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) b c x +2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {-\frac {2 c x +b}{\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {-2 c x +\sqrt {-4 a c +b^{2}}-b}{\sqrt {-4 a c +b^{2}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {2 c x +\sqrt {-4 a c +b^{2}}+b}{\sqrt {-4 a c +b^{2}}}}}{2}, \sqrt {2}\right ) a c +4 c^{3} x^{3}+6 b \,c^{2} x^{2}-4 a \,c^{2} x +4 b^{2} c x -2 a b c +b^{3}\right ) d \sqrt {d \left (2 c x +b \right )}}{3 \left (4 a c -b^{2}\right ) \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(487\) |
| elliptic | \(\frac {\sqrt {d \left (2 c x +b \right ) \left (c \,x^{2}+b x +a \right )}\, \sqrt {d \left (2 c x +b \right )}\, \left (-\frac {2 d \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}{3 c^{2} \left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right )^{2}}+\frac {4 \left (2 c^{2} d x +d b c \right ) d}{3 \left (4 a c -b^{2}\right ) \sqrt {\left (\frac {a}{c}+\frac {b x}{c}+x^{2}\right ) \left (2 c^{2} d x +d b c \right )}}+\frac {8 c^{2} d^{2} \left (\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}, \sqrt {\frac {-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {b}{2 c}}}\right )}{3 \left (4 a c -b^{2}\right ) \sqrt {2 c^{2} d \,x^{3}+3 b c d \,x^{2}+2 a d x c +b^{2} d x +d a b}}\right )}{\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}\, d}\) | \(554\) |
Input:
int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/3*(2*(-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))*c^2*x^2+2*(-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*c*x+2*(-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* c^3*x^3+6*b*c^2*x^2-4*a*c^2*x+4*b^2*c*x-2*a*b*c+b^3)*d*(d*(2*c*x+b))^(1/2) /(4*a*c-b^2)/(2*c*x+b)/(c*x^2+b*x+a)^(3/2)
Time = 0.10 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.28 \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} {\left (c^{2} d x^{4} + 2 \, b c d x^{3} + 2 \, a b d x + {\left (b^{2} + 2 \, a c\right )} d x^{2} + a^{2} d\right )} \sqrt {c^{2} d} {\rm weierstrassPInverse}\left (\frac {b^{2} - 4 \, a c}{c^{2}}, 0, \frac {2 \, c x + b}{2 \, c}\right ) + {\left (2 \, c^{2} d x^{2} + 2 \, b c d x + {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {2 \, c d x + b d} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
-2/3*(2*sqrt(2)*(c^2*d*x^4 + 2*b*c*d*x^3 + 2*a*b*d*x + (b^2 + 2*a*c)*d*x^2 + a^2*d)*sqrt(c^2*d)*weierstrassPInverse((b^2 - 4*a*c)/c^2, 0, 1/2*(2*c*x + b)/c) + (2*c^2*d*x^2 + 2*b*c*d*x + (b^2 - 2*a*c)*d)*sqrt(2*c*d*x + b*d) *sqrt(c*x^2 + b*x + a))/((b^2*c^2 - 4*a*c^3)*x^4 + a^2*b^2 - 4*a^3*c + 2*( b^3*c - 4*a*b*c^2)*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*x^2 + 2*(a*b^3 - 4* a^2*b*c)*x)
Timed out. \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2*c*d*x+b*d)**(3/2)/(c*x**2+b*x+a)**(5/2),x)
Output:
Timed out
\[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2), x)
\[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (2 \, c d x + b d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate((2*c*d*x + b*d)^(3/2)/(c*x^2 + b*x + a)^(5/2), x)
Timed out. \[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int((b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2),x)
Output:
int((b*d + 2*c*d*x)^(3/2)/(a + b*x + c*x^2)^(5/2), x)
\[ \int \frac {(b d+2 c d x)^{3/2}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((2*c*d*x+b*d)^(3/2)/(c*x^2+b*x+a)^(5/2),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**4*b*c + 4*a**4*c**2*x - 3*a**3 *b**3 + 18*a**3*b*c**2*x**2 + 12*a**3*c**3*x**3 - 9*a**2*b**4*x - 21*a**2* b**3*c*x**2 + 6*a**2*b**2*c**2*x**3 + 30*a**2*b*c**3*x**4 + 12*a**2*c**4*x **5 - 9*a*b**5*x**2 - 34*a*b**4*c*x**3 - 35*a*b**3*c**2*x**4 + 14*a*b*c**4 *x**6 + 4*a*c**5*x**7 - 3*b**6*x**3 - 15*b**5*c*x**4 - 27*b**4*c**2*x**5 - 21*b**3*c**3*x**6 - 6*b**2*c**4*x**7),x)*a**4*c**4 - 14*int((sqrt(b + 2*c *x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**4*b*c + 4*a**4*c**2*x - 3*a**3*b**3 + 18*a**3*b*c**2*x**2 + 12*a**3*c**3*x**3 - 9*a**2*b**4*x - 21*a**2*b**3* c*x**2 + 6*a**2*b**2*c**2*x**3 + 30*a**2*b*c**3*x**4 + 12*a**2*c**4*x**5 - 9*a*b**5*x**2 - 34*a*b**4*c*x**3 - 35*a*b**3*c**2*x**4 + 14*a*b*c**4*x**6 + 4*a*c**5*x**7 - 3*b**6*x**3 - 15*b**5*c*x**4 - 27*b**4*c**2*x**5 - 21*b **3*c**3*x**6 - 6*b**2*c**4*x**7),x)*a**3*b**2*c**3 + 16*int((sqrt(b + 2*c *x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**4*b*c + 4*a**4*c**2*x - 3*a**3*b**3 + 18*a**3*b*c**2*x**2 + 12*a**3*c**3*x**3 - 9*a**2*b**4*x - 21*a**2*b**3* c*x**2 + 6*a**2*b**2*c**2*x**3 + 30*a**2*b*c**3*x**4 + 12*a**2*c**4*x**5 - 9*a*b**5*x**2 - 34*a*b**4*c*x**3 - 35*a*b**3*c**2*x**4 + 14*a*b*c**4*x**6 + 4*a*c**5*x**7 - 3*b**6*x**3 - 15*b**5*c*x**4 - 27*b**4*c**2*x**5 - 21*b **3*c**3*x**6 - 6*b**2*c**4*x**7),x)*a**3*b*c**4*x + 16*int((sqrt(b + 2*c* x)*sqrt(a + b*x + c*x**2)*x**2)/(2*a**4*b*c + 4*a**4*c**2*x - 3*a**3*b*...