Integrand size = 27, antiderivative size = 121 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (b B d-2 A c d+A b e-2 a B e) (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
-2/3*(A*b-2*B*a-(-2*A*c+B*b)*x)*(e*x+d)^2/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2) -8/3*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2) ^2/(c*x^2+b*x+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(121)=242\).
Time = 1.91 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 A \left (-b^3 \left (d^2+6 d e x-3 e^2 x^2\right )+4 b \left (2 a^2 e^2+2 c^2 d x^2 (3 d-2 e x)+3 a c (d-e x)^2\right )+8 c \left (-2 a^2 d e+2 c^2 d^2 x^3+a c x \left (3 d^2+e^2 x^2\right )\right )+b^2 \left (-4 a e (d-3 e x)+2 c x \left (3 d^2-12 d e x+e^2 x^2\right )\right )\right )-2 B \left (16 a^3 e^2+b x \left (8 c^2 d^2 x^2+4 b c d x (3 d-e x)+b^2 \left (3 d^2-6 d e x-e^2 x^2\right )\right )+8 a^2 \left (b e (-2 d+3 e x)+c \left (d^2+3 e^2 x^2\right )\right )+2 a \left (-8 c^2 d e x^3+6 b c x (d-e x)^2+b^2 \left (d^2-12 d e x+3 e^2 x^2\right )\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
(2*A*(-(b^3*(d^2 + 6*d*e*x - 3*e^2*x^2)) + 4*b*(2*a^2*e^2 + 2*c^2*d*x^2*(3 *d - 2*e*x) + 3*a*c*(d - e*x)^2) + 8*c*(-2*a^2*d*e + 2*c^2*d^2*x^3 + a*c*x *(3*d^2 + e^2*x^2)) + b^2*(-4*a*e*(d - 3*e*x) + 2*c*x*(3*d^2 - 12*d*e*x + e^2*x^2))) - 2*B*(16*a^3*e^2 + b*x*(8*c^2*d^2*x^2 + 4*b*c*d*x*(3*d - e*x) + b^2*(3*d^2 - 6*d*e*x - e^2*x^2)) + 8*a^2*(b*e*(-2*d + 3*e*x) + c*(d^2 + 3*e^2*x^2)) + 2*a*(-8*c^2*d*e*x^3 + 6*b*c*x*(d - e*x)^2 + b^2*(d^2 - 12*d* e*x + 3*e^2*x^2))))/(3*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2))
Time = 0.25 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1227, 1158}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1227 |
| \(\displaystyle \frac {4 (-2 a B e+A b e-2 A c d+b B d) \int \frac {d+e x}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1158 |
| \(\displaystyle -\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (-2 a e+x (2 c d-b e)+b d) (-2 a B e+A b e-2 A c d+b B d)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\) |
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - (8*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*(b*d - 2*a*e + ( 2*c*d - b*e)*x))/(3*(b^2 - 4*a*c)^2*Sqrt[a + b*x + c*x^2])
3.25.82.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[-2*((b*d - 2*a*e + (2*c*d - b*e)*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( (b*f - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] - Simp[m*((b*( e*f + d*g) - 2*(c*d*f + a*e*g))/((p + 1)*(b^2 - 4*a*c))) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[Simplify[m + 2*p + 3], 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(423\) vs. \(2(113)=226\).
Time = 0.55 (sec) , antiderivative size = 424, normalized size of antiderivative = 3.50
| method | result | size |
| trager | \(\frac {-\frac {32}{3} B \,e^{2} a^{3}-\frac {2}{3} A \,d^{2} b^{3}-\frac {32}{3} A b \,c^{2} d e \,x^{3}+\frac {8}{3} B \,b^{2} c d e \,x^{3}+\frac {4}{3} A \,b^{2} c \,e^{2} x^{3}+16 A b \,c^{2} d^{2} x^{2}-8 B \,b^{2} c \,d^{2} x^{2}+4 A \,b^{2} c \,d^{2} x +4 B \,b^{3} d e \,x^{2}-4 A \,b^{3} d e x +2 A \,b^{3} e^{2} x^{2}+\frac {32}{3} A \,c^{3} d^{2} x^{3}-2 B \,b^{3} d^{2} x +\frac {2}{3} B \,e^{2} x^{3} b^{3}+16 B x a \,b^{2} d e -\frac {16}{3} B \,a^{2} c \,d^{2}+\frac {32}{3} B a \,c^{2} d e \,x^{3}+8 A a b c \,e^{2} x^{2}-8 B a b c \,d^{2} x -\frac {4}{3} B \,b^{2} d^{2} a -8 B a b c \,e^{2} x^{3}+\frac {16}{3} A \,a^{2} b \,e^{2}+\frac {32}{3} B \,a^{2} b d e -16 A \,b^{2} c d e \,x^{2}+8 A x a \,b^{2} e^{2}-4 B \,x^{2} a \,b^{2} e^{2}+16 B a b c d e \,x^{2}-16 A a b c d e x -16 B \,a^{2} b \,e^{2} x +\frac {16}{3} A a \,c^{2} e^{2} x^{3}-\frac {16}{3} B b \,c^{2} d^{2} x^{3}-16 B \,a^{2} c \,e^{2} x^{2}+16 A a \,c^{2} d^{2} x -\frac {32}{3} A \,a^{2} c d e +8 A a b c \,d^{2}-\frac {8}{3} A \,b^{2} d e a}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(424\) |
| gosper | \(\frac {-\frac {32}{3} B \,e^{2} a^{3}-\frac {2}{3} A \,d^{2} b^{3}-\frac {32}{3} A b \,c^{2} d e \,x^{3}+\frac {8}{3} B \,b^{2} c d e \,x^{3}+\frac {4}{3} A \,b^{2} c \,e^{2} x^{3}+16 A b \,c^{2} d^{2} x^{2}-8 B \,b^{2} c \,d^{2} x^{2}+4 A \,b^{2} c \,d^{2} x +4 B \,b^{3} d e \,x^{2}-4 A \,b^{3} d e x +2 A \,b^{3} e^{2} x^{2}+\frac {32}{3} A \,c^{3} d^{2} x^{3}-2 B \,b^{3} d^{2} x +\frac {2}{3} B \,e^{2} x^{3} b^{3}+16 B x a \,b^{2} d e -\frac {16}{3} B \,a^{2} c \,d^{2}+\frac {32}{3} B a \,c^{2} d e \,x^{3}+8 A a b c \,e^{2} x^{2}-8 B a b c \,d^{2} x -\frac {4}{3} B \,b^{2} d^{2} a -8 B a b c \,e^{2} x^{3}+\frac {16}{3} A \,a^{2} b \,e^{2}+\frac {32}{3} B \,a^{2} b d e -16 A \,b^{2} c d e \,x^{2}+8 A x a \,b^{2} e^{2}-4 B \,x^{2} a \,b^{2} e^{2}+16 B a b c d e \,x^{2}-16 A a b c d e x -16 B \,a^{2} b \,e^{2} x +\frac {16}{3} A a \,c^{2} e^{2} x^{3}-\frac {16}{3} B b \,c^{2} d^{2} x^{3}-16 B \,a^{2} c \,e^{2} x^{2}+16 A a \,c^{2} d^{2} x -\frac {32}{3} A \,a^{2} c d e +8 A a b c \,d^{2}-\frac {8}{3} A \,b^{2} d e a}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(433\) |
| default | \(A \,d^{2} \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 )+B \,e^{2} \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \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 c}\right )}{4 c}+\frac {a \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 c}\right )}{2 c}+\frac {2 a \left (-\frac {1}{3 c \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 c}\right )}{c}\right )+\left (A \,e^{2}+2 B d e \right ) \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \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 c}\right )}{4 c}+\frac {a \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 c}\right )+\left (2 A d e +B \,d^{2}\right ) \left (-\frac {1}{3 c \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 c}\right )\) | \(689\) |
2/3*(8*A*a*c^2*e^2*x^3+2*A*b^2*c*e^2*x^3-16*A*b*c^2*d*e*x^3+16*A*c^3*d^2*x ^3-12*B*a*b*c*e^2*x^3+16*B*a*c^2*d*e*x^3+B*b^3*e^2*x^3+4*B*b^2*c*d*e*x^3-8 *B*b*c^2*d^2*x^3+12*A*a*b*c*e^2*x^2+3*A*b^3*e^2*x^2-24*A*b^2*c*d*e*x^2+24* A*b*c^2*d^2*x^2-24*B*a^2*c*e^2*x^2-6*B*a*b^2*e^2*x^2+24*B*a*b*c*d*e*x^2+6* B*b^3*d*e*x^2-12*B*b^2*c*d^2*x^2+12*A*a*b^2*e^2*x-24*A*a*b*c*d*e*x+24*A*a* c^2*d^2*x-6*A*b^3*d*e*x+6*A*b^2*c*d^2*x-24*B*a^2*b*e^2*x+24*B*a*b^2*d*e*x- 12*B*a*b*c*d^2*x-3*B*b^3*d^2*x+8*A*a^2*b*e^2-16*A*a^2*c*d*e-4*A*a*b^2*d*e+ 12*A*a*b*c*d^2-A*b^3*d^2-16*B*a^3*e^2+16*B*a^2*b*d*e-8*B*a^2*c*d^2-2*B*a*b ^2*d^2)/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (113) = 226\).
Time = 7.40 (sec) , antiderivative size = 469, normalized size of antiderivative = 3.88 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} d^{2} - 4 \, {\left (B b^{2} c + 4 \, {\left (B a - A b\right )} c^{2}\right )} d e - {\left (B b^{3} + 8 \, A a c^{2} - 2 \, {\left (6 \, B a b - A b^{2}\right )} c\right )} e^{2}\right )} x^{3} + {\left (2 \, B a b^{2} + A b^{3} + 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} d^{2} - 4 \, {\left (4 \, B a^{2} b - A a b^{2} - 4 \, A a^{2} c\right )} d e + 8 \, {\left (2 \, B a^{3} - A a^{2} b\right )} e^{2} + 3 \, {\left (4 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} d^{2} - 2 \, {\left (B b^{3} + 4 \, {\left (B a b - A b^{2}\right )} c\right )} d e + {\left (2 \, B a b^{2} - A b^{3} + 4 \, {\left (2 \, B a^{2} - A a b\right )} c\right )} e^{2}\right )} x^{2} + 3 \, {\left ({\left (B b^{3} - 8 \, A a c^{2} + 2 \, {\left (2 \, B a b - A b^{2}\right )} c\right )} d^{2} - 2 \, {\left (4 \, B a b^{2} - A b^{3} - 4 \, A a b c\right )} d e + 4 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
-2/3*((8*(B*b*c^2 - 2*A*c^3)*d^2 - 4*(B*b^2*c + 4*(B*a - A*b)*c^2)*d*e - ( B*b^3 + 8*A*a*c^2 - 2*(6*B*a*b - A*b^2)*c)*e^2)*x^3 + (2*B*a*b^2 + A*b^3 + 4*(2*B*a^2 - 3*A*a*b)*c)*d^2 - 4*(4*B*a^2*b - A*a*b^2 - 4*A*a^2*c)*d*e + 8*(2*B*a^3 - A*a^2*b)*e^2 + 3*(4*(B*b^2*c - 2*A*b*c^2)*d^2 - 2*(B*b^3 + 4* (B*a*b - A*b^2)*c)*d*e + (2*B*a*b^2 - A*b^3 + 4*(2*B*a^2 - A*a*b)*c)*e^2)* x^2 + 3*((B*b^3 - 8*A*a*c^2 + 2*(2*B*a*b - A*b^2)*c)*d^2 - 2*(4*B*a*b^2 - A*b^3 - 4*A*a*b*c)*d*e + 4*(2*B*a^2*b - A*a*b^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)/(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2 *c^4)*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x)
Timed out. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (113) = 226\).
Time = 0.30 (sec) , antiderivative size = 448, normalized size of antiderivative = 3.70 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (\frac {{\left (8 \, B b c^{2} d^{2} - 16 \, A c^{3} d^{2} - 4 \, B b^{2} c d e - 16 \, B a c^{2} d e + 16 \, A b c^{2} d e - B b^{3} e^{2} + 12 \, B a b c e^{2} - 2 \, A b^{2} c e^{2} - 8 \, A a c^{2} e^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (4 \, B b^{2} c d^{2} - 8 \, A b c^{2} d^{2} - 2 \, B b^{3} d e - 8 \, B a b c d e + 8 \, A b^{2} c d e + 2 \, B a b^{2} e^{2} - A b^{3} e^{2} + 8 \, B a^{2} c e^{2} - 4 \, A a b c e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (B b^{3} d^{2} + 4 \, B a b c d^{2} - 2 \, A b^{2} c d^{2} - 8 \, A a c^{2} d^{2} - 8 \, B a b^{2} d e + 2 \, A b^{3} d e + 8 \, A a b c d e + 8 \, B a^{2} b e^{2} - 4 \, A a b^{2} e^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, B a b^{2} d^{2} + A b^{3} d^{2} + 8 \, B a^{2} c d^{2} - 12 \, A a b c d^{2} - 16 \, B a^{2} b d e + 4 \, A a b^{2} d e + 16 \, A a^{2} c d e + 16 \, B a^{3} e^{2} - 8 \, A a^{2} b e^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
-2/3*((((8*B*b*c^2*d^2 - 16*A*c^3*d^2 - 4*B*b^2*c*d*e - 16*B*a*c^2*d*e + 1 6*A*b*c^2*d*e - B*b^3*e^2 + 12*B*a*b*c*e^2 - 2*A*b^2*c*e^2 - 8*A*a*c^2*e^2 )*x/(b^4 - 8*a*b^2*c + 16*a^2*c^2) + 3*(4*B*b^2*c*d^2 - 8*A*b*c^2*d^2 - 2* B*b^3*d*e - 8*B*a*b*c*d*e + 8*A*b^2*c*d*e + 2*B*a*b^2*e^2 - A*b^3*e^2 + 8* B*a^2*c*e^2 - 4*A*a*b*c*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + 3*(B*b^3* d^2 + 4*B*a*b*c*d^2 - 2*A*b^2*c*d^2 - 8*A*a*c^2*d^2 - 8*B*a*b^2*d*e + 2*A* b^3*d*e + 8*A*a*b*c*d*e + 8*B*a^2*b*e^2 - 4*A*a*b^2*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))*x + (2*B*a*b^2*d^2 + A*b^3*d^2 + 8*B*a^2*c*d^2 - 12*A*a*b*c *d^2 - 16*B*a^2*b*d*e + 4*A*a*b^2*d*e + 16*A*a^2*c*d*e + 16*B*a^3*e^2 - 8* A*a^2*b*e^2)/(b^4 - 8*a*b^2*c + 16*a^2*c^2))/(c*x^2 + b*x + a)^(3/2)
Time = 11.71 (sec) , antiderivative size = 423, normalized size of antiderivative = 3.50 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (16\,B\,a^3\,e^2-16\,B\,a^2\,b\,d\,e+24\,B\,a^2\,b\,e^2\,x-8\,A\,a^2\,b\,e^2+8\,B\,a^2\,c\,d^2+16\,A\,a^2\,c\,d\,e+24\,B\,a^2\,c\,e^2\,x^2+2\,B\,a\,b^2\,d^2-24\,B\,a\,b^2\,d\,e\,x+4\,A\,a\,b^2\,d\,e+6\,B\,a\,b^2\,e^2\,x^2-12\,A\,a\,b^2\,e^2\,x+12\,B\,a\,b\,c\,d^2\,x-12\,A\,a\,b\,c\,d^2-24\,B\,a\,b\,c\,d\,e\,x^2+24\,A\,a\,b\,c\,d\,e\,x+12\,B\,a\,b\,c\,e^2\,x^3-12\,A\,a\,b\,c\,e^2\,x^2-24\,A\,a\,c^2\,d^2\,x-16\,B\,a\,c^2\,d\,e\,x^3-8\,A\,a\,c^2\,e^2\,x^3+3\,B\,b^3\,d^2\,x+A\,b^3\,d^2-6\,B\,b^3\,d\,e\,x^2+6\,A\,b^3\,d\,e\,x-B\,b^3\,e^2\,x^3-3\,A\,b^3\,e^2\,x^2+12\,B\,b^2\,c\,d^2\,x^2-6\,A\,b^2\,c\,d^2\,x-4\,B\,b^2\,c\,d\,e\,x^3+24\,A\,b^2\,c\,d\,e\,x^2-2\,A\,b^2\,c\,e^2\,x^3+8\,B\,b\,c^2\,d^2\,x^3-24\,A\,b\,c^2\,d^2\,x^2+16\,A\,b\,c^2\,d\,e\,x^3-16\,A\,c^3\,d^2\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
-(2*(A*b^3*d^2 + 16*B*a^3*e^2 - 8*A*a^2*b*e^2 + 2*B*a*b^2*d^2 + 8*B*a^2*c* d^2 + 3*B*b^3*d^2*x - 3*A*b^3*e^2*x^2 - 16*A*c^3*d^2*x^3 - B*b^3*e^2*x^3 - 6*A*b^2*c*d^2*x + 24*B*a^2*b*e^2*x - 6*B*b^3*d*e*x^2 - 24*A*b*c^2*d^2*x^2 + 6*B*a*b^2*e^2*x^2 - 8*A*a*c^2*e^2*x^3 + 24*B*a^2*c*e^2*x^2 + 12*B*b^2*c *d^2*x^2 - 2*A*b^2*c*e^2*x^3 + 8*B*b*c^2*d^2*x^3 - 12*A*a*b*c*d^2 + 4*A*a* b^2*d*e + 16*A*a^2*c*d*e - 16*B*a^2*b*d*e + 6*A*b^3*d*e*x - 12*A*a*b^2*e^2 *x - 24*A*a*c^2*d^2*x + 12*B*a*b*c*d^2*x - 24*B*a*b^2*d*e*x - 12*A*a*b*c*e ^2*x^2 + 12*B*a*b*c*e^2*x^3 + 24*A*b^2*c*d*e*x^2 + 16*A*b*c^2*d*e*x^3 - 16 *B*a*c^2*d*e*x^3 - 4*B*b^2*c*d*e*x^3 + 24*A*a*b*c*d*e*x - 24*B*a*b*c*d*e*x ^2))/(3*(4*a*c - b^2)^2*(a + b*x + c*x^2)^(3/2))