Integrand size = 27, antiderivative size = 324 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (A b-2 a B-(b B-2 A c) x) (d+e x)^2}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}-\frac {8 \left (4 a c e (3 A c d+a B e)-4 b c \left (A c d^2+2 a B d e+a A e^2\right )+b^2 \left (2 B c d^2+A c d e+a B e^2\right )+\left (b^3 B e^2-3 b^2 c e (B d+A e)+4 b c^2 d (B d+2 A e)-4 c^2 \left (2 A c d^2+a B d e-a A e^2\right )\right ) x\right )}{15 c \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}}+\frac {8 \left (b^3 B e^2-6 b^2 c e (2 B d+A e)-8 c^2 \left (4 A c d^2+2 a B d e+a A e^2\right )+4 b c \left (4 B c d^2+8 A c d e+3 a B e^2\right )\right ) (b+2 c x)}{15 c \left (b^2-4 a c\right )^3 \sqrt {a+b x+c x^2}} \]
-2/5*(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)^(5/2) -8/15*(4*a*c*e*(3*A*c*d+B*a*e)-4*b*c*(A*a*e^2+A*c*d^2+2*B*a*d*e)+b^2*(A*c* d*e+B*a*e^2+2*B*c*d^2)+(b^3*B*e^2-3*b^2*c*e*(A*e+B*d)+4*b*c^2*d*(2*A*e+B*d )-4*c^2*(-A*a*e^2+2*A*c*d^2+B*a*d*e))*x)/c/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(3 /2)+8/15*(b^3*B*e^2-6*b^2*c*e*(A*e+2*B*d)-8*c^2*(A*a*e^2+4*A*c*d^2+2*B*a*d *e)+4*b*c*(8*A*c*d*e+3*B*a*e^2+4*B*c*d^2))*(2*c*x+b)/c/(-4*a*c+b^2)^3/(c*x ^2+b*x+a)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(711\) vs. \(2(324)=648\).
Time = 6.55 (sec) , antiderivative size = 711, normalized size of antiderivative = 2.19 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {-2 A \left (b^5 \left (3 d^2+10 d e x+15 e^2 x^2\right )+2 b^4 \left (2 a e (d+5 e x)-5 c x \left (d^2+8 d e x-9 e^2 x^2\right )\right )+32 c^2 \left (-6 a^3 d e+8 c^3 d^2 x^5+5 a^2 c x \left (3 d^2+e^2 x^2\right )+2 a c^2 x^3 \left (10 d^2+e^2 x^2\right )\right )+16 b c \left (6 a^3 e^2+8 c^3 d x^4 (5 d-2 e x)+15 a^2 c (d-e x)^2+10 a c^2 x^2 \left (6 d^2-4 d e x+e^2 x^2\right )\right )+16 b^2 c \left (3 a^2 e (-2 d+5 e x)+15 a c x \left (d^2-4 d e x+e^2 x^2\right )+c^2 x^3 \left (30 d^2-40 d e x+3 e^2 x^2\right )\right )+8 b^3 \left (a^2 e^2-5 a c \left (d^2+6 d e x-5 e^2 x^2\right )+5 c^2 x^2 \left (2 d^2-12 d e x+3 e^2 x^2\right )\right )\right )+2 B \left (64 a^4 c e^2+b x \left (128 c^4 d^2 x^4+32 b c^3 d x^3 (10 d-3 e x)-5 b^4 \left (d^2+6 d e x-3 e^2 x^2\right )+8 b^2 c^2 x^2 \left (30 d^2-30 d e x+e^2 x^2\right )+20 b^3 c x \left (2 d^2-9 d e x+e^2 x^2\right )\right )+16 a^3 \left (3 b^2 e^2+2 b c e (-6 d+5 e x)+2 c^2 \left (3 d^2+5 e^2 x^2\right )\right )-8 a^2 \left (40 c^3 d e x^3+b^3 e (2 d-15 e x)-30 b c^2 x (d-e x)^2-6 b^2 c \left (d^2-10 d e x+5 e^2 x^2\right )\right )-2 a \left (64 c^4 d e x^5+b^4 \left (d^2+20 d e x-45 e^2 x^2\right )-120 b^2 c^2 x^2 \left (2 d^2-2 d e x+e^2 x^2\right )-16 b c^3 x^3 \left (10 d^2-10 d e x+3 e^2 x^2\right )-20 b^3 c x \left (3 d^2-10 d e x+5 e^2 x^2\right )\right )\right )}{15 \left (b^2-4 a c\right )^3 (a+x (b+c x))^{5/2}} \]
(-2*A*(b^5*(3*d^2 + 10*d*e*x + 15*e^2*x^2) + 2*b^4*(2*a*e*(d + 5*e*x) - 5* c*x*(d^2 + 8*d*e*x - 9*e^2*x^2)) + 32*c^2*(-6*a^3*d*e + 8*c^3*d^2*x^5 + 5* a^2*c*x*(3*d^2 + e^2*x^2) + 2*a*c^2*x^3*(10*d^2 + e^2*x^2)) + 16*b*c*(6*a^ 3*e^2 + 8*c^3*d*x^4*(5*d - 2*e*x) + 15*a^2*c*(d - e*x)^2 + 10*a*c^2*x^2*(6 *d^2 - 4*d*e*x + e^2*x^2)) + 16*b^2*c*(3*a^2*e*(-2*d + 5*e*x) + 15*a*c*x*( d^2 - 4*d*e*x + e^2*x^2) + c^2*x^3*(30*d^2 - 40*d*e*x + 3*e^2*x^2)) + 8*b^ 3*(a^2*e^2 - 5*a*c*(d^2 + 6*d*e*x - 5*e^2*x^2) + 5*c^2*x^2*(2*d^2 - 12*d*e *x + 3*e^2*x^2))) + 2*B*(64*a^4*c*e^2 + b*x*(128*c^4*d^2*x^4 + 32*b*c^3*d* x^3*(10*d - 3*e*x) - 5*b^4*(d^2 + 6*d*e*x - 3*e^2*x^2) + 8*b^2*c^2*x^2*(30 *d^2 - 30*d*e*x + e^2*x^2) + 20*b^3*c*x*(2*d^2 - 9*d*e*x + e^2*x^2)) + 16* a^3*(3*b^2*e^2 + 2*b*c*e*(-6*d + 5*e*x) + 2*c^2*(3*d^2 + 5*e^2*x^2)) - 8*a ^2*(40*c^3*d*e*x^3 + b^3*e*(2*d - 15*e*x) - 30*b*c^2*x*(d - e*x)^2 - 6*b^2 *c*(d^2 - 10*d*e*x + 5*e^2*x^2)) - 2*a*(64*c^4*d*e*x^5 + b^4*(d^2 + 20*d*e *x - 45*e^2*x^2) - 120*b^2*c^2*x^2*(2*d^2 - 2*d*e*x + e^2*x^2) - 16*b*c^3* x^3*(10*d^2 - 10*d*e*x + 3*e^2*x^2) - 20*b^3*c*x*(3*d^2 - 10*d*e*x + 5*e^2 *x^2))))/(15*(b^2 - 4*a*c)^3*(a + x*(b + c*x))^(5/2))
Time = 0.53 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1234, 27, 25, 1224, 1088}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1234 |
| \(\displaystyle -\frac {2 \int \frac {2 (d+e x) (4 A c d+2 a B e-b (2 B d+A e)-(b B-2 A c) e x)}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {4 \int -\frac {(d+e x) (2 b B d-4 A c d+A b e-2 a B e+(b B-2 A c) e x)}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \int \frac {(d+e x) (2 b B d-4 A c d+A b e-2 a B e+(b B-2 A c) e x)}{\left (c x^2+b x+a\right )^{5/2}}dx}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1224 |
| \(\displaystyle \frac {4 \left (-\frac {\left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right ) \int \frac {1}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}-\frac {2 \left (b^2 \left (a B e^2+A c d e+2 B c d^2\right )+x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
| \(\Big \downarrow \) 1088 |
| \(\displaystyle \frac {4 \left (\frac {2 (b+2 c x) \left (4 b c \left (3 a B e^2+8 A c d e+4 B c d^2\right )-8 c^2 \left (a A e^2+2 a B d e+4 A c d^2\right )-6 b^2 c e (A e+2 B d)+b^3 B e^2\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \left (b^2 \left (a B e^2+A c d e+2 B c d^2\right )+x \left (-4 c^2 \left (-a A e^2+a B d e+2 A c d^2\right )-3 b^2 c e (A e+B d)+4 b c^2 d (2 A e+B d)+b^3 B e^2\right )-4 b c \left (a A e^2+2 a B d e+A c d^2\right )+4 a c e (a B e+3 A c d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\right )}{5 \left (b^2-4 a c\right )}-\frac {2 (d+e x)^2 (-2 a B-x (b B-2 A c)+A b)}{5 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{5/2}}\) |
(-2*(A*b - 2*a*B - (b*B - 2*A*c)*x)*(d + e*x)^2)/(5*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(5/2)) + (4*((-2*(4*a*c*e*(3*A*c*d + a*B*e) - 4*b*c*(A*c*d^2 + 2 *a*B*d*e + a*A*e^2) + b^2*(2*B*c*d^2 + A*c*d*e + a*B*e^2) + (b^3*B*e^2 - 3 *b^2*c*e*(B*d + A*e) + 4*b*c^2*d*(B*d + 2*A*e) - 4*c^2*(2*A*c*d^2 + a*B*d* e - a*A*e^2))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) + (2*(b^3*B* e^2 - 6*b^2*c*e*(2*B*d + A*e) - 8*c^2*(4*A*c*d^2 + 2*a*B*d*e + a*A*e^2) + 4*b*c*(4*B*c*d^2 + 8*A*c*d*e + 3*a*B*e^2))*(b + 2*c*x))/(3*c*(b^2 - 4*a*c) ^2*Sqrt[a + b*x + c*x^2])))/(5*(b^2 - 4*a*c))
3.25.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[-2*((b + 2*c*x)/((b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2])), x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c *(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 1] && !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
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)*( (f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g *(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* (m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 ] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(1024\) vs. \(2(312)=624\).
Time = 0.59 (sec) , antiderivative size = 1025, normalized size of antiderivative = 3.16
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1025\) |
| trager | \(\text {Expression too large to display}\) | \(1042\) |
| gosper | \(\text {Expression too large to display}\) | \(1064\) |
A*d^2*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2 /3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b )/(c*x^2+b*x+a)^(1/2)))+B*e^2*(-1/3*x^2/c/(c*x^2+b*x+a)^(5/2)-1/6*b/c*(-1/ 4*x/c/(c*x^2+b*x+a)^(5/2)-3/8*b/c*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2/5 *(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+ b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b *x+a)^(1/2))))+1/4*a/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5 *c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a* c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+2/3*a/c*(-1/5/c/(c*x^2+b*x+a)^(5 /2)-1/2*b/c*(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b ^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2 *c*x+b)/(c*x^2+b*x+a)^(1/2)))))+(A*e^2+2*B*d*e)*(-1/4*x/c/(c*x^2+b*x+a)^(5 /2)-3/8*b/c*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2/5*(2*c*x+b)/(4*a*c-b^2) /(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+ b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*x^2+b*x+a)^(1/2))))+1/4*a/c *(2/5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2 *c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c* x^2+b*x+a)^(1/2))))+(2*A*d*e+B*d^2)*(-1/5/c/(c*x^2+b*x+a)^(5/2)-1/2*b/c*(2 /5*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(5/2)+16/5*c/(4*a*c-b^2)*(2/3*(2*c* x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+16/3*c/(4*a*c-b^2)^2*(2*c*x+b)/(c*...
Leaf count of result is larger than twice the leaf count of optimal. 1095 vs. \(2 (312) = 624\).
Time = 81.77 (sec) , antiderivative size = 1095, normalized size of antiderivative = 3.38 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
2/15*(8*(16*(B*b*c^4 - 2*A*c^5)*d^2 - 4*(3*B*b^2*c^3 + 4*(B*a - 2*A*b)*c^4 )*d*e + (B*b^3*c^2 - 8*A*a*c^4 + 6*(2*B*a*b - A*b^2)*c^3)*e^2)*x^5 + 20*(1 6*(B*b^2*c^3 - 2*A*b*c^4)*d^2 - 4*(3*B*b^3*c^2 + 4*(B*a*b - 2*A*b^2)*c^3)* d*e + (B*b^4*c - 8*A*a*b*c^3 + 6*(2*B*a*b^2 - A*b^3)*c^2)*e^2)*x^4 + 5*(16 *(3*B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - 3*A*b^2)*c^3)*d^2 - 4*(9*B*b^4*c + 16*(B*a^2 - 2*A*a*b)*c^3 + 24*(B*a*b^2 - A*b^3)*c^2)*d*e + (3*B*b^5 - 32 *A*a^2*c^3 + 48*(B*a^2*b - A*a*b^2)*c^2 + 2*(20*B*a*b^3 - 9*A*b^4)*c)*e^2) *x^3 - (2*B*a*b^4 + 3*A*b^5 - 48*(2*B*a^3 - 5*A*a^2*b)*c^2 - 8*(6*B*a^2*b^ 2 + 5*A*a*b^3)*c)*d^2 - 4*(4*B*a^2*b^3 + A*a*b^4 - 48*A*a^3*c^2 + 24*(2*B* a^3*b - A*a^2*b^2)*c)*d*e + 8*(6*B*a^3*b^2 - A*a^2*b^3 + 4*(2*B*a^4 - 3*A* a^3*b)*c)*e^2 + 5*(8*(B*b^4*c - 24*A*a*b*c^3 + 2*(6*B*a*b^2 - A*b^3)*c^2)* d^2 - 2*(3*B*b^5 + 48*(B*a^2*b - 2*A*a*b^2)*c^2 + 8*(5*B*a*b^3 - A*b^4)*c) *d*e + (18*B*a*b^4 - 3*A*b^5 + 16*(2*B*a^3 - 3*A*a^2*b)*c^2 + 8*(6*B*a^2*b ^2 - 5*A*a*b^3)*c)*e^2)*x^2 - 5*((B*b^5 + 96*A*a^2*c^3 - 48*(B*a^2*b - A*a *b^2)*c^2 - 2*(12*B*a*b^3 + A*b^4)*c)*d^2 + 2*(4*B*a*b^4 + A*b^5 - 48*A*a^ 2*b*c^2 + 24*(2*B*a^2*b^2 - A*a*b^3)*c)*d*e - 4*(6*B*a^2*b^3 - A*a*b^4 + 4 *(2*B*a^3*b - 3*A*a^2*b^2)*c)*e^2)*x)*sqrt(c*x^2 + b*x + a)/(a^3*b^6 - 12* a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2 *b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b...
Timed out. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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. 1091 vs. \(2 (312) = 624\).
Time = 0.30 (sec) , antiderivative size = 1091, normalized size of antiderivative = 3.37 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (4 \, {\left (\frac {2 \, {\left (16 \, B b c^{4} d^{2} - 32 \, A c^{5} d^{2} - 12 \, B b^{2} c^{3} d e - 16 \, B a c^{4} d e + 32 \, A b c^{4} d e + B b^{3} c^{2} e^{2} + 12 \, B a b c^{3} e^{2} - 6 \, A b^{2} c^{3} e^{2} - 8 \, A a c^{4} e^{2}\right )} x}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}} + \frac {5 \, {\left (16 \, B b^{2} c^{3} d^{2} - 32 \, A b c^{4} d^{2} - 12 \, B b^{3} c^{2} d e - 16 \, B a b c^{3} d e + 32 \, A b^{2} c^{3} d e + B b^{4} c e^{2} + 12 \, B a b^{2} c^{2} e^{2} - 6 \, A b^{3} c^{2} e^{2} - 8 \, A a b c^{3} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (48 \, B b^{3} c^{2} d^{2} + 64 \, B a b c^{3} d^{2} - 96 \, A b^{2} c^{3} d^{2} - 128 \, A a c^{4} d^{2} - 36 \, B b^{4} c d e - 96 \, B a b^{2} c^{2} d e + 96 \, A b^{3} c^{2} d e - 64 \, B a^{2} c^{3} d e + 128 \, A a b c^{3} d e + 3 \, B b^{5} e^{2} + 40 \, B a b^{3} c e^{2} - 18 \, A b^{4} c e^{2} + 48 \, B a^{2} b c^{2} e^{2} - 48 \, A a b^{2} c^{2} e^{2} - 32 \, A a^{2} c^{3} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x + \frac {5 \, {\left (8 \, B b^{4} c d^{2} + 96 \, B a b^{2} c^{2} d^{2} - 16 \, A b^{3} c^{2} d^{2} - 192 \, A a b c^{3} d^{2} - 6 \, B b^{5} d e - 80 \, B a b^{3} c d e + 16 \, A b^{4} c d e - 96 \, B a^{2} b c^{2} d e + 192 \, A a b^{2} c^{2} d e + 18 \, B a b^{4} e^{2} - 3 \, A b^{5} e^{2} + 48 \, B a^{2} b^{2} c e^{2} - 40 \, A a b^{3} c e^{2} + 32 \, B a^{3} c^{2} e^{2} - 48 \, A a^{2} b c^{2} e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {5 \, {\left (B b^{5} d^{2} - 24 \, B a b^{3} c d^{2} - 2 \, A b^{4} c d^{2} - 48 \, B a^{2} b c^{2} d^{2} + 48 \, A a b^{2} c^{2} d^{2} + 96 \, A a^{2} c^{3} d^{2} + 8 \, B a b^{4} d e + 2 \, A b^{5} d e + 96 \, B a^{2} b^{2} c d e - 48 \, A a b^{3} c d e - 96 \, A a^{2} b c^{2} d e - 24 \, B a^{2} b^{3} e^{2} + 4 \, A a b^{4} e^{2} - 32 \, B a^{3} b c e^{2} + 48 \, A a^{2} b^{2} c e^{2}\right )}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )} x - \frac {2 \, B a b^{4} d^{2} + 3 \, A b^{5} d^{2} - 48 \, B a^{2} b^{2} c d^{2} - 40 \, A a b^{3} c d^{2} - 96 \, B a^{3} c^{2} d^{2} + 240 \, A a^{2} b c^{2} d^{2} + 16 \, B a^{2} b^{3} d e + 4 \, A a b^{4} d e + 192 \, B a^{3} b c d e - 96 \, A a^{2} b^{2} c d e - 192 \, A a^{3} c^{2} d e - 48 \, B a^{3} b^{2} e^{2} + 8 \, A a^{2} b^{3} e^{2} - 64 \, B a^{4} c e^{2} + 96 \, A a^{3} b c e^{2}}{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}\right )}}{15 \, {\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \]
2/15*((((4*(2*(16*B*b*c^4*d^2 - 32*A*c^5*d^2 - 12*B*b^2*c^3*d*e - 16*B*a*c ^4*d*e + 32*A*b*c^4*d*e + B*b^3*c^2*e^2 + 12*B*a*b*c^3*e^2 - 6*A*b^2*c^3*e ^2 - 8*A*a*c^4*e^2)*x/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3) + 5 *(16*B*b^2*c^3*d^2 - 32*A*b*c^4*d^2 - 12*B*b^3*c^2*d*e - 16*B*a*b*c^3*d*e + 32*A*b^2*c^3*d*e + B*b^4*c*e^2 + 12*B*a*b^2*c^2*e^2 - 6*A*b^3*c^2*e^2 - 8*A*a*b*c^3*e^2)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*( 48*B*b^3*c^2*d^2 + 64*B*a*b*c^3*d^2 - 96*A*b^2*c^3*d^2 - 128*A*a*c^4*d^2 - 36*B*b^4*c*d*e - 96*B*a*b^2*c^2*d*e + 96*A*b^3*c^2*d*e - 64*B*a^2*c^3*d*e + 128*A*a*b*c^3*d*e + 3*B*b^5*e^2 + 40*B*a*b^3*c*e^2 - 18*A*b^4*c*e^2 + 4 8*B*a^2*b*c^2*e^2 - 48*A*a*b^2*c^2*e^2 - 32*A*a^2*c^3*e^2)/(b^6 - 12*a*b^4 *c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x + 5*(8*B*b^4*c*d^2 + 96*B*a*b^2*c^2*d ^2 - 16*A*b^3*c^2*d^2 - 192*A*a*b*c^3*d^2 - 6*B*b^5*d*e - 80*B*a*b^3*c*d*e + 16*A*b^4*c*d*e - 96*B*a^2*b*c^2*d*e + 192*A*a*b^2*c^2*d*e + 18*B*a*b^4* e^2 - 3*A*b^5*e^2 + 48*B*a^2*b^2*c*e^2 - 40*A*a*b^3*c*e^2 + 32*B*a^3*c^2*e ^2 - 48*A*a^2*b*c^2*e^2)/(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)) *x - 5*(B*b^5*d^2 - 24*B*a*b^3*c*d^2 - 2*A*b^4*c*d^2 - 48*B*a^2*b*c^2*d^2 + 48*A*a*b^2*c^2*d^2 + 96*A*a^2*c^3*d^2 + 8*B*a*b^4*d*e + 2*A*b^5*d*e + 96 *B*a^2*b^2*c*d*e - 48*A*a*b^3*c*d*e - 96*A*a^2*b*c^2*d*e - 24*B*a^2*b^3*e^ 2 + 4*A*a*b^4*e^2 - 32*B*a^3*b*c*e^2 + 48*A*a^2*b^2*c*e^2)/(b^6 - 12*a*b^4 *c + 48*a^2*b^2*c^2 - 64*a^3*c^3))*x - (2*B*a*b^4*d^2 + 3*A*b^5*d^2 - 4...
Time = 12.61 (sec) , antiderivative size = 1996, normalized size of antiderivative = 6.16 \[ \int \frac {(A+B x) (d+e x)^2}{\left (a+b x+c x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
(x*((2*c^2*(8*A*e^2 + 16*B*d*e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (8 *B*b*c*e^2)/(5*(4*a*c^2 - b^2*c)*(4*a*c - b^2))) + (b*c*(8*A*e^2 + 16*B*d* e))/(15*(4*a*c^2 - b^2*c)*(4*a*c - b^2)) - (16*B*a*c*e^2)/(5*(4*a*c^2 - b^ 2*c)*(4*a*c - b^2)))/(a + b*x + c*x^2)^(1/2) - (x*((a*((2*c^2*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c)))) /c + (b*((2*c^2*((2*B*d^2)/5 + (4*A*d*e)/5))/(4*a*c^2 - b^2*c) - (b*((2*c^ 2*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^2)/(5*(4*a*c ^2 - b^2*c))))/c + (b*c*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - ( 4*B*a*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c - (b*c*((2*B*d^2)/5 + (4*A*d*e)/5)) /(4*a*c^2 - b^2*c) - (4*A*c^2*d^2)/(5*(4*a*c^2 - b^2*c))) + (a*((2*c^2*((2 *B*d^2)/5 + (4*A*d*e)/5))/(4*a*c^2 - b^2*c) - (b*((2*c^2*((2*A*e^2)/5 + (4 *B*d*e)/5))/(4*a*c^2 - b^2*c) - (2*B*b*c*e^2)/(5*(4*a*c^2 - b^2*c))))/c + (b*c*((2*A*e^2)/5 + (4*B*d*e)/5))/(4*a*c^2 - b^2*c) - (4*B*a*c*e^2)/(5*(4* a*c^2 - b^2*c))))/c - (2*A*b*c*d^2)/(5*(4*a*c^2 - b^2*c)))/(a + b*x + c*x^ 2)^(5/2) - (x*((2*e*(2*A*c*e - B*b*e + 4*B*c*d))/(15*c*(4*a*c - b^2)) - (4 *B*b*e^2)/(15*c*(4*a*c - b^2))) + (2*B*b^2*e^2 + 4*B*c^2*d^2 - 2*A*b*c*e^2 - 4*B*a*c*e^2 + 8*A*c^2*d*e - 4*B*b*c*d*e)/(15*c^2*(4*a*c - b^2)) - (4*B* a*e^2)/(15*c*(4*a*c - b^2)))/(a + b*x + c*x^2)^(3/2) + (x*((b*((2*(8*B*c^3 *d^2 + 16*A*c^3*d*e - 48*B*a*c^2*e^2 + 12*B*b^2*c*e^2))/(15*(4*a*c^2 - b^2 *c)*(4*a*c - b^2)) - (b*((16*c^3*e*(A*e + 2*B*d))/(15*(4*a*c^2 - b^2*c)...