Integrand size = 27, antiderivative size = 397 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 (d+e x)^2 \left (2 a c (B d+A e)-b (A c d+a B e)-\left (b^2 B e-b c (B d+A e)+2 c (A c d-a B e)\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \left (4 b^2 c^2 d^2 (B d+2 A e)+16 a c^2 e \left (A c d^2+3 a B d e+a A e^2\right )-b^3 B \left (c d^2 e-3 a e^3\right )-4 b c \left (5 a B e \left (c d^2+a e^2\right )+2 A c d \left (c d^2+3 a e^2\right )\right )-\left (2 b^3 B c d e^2-3 b^4 B e^3+2 b^2 c e \left (3 B c d^2+4 A c d e+11 a B e^2\right )-8 b c^2 \left (B c d^3+3 A c d^2 e+4 a B d e^2+a A e^3\right )+8 c^2 \left (3 a B e \left (c d^2-a e^2\right )+2 A c d \left (c d^2+a e^2\right )\right )\right ) x\right )}{3 c^2 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}+\frac {B e^3 \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{5/2}} \]
2/3*(e*x+d)^2*(2*a*c*(A*e+B*d)-b*(A*c*d+B*a*e)-(b^2*B*e-b*c*(A*e+B*d)+2*c* (A*c*d-B*a*e))*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)+B*e^3*arctanh(1/2*(2* c*x+b)/c^(1/2)/(c*x^2+b*x+a)^(1/2))/c^(5/2)-2/3*(4*b^2*c^2*d^2*(2*A*e+B*d) +16*a*c^2*e*(A*a*e^2+A*c*d^2+3*B*a*d*e)-b^3*B*(-3*a*e^3+c*d^2*e)-4*b*c*(5* a*B*e*(a*e^2+c*d^2)+2*A*c*d*(3*a*e^2+c*d^2))-(2*b^3*B*c*d*e^2-3*b^4*B*e^3+ 2*b^2*c*e*(4*A*c*d*e+11*B*a*e^2+3*B*c*d^2)-8*b*c^2*(A*a*e^3+3*A*c*d^2*e+4* B*a*d*e^2+B*c*d^3)+8*c^2*(3*a*B*e*(-a*e^2+c*d^2)+2*A*c*d*(a*e^2+c*d^2)))*x )/c^2/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/2)
Time = 3.54 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.28 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (A c^2 \left (6 b^2 (a e-c d x) \left (d^2-6 d e x+e^2 x^2\right )+b^3 \left (d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )-12 b (d-e x) \left (2 a^2 e^2+2 c^2 d^2 x^2+a c (d-e x)^2\right )+8 \left (2 a^3 e^3-2 c^3 d^3 x^3+3 a^2 c e \left (d^2+e^2 x^2\right )-3 a c^2 d x \left (d^2+e^2 x^2\right )\right )\right )+B \left (4 a^3 c e^2 (-5 b e+6 c (2 d+e x))+b x \left (3 b^4 e^3 x+8 c^4 d^3 x^2+4 b^3 c e^3 x^2+6 b c^3 d^2 x (2 d-e x)+3 b^2 c^2 d \left (d^2-3 d e x-e^2 x^2\right )\right )+2 a \left (3 b^4 e^3 x-9 b^3 c e^3 x^2-12 c^4 d^2 e x^3+6 b c^3 d x \left (d^2-3 d e x+3 e^2 x^2\right )+b^2 c^2 \left (d^3-18 d^2 e x+9 d e^2 x^2-14 e^3 x^3\right )\right )+a^2 \left (3 b^3 e^3-42 b^2 c e^3 x-24 b c^2 d e (d-3 e x)+8 c^3 \left (d^3+9 d e^2 x^2+4 e^3 x^3\right )\right )\right )\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}}-\frac {B e^3 \log \left (c^2 \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )\right )}{c^{5/2}} \]
(-2*(A*c^2*(6*b^2*(a*e - c*d*x)*(d^2 - 6*d*e*x + e^2*x^2) + b^3*(d^3 + 9*d ^2*e*x - 9*d*e^2*x^2 - e^3*x^3) - 12*b*(d - e*x)*(2*a^2*e^2 + 2*c^2*d^2*x^ 2 + a*c*(d - e*x)^2) + 8*(2*a^3*e^3 - 2*c^3*d^3*x^3 + 3*a^2*c*e*(d^2 + e^2 *x^2) - 3*a*c^2*d*x*(d^2 + e^2*x^2))) + B*(4*a^3*c*e^2*(-5*b*e + 6*c*(2*d + e*x)) + b*x*(3*b^4*e^3*x + 8*c^4*d^3*x^2 + 4*b^3*c*e^3*x^2 + 6*b*c^3*d^2 *x*(2*d - e*x) + 3*b^2*c^2*d*(d^2 - 3*d*e*x - e^2*x^2)) + 2*a*(3*b^4*e^3*x - 9*b^3*c*e^3*x^2 - 12*c^4*d^2*e*x^3 + 6*b*c^3*d*x*(d^2 - 3*d*e*x + 3*e^2 *x^2) + b^2*c^2*(d^3 - 18*d^2*e*x + 9*d*e^2*x^2 - 14*e^3*x^3)) + a^2*(3*b^ 3*e^3 - 42*b^2*c*e^3*x - 24*b*c^2*d*e*(d - 3*e*x) + 8*c^3*(d^3 + 9*d*e^2*x ^2 + 4*e^3*x^3)))))/(3*c^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(3/2)) - (B*e ^3*Log[c^2*(b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b + c*x)])])/c^(5/2)
Time = 0.69 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1233, 27, 1224, 1092, 219}
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2 \int -\frac {(d+e x) \left (8 A c^2 d^2-4 b c (B d+2 A e) d+b B e (b d-4 a e)+4 a c e (3 B d+2 A e)-3 B \left (b^2-4 a c\right ) e^2 x\right )}{2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}+\frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\int \frac {(d+e x) \left (8 A c^2 d^2-4 b c (B d+2 A e) d+b B e (b d-4 a e)+4 a c e (3 B d+2 A e)-3 B \left (b^2-4 a c\right ) e^2 x\right )}{\left (c x^2+b x+a\right )^{3/2}}dx}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1224 |
| \(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-3 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) e \left (c d^2-3 a e^2\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {3 B e^3 \left (b^2-4 a c\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{c}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-3 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) e \left (c d^2-3 a e^2\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {6 B e^3 \left (b^2-4 a c\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{c}}{3 c \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 (d+e x)^2 \left (-x \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )-b (a B e+A c d)+2 a c (A e+B d)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\frac {2 \left (-x \left (2 b^2 c e \left (11 a B e^2+4 A c d e+3 B c d^2\right )-8 b c^2 \left (a A e^3+4 a B d e^2+3 A c d^2 e+B c d^3\right )+8 c^2 \left (2 A c d \left (a e^2+c d^2\right )+3 a B e \left (c d^2-a e^2\right )\right )-3 b^4 B e^3+2 b^3 B c d e^2\right )-4 b c \left (2 A c d \left (3 a e^2+c d^2\right )+5 a B e \left (a e^2+c d^2\right )\right )+16 a c^2 e \left (a A e^2+3 a B d e+A c d^2\right )+b^3 (-B) e \left (c d^2-3 a e^2\right )+4 b^2 c^2 d^2 (2 A e+B d)\right )}{c \left (b^2-4 a c\right ) \sqrt {a+b x+c x^2}}-\frac {3 B e^3 \left (b^2-4 a c\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{c^{3/2}}}{3 c \left (b^2-4 a c\right )}\) |
(2*(d + e*x)^2*(2*a*c*(B*d + A*e) - b*(A*c*d + a*B*e) - (b^2*B*e - b*c*(B* d + A*e) + 2*c*(A*c*d - a*B*e))*x))/(3*c*(b^2 - 4*a*c)*(a + b*x + c*x^2)^( 3/2)) - ((2*(4*b^2*c^2*d^2*(B*d + 2*A*e) - b^3*B*e*(c*d^2 - 3*a*e^2) + 16* a*c^2*e*(A*c*d^2 + 3*a*B*d*e + a*A*e^2) - 4*b*c*(5*a*B*e*(c*d^2 + a*e^2) + 2*A*c*d*(c*d^2 + 3*a*e^2)) - (2*b^3*B*c*d*e^2 - 3*b^4*B*e^3 + 2*b^2*c*e*( 3*B*c*d^2 + 4*A*c*d*e + 11*a*B*e^2) - 8*b*c^2*(B*c*d^3 + 3*A*c*d^2*e + 4*a *B*d*e^2 + a*A*e^3) + 8*c^2*(3*a*B*e*(c*d^2 - a*e^2) + 2*A*c*d*(c*d^2 + a* e^2)))*x))/(c*(b^2 - 4*a*c)*Sqrt[a + b*x + c*x^2]) - (3*B*(b^2 - 4*a*c)*e^ 3*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/c^(3/2))/(3*c*(b ^2 - 4*a*c))
3.25.81.3.1 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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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 - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Leaf count of result is larger than twice the leaf count of optimal. \(1157\) vs. \(2(381)=762\).
Time = 0.70 (sec) , antiderivative size = 1158, normalized size of antiderivative = 2.92
A*d^3*(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^3*(-1/3*x^3/c/(c*x^2+b*x+a)^(3/2)-1/2*b /c*(-x^2/c/(c*x^2+b*x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b /c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(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)))+1/2*a/c*( 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/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(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/c*(-x/c/(c*x^2+b*x+a)^(1/2)-1/2*b/c*(-1/c/(c*x^2+b*x +a)^(1/2)-b/c*(2*c*x+b)/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2))+1/c^(3/2)*ln((1/2 *b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))+(A*e^3+3*B*d*e^2)*(-x^2/c/(c*x^2+b* x+a)^(3/2)+1/2*b/c*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3/c/(c*x^2+b* x+a)^(3/2)-1/2*b/c*(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)))+1/2*a/c*(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/c*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(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))))+ (3*A*d^2*e+B*d^3)*(-1/3/c/(c*x^2+b*x+a)^(3/2)-1/2*b/c*(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)))+(3*A*d*e^2+3*B*d^2*e)*(-1/2*x/c/(c*x^2+b*x+a)^(3/2)-1/4*b/c*(-1/3...
Leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (381) = 762\).
Time = 6.92 (sec) , antiderivative size = 1881, normalized size of antiderivative = 4.74 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
[1/6*(3*((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*e^3*x^4 + 2*(B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^3)*e^3*x^3 + (B*b^6 - 6*B*a*b^4*c + 32*B*a^3 *c^3)*e^3*x^2 + 2*(B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*e^3*x + (B*a^ 2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*e^3)*sqrt(c)*log(-8*c^2*x^2 - 8*b*c* x - b^2 - 4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - 4*a*c) - 4*(24*(2* B*a^3 - A*a^2*b)*c^3*d*e^2 + (4*(2*B*a^2 - 3*A*a*b)*c^4 + (2*B*a*b^2 + A*b ^3)*c^3)*d^3 + 6*(4*A*a^2*c^4 - (4*B*a^2*b - A*a*b^2)*c^3)*d^2*e + (3*B*a^ 2*b^3*c - 20*B*a^3*b*c^2 + 16*A*a^3*c^3)*e^3 + (8*(B*b*c^5 - 2*A*c^6)*d^3 - 6*(B*b^2*c^4 + 4*(B*a - A*b)*c^5)*d^2*e - 3*(B*b^3*c^3 + 8*A*a*c^5 - 2*( 6*B*a*b - A*b^2)*c^4)*d*e^2 + (4*B*b^4*c^2 + 4*(8*B*a^2 + 3*A*a*b)*c^4 - ( 28*B*a*b^2 + A*b^3)*c^3)*e^3)*x^3 + 3*(4*(B*b^2*c^4 - 2*A*b*c^5)*d^3 - 3*( B*b^3*c^3 + 4*(B*a*b - A*b^2)*c^4)*d^2*e + 3*(4*(2*B*a^2 - A*a*b)*c^4 + (2 *B*a*b^2 - A*b^3)*c^3)*d*e^2 + (B*b^5*c - 6*B*a*b^3*c^2 + 2*A*a*b^2*c^3 + 8*A*a^2*c^4)*e^3)*x^2 + 3*(12*(2*B*a^2*b - A*a*b^2)*c^3*d*e^2 + (B*b^3*c^3 - 8*A*a*c^5 + 2*(2*B*a*b - A*b^2)*c^4)*d^3 + 3*(4*A*a*b*c^4 - (4*B*a*b^2 - A*b^3)*c^3)*d^2*e + 2*(B*a*b^4*c - 7*B*a^2*b^2*c^2 + 4*(B*a^3 + A*a^2*b) *c^3)*e^3)*x)*sqrt(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4 *c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5 *c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x), -1/3*(3*((B*b^4*c^2 - 8*B*a*b^...
Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 805 vs. \(2 (381) = 762\).
Time = 0.31 (sec) , antiderivative size = 805, normalized size of antiderivative = 2.03 \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {B e^{3} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} \sqrt {c} + b \right |}\right )}{c^{\frac {5}{2}}} - \frac {2 \, {\left ({\left ({\left (\frac {{\left (8 \, B b c^{4} d^{3} - 16 \, A c^{5} d^{3} - 6 \, B b^{2} c^{3} d^{2} e - 24 \, B a c^{4} d^{2} e + 24 \, A b c^{4} d^{2} e - 3 \, B b^{3} c^{2} d e^{2} + 36 \, B a b c^{3} d e^{2} - 6 \, A b^{2} c^{3} d e^{2} - 24 \, A a c^{4} d e^{2} + 4 \, B b^{4} c e^{3} - 28 \, B a b^{2} c^{2} e^{3} - A b^{3} c^{2} e^{3} + 32 \, B a^{2} c^{3} e^{3} + 12 \, A a b c^{3} e^{3}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac {3 \, {\left (4 \, B b^{2} c^{3} d^{3} - 8 \, A b c^{4} d^{3} - 3 \, B b^{3} c^{2} d^{2} e - 12 \, B a b c^{3} d^{2} e + 12 \, A b^{2} c^{3} d^{2} e + 6 \, B a b^{2} c^{2} d e^{2} - 3 \, A b^{3} c^{2} d e^{2} + 24 \, B a^{2} c^{3} d e^{2} - 12 \, A a b c^{3} d e^{2} + B b^{5} e^{3} - 6 \, B a b^{3} c e^{3} + 2 \, A a b^{2} c^{2} e^{3} + 8 \, A a^{2} c^{3} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {3 \, {\left (B b^{3} c^{2} d^{3} + 4 \, B a b c^{3} d^{3} - 2 \, A b^{2} c^{3} d^{3} - 8 \, A a c^{4} d^{3} - 12 \, B a b^{2} c^{2} d^{2} e + 3 \, A b^{3} c^{2} d^{2} e + 12 \, A a b c^{3} d^{2} e + 24 \, B a^{2} b c^{2} d e^{2} - 12 \, A a b^{2} c^{2} d e^{2} + 2 \, B a b^{4} e^{3} - 14 \, B a^{2} b^{2} c e^{3} + 8 \, B a^{3} c^{2} e^{3} + 8 \, A a^{2} b c^{2} e^{3}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac {2 \, B a b^{2} c^{2} d^{3} + A b^{3} c^{2} d^{3} + 8 \, B a^{2} c^{3} d^{3} - 12 \, A a b c^{3} d^{3} - 24 \, B a^{2} b c^{2} d^{2} e + 6 \, A a b^{2} c^{2} d^{2} e + 24 \, A a^{2} c^{3} d^{2} e + 48 \, B a^{3} c^{2} d e^{2} - 24 \, A a^{2} b c^{2} d e^{2} + 3 \, B a^{2} b^{3} e^{3} - 20 \, B a^{3} b c e^{3} + 16 \, A a^{3} c^{2} e^{3}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
-B*e^3*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) + b))/c^(5/2) - 2/3*((((8*B*b*c^4*d^3 - 16*A*c^5*d^3 - 6*B*b^2*c^3*d^2*e - 24*B*a*c^4*d ^2*e + 24*A*b*c^4*d^2*e - 3*B*b^3*c^2*d*e^2 + 36*B*a*b*c^3*d*e^2 - 6*A*b^2 *c^3*d*e^2 - 24*A*a*c^4*d*e^2 + 4*B*b^4*c*e^3 - 28*B*a*b^2*c^2*e^3 - A*b^3 *c^2*e^3 + 32*B*a^2*c^3*e^3 + 12*A*a*b*c^3*e^3)*x/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(4*B*b^2*c^3*d^3 - 8*A*b*c^4*d^3 - 3*B*b^3*c^2*d^2*e - 12 *B*a*b*c^3*d^2*e + 12*A*b^2*c^3*d^2*e + 6*B*a*b^2*c^2*d*e^2 - 3*A*b^3*c^2* d*e^2 + 24*B*a^2*c^3*d*e^2 - 12*A*a*b*c^3*d*e^2 + B*b^5*e^3 - 6*B*a*b^3*c* e^3 + 2*A*a*b^2*c^2*e^3 + 8*A*a^2*c^3*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2 *c^4))*x + 3*(B*b^3*c^2*d^3 + 4*B*a*b*c^3*d^3 - 2*A*b^2*c^3*d^3 - 8*A*a*c^ 4*d^3 - 12*B*a*b^2*c^2*d^2*e + 3*A*b^3*c^2*d^2*e + 12*A*a*b*c^3*d^2*e + 24 *B*a^2*b*c^2*d*e^2 - 12*A*a*b^2*c^2*d*e^2 + 2*B*a*b^4*e^3 - 14*B*a^2*b^2*c *e^3 + 8*B*a^3*c^2*e^3 + 8*A*a^2*b*c^2*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^ 2*c^4))*x + (2*B*a*b^2*c^2*d^3 + A*b^3*c^2*d^3 + 8*B*a^2*c^3*d^3 - 12*A*a* b*c^3*d^3 - 24*B*a^2*b*c^2*d^2*e + 6*A*a*b^2*c^2*d^2*e + 24*A*a^2*c^3*d^2* e + 48*B*a^3*c^2*d*e^2 - 24*A*a^2*b*c^2*d*e^2 + 3*B*a^2*b^3*e^3 - 20*B*a^3 *b*c*e^3 + 16*A*a^3*c^2*e^3)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]