Integrand size = 20, antiderivative size = 126 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=-\frac {(B d-A e) \left (c d^2+a e^2\right ) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {\left (3 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {c (3 B d-A e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {B c (d+e x)^{4+m}}{e^4 (4+m)} \]
-(-A*e+B*d)*(a*e^2+c*d^2)*(e*x+d)^(1+m)/e^4/(1+m)+(-2*A*c*d*e+B*a*e^2+3*B* c*d^2)*(e*x+d)^(2+m)/e^4/(2+m)-c*(-A*e+3*B*d)*(e*x+d)^(3+m)/e^4/(3+m)+B*c* (e*x+d)^(4+m)/e^4/(4+m)
Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.97 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\frac {(d+e x)^{1+m} \left ((-B d+A e) \left (\frac {c d^2+a e^2}{1+m}-\frac {2 c d (d+e x)}{2+m}+\frac {c (d+e x)^2}{3+m}\right )+B (d+e x) \left (\frac {c d^2+a e^2}{2+m}-\frac {2 c d (d+e x)}{3+m}+\frac {c (d+e x)^2}{4+m}\right )\right )}{e^4} \]
((d + e*x)^(1 + m)*((-(B*d) + A*e)*((c*d^2 + a*e^2)/(1 + m) - (2*c*d*(d + e*x))/(2 + m) + (c*(d + e*x)^2)/(3 + m)) + B*(d + e*x)*((c*d^2 + a*e^2)/(2 + m) - (2*c*d*(d + e*x))/(3 + m) + (c*(d + e*x)^2)/(4 + m))))/e^4
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {652, 2009}
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.
| \(\displaystyle \int \left (a+c x^2\right ) (A+B x) (d+e x)^m \, dx\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \int \left (\frac {\left (a e^2+c d^2\right ) (A e-B d) (d+e x)^m}{e^3}+\frac {(d+e x)^{m+1} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^3}+\frac {c (A e-3 B d) (d+e x)^{m+2}}{e^3}+\frac {B c (d+e x)^{m+3}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a e^2+c d^2\right ) (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(d+e x)^{m+2} \left (a B e^2-2 A c d e+3 B c d^2\right )}{e^4 (m+2)}-\frac {c (3 B d-A e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac {B c (d+e x)^{m+4}}{e^4 (m+4)}\) |
-(((B*d - A*e)*(c*d^2 + a*e^2)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((3*B*c *d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (c*(3*B*d - A*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (B*c*(d + e*x)^(4 + m))/(e^4*(4 + m))
3.15.89.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(337\) vs. \(2(126)=252\).
Time = 0.51 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.68
| method | result | size |
| gosper | \(\frac {\left (e x +d \right )^{1+m} \left (B c \,e^{3} m^{3} x^{3}+A c \,e^{3} m^{3} x^{2}+6 B c \,e^{3} m^{2} x^{3}+7 A c \,e^{3} m^{2} x^{2}+B a \,e^{3} m^{3} x -3 B c d \,e^{2} m^{2} x^{2}+11 B c \,e^{3} m \,x^{3}+A a \,e^{3} m^{3}-2 A c d \,e^{2} m^{2} x +14 A c \,e^{3} m \,x^{2}+8 B a \,e^{3} m^{2} x -9 B c d \,e^{2} m \,x^{2}+6 B c \,x^{3} e^{3}+9 A a \,e^{3} m^{2}-10 A c d \,e^{2} m x +8 A c \,e^{3} x^{2}-B a d \,e^{2} m^{2}+19 B a \,e^{3} m x +6 B c \,d^{2} e m x -6 B \,x^{2} c d \,e^{2}+26 A a \,e^{3} m +2 A c \,d^{2} e m -8 A c d \,e^{2} x -7 B a d \,e^{2} m +12 B x a \,e^{3}+6 B c \,d^{2} e x +24 A a \,e^{3}+8 A c \,d^{2} e -12 B a d \,e^{2}-6 B c \,d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(338\) |
| norman | \(\frac {B c \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {d \left (A a \,e^{3} m^{3}+9 A a \,e^{3} m^{2}-B a d \,e^{2} m^{2}+26 A a \,e^{3} m +2 A c \,d^{2} e m -7 B a d \,e^{2} m +24 A a \,e^{3}+8 A c \,d^{2} e -12 B a d \,e^{2}-6 B c \,d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (A c d e \,m^{2}+B a \,e^{2} m^{2}+4 A c d e m +7 B a \,e^{2} m -3 B c \,d^{2} m +12 B a \,e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (A a \,e^{3} m^{3}+B a d \,e^{2} m^{3}+9 A a \,e^{3} m^{2}-2 A c \,d^{2} e \,m^{2}+7 B a d \,e^{2} m^{2}+26 A a \,e^{3} m -8 A c \,d^{2} e m +12 B a d \,e^{2} m +6 B c \,d^{3} m +24 A a \,e^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (A e m +B d m +4 A e \right ) c \,x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}\) | \(389\) |
| risch | \(\frac {\left (B c \,e^{4} m^{3} x^{4}+A c \,e^{4} m^{3} x^{3}+B c d \,e^{3} m^{3} x^{3}+6 B c \,e^{4} m^{2} x^{4}+A c d \,e^{3} m^{3} x^{2}+7 A c \,e^{4} m^{2} x^{3}+B a \,e^{4} m^{3} x^{2}+3 B c d \,e^{3} m^{2} x^{3}+11 B c \,e^{4} m \,x^{4}+A a \,e^{4} m^{3} x +5 A c d \,e^{3} m^{2} x^{2}+14 A c \,e^{4} m \,x^{3}+B a d \,e^{3} m^{3} x +8 B a \,e^{4} m^{2} x^{2}-3 B c \,d^{2} e^{2} m^{2} x^{2}+2 B c d \,e^{3} m \,x^{3}+6 B c \,x^{4} e^{4}+A a d \,e^{3} m^{3}+9 A a \,e^{4} m^{2} x -2 A c \,d^{2} e^{2} m^{2} x +4 A c d \,e^{3} m \,x^{2}+8 A c \,e^{4} x^{3}+7 B a d \,e^{3} m^{2} x +19 B a \,e^{4} m \,x^{2}-3 B c \,d^{2} e^{2} m \,x^{2}+9 A a d \,e^{3} m^{2}+26 A a \,e^{4} m x -8 A c \,d^{2} e^{2} m x -B a \,d^{2} e^{2} m^{2}+12 B a d \,e^{3} m x +12 B a \,e^{4} x^{2}+6 B c \,d^{3} e m x +26 A a d \,e^{3} m +24 A a \,e^{4} x +2 A c \,d^{3} e m -7 B a \,d^{2} e^{2} m +24 A a d \,e^{3}+8 A c \,d^{3} e -12 B a \,d^{2} e^{2}-6 B c \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\) | \(482\) |
| parallelrisch | \(\frac {A x \left (e x +d \right )^{m} a \,e^{4} m^{3}+8 B \,x^{2} \left (e x +d \right )^{m} a \,e^{4} m^{2}+9 A x \left (e x +d \right )^{m} a \,e^{4} m^{2}+A \left (e x +d \right )^{m} a d \,e^{3} m^{3}+19 B \,x^{2} \left (e x +d \right )^{m} a \,e^{4} m +26 A x \left (e x +d \right )^{m} a \,e^{4} m +9 A \left (e x +d \right )^{m} a d \,e^{3} m^{2}-B \left (e x +d \right )^{m} a \,d^{2} e^{2} m^{2}+26 A \left (e x +d \right )^{m} a d \,e^{3} m -7 B \left (e x +d \right )^{m} a \,d^{2} e^{2} m +B \,x^{2} \left (e x +d \right )^{m} a \,e^{4} m^{3}+14 A \,x^{3} \left (e x +d \right )^{m} c \,e^{4} m +2 A \left (e x +d \right )^{m} c \,d^{3} e m +B \,x^{4} \left (e x +d \right )^{m} c \,e^{4} m^{3}+A \,x^{3} \left (e x +d \right )^{m} c \,e^{4} m^{3}+6 B \,x^{4} \left (e x +d \right )^{m} c \,e^{4} m^{2}+7 A \,x^{3} \left (e x +d \right )^{m} c \,e^{4} m^{2}+11 B \,x^{4} \left (e x +d \right )^{m} c \,e^{4} m -6 B \left (e x +d \right )^{m} c \,d^{4}+6 B \,x^{4} \left (e x +d \right )^{m} c \,e^{4}+8 A \,x^{3} \left (e x +d \right )^{m} c \,e^{4}+8 A \left (e x +d \right )^{m} c \,d^{3} e +12 B \,x^{2} \left (e x +d \right )^{m} a \,e^{4}+24 A x \left (e x +d \right )^{m} a \,e^{4}+24 A \left (e x +d \right )^{m} a d \,e^{3}-12 B \left (e x +d \right )^{m} a \,d^{2} e^{2}+B \,x^{3} \left (e x +d \right )^{m} c d \,e^{3} m^{3}+A \,x^{2} \left (e x +d \right )^{m} c d \,e^{3} m^{3}+3 B \,x^{3} \left (e x +d \right )^{m} c d \,e^{3} m^{2}+5 A \,x^{2} \left (e x +d \right )^{m} c d \,e^{3} m^{2}+2 B \,x^{3} \left (e x +d \right )^{m} c d \,e^{3} m -3 B \,x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m^{2}+4 A \,x^{2} \left (e x +d \right )^{m} c d \,e^{3} m -2 A x \left (e x +d \right )^{m} c \,d^{2} e^{2} m^{2}-3 B \,x^{2} \left (e x +d \right )^{m} c \,d^{2} e^{2} m -8 A x \left (e x +d \right )^{m} c \,d^{2} e^{2} m +6 B x \left (e x +d \right )^{m} c \,d^{3} e m +B x \left (e x +d \right )^{m} a d \,e^{3} m^{3}+7 B x \left (e x +d \right )^{m} a d \,e^{3} m^{2}+12 B x \left (e x +d \right )^{m} a d \,e^{3} m}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\) | \(755\) |
1/e^4*(e*x+d)^(1+m)/(m^4+10*m^3+35*m^2+50*m+24)*(B*c*e^3*m^3*x^3+A*c*e^3*m ^3*x^2+6*B*c*e^3*m^2*x^3+7*A*c*e^3*m^2*x^2+B*a*e^3*m^3*x-3*B*c*d*e^2*m^2*x ^2+11*B*c*e^3*m*x^3+A*a*e^3*m^3-2*A*c*d*e^2*m^2*x+14*A*c*e^3*m*x^2+8*B*a*e ^3*m^2*x-9*B*c*d*e^2*m*x^2+6*B*c*e^3*x^3+9*A*a*e^3*m^2-10*A*c*d*e^2*m*x+8* A*c*e^3*x^2-B*a*d*e^2*m^2+19*B*a*e^3*m*x+6*B*c*d^2*e*m*x-6*B*c*d*e^2*x^2+2 6*A*a*e^3*m+2*A*c*d^2*e*m-8*A*c*d*e^2*x-7*B*a*d*e^2*m+12*B*a*e^3*x+6*B*c*d ^2*e*x+24*A*a*e^3+8*A*c*d^2*e-12*B*a*d*e^2-6*B*c*d^3)
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (126) = 252\).
Time = 0.34 (sec) , antiderivative size = 434, normalized size of antiderivative = 3.44 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\frac {{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 8 \, A c d^{3} e - 12 \, B a d^{2} e^{2} + 24 \, A a d e^{3} + {\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} + {\left (8 \, A c e^{4} + {\left (B c d e^{3} + A c e^{4}\right )} m^{3} + {\left (3 \, B c d e^{3} + 7 \, A c e^{4}\right )} m^{2} + 2 \, {\left (B c d e^{3} + 7 \, A c e^{4}\right )} m\right )} x^{3} - {\left (B a d^{2} e^{2} - 9 \, A a d e^{3}\right )} m^{2} + {\left (12 \, B a e^{4} + {\left (A c d e^{3} + B a e^{4}\right )} m^{3} - {\left (3 \, B c d^{2} e^{2} - 5 \, A c d e^{3} - 8 \, B a e^{4}\right )} m^{2} - {\left (3 \, B c d^{2} e^{2} - 4 \, A c d e^{3} - 19 \, B a e^{4}\right )} m\right )} x^{2} + {\left (2 \, A c d^{3} e - 7 \, B a d^{2} e^{2} + 26 \, A a d e^{3}\right )} m + {\left (24 \, A a e^{4} + {\left (B a d e^{3} + A a e^{4}\right )} m^{3} - {\left (2 \, A c d^{2} e^{2} - 7 \, B a d e^{3} - 9 \, A a e^{4}\right )} m^{2} + 2 \, {\left (3 \, B c d^{3} e - 4 \, A c d^{2} e^{2} + 6 \, B a d e^{3} + 13 \, A a e^{4}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
(A*a*d*e^3*m^3 - 6*B*c*d^4 + 8*A*c*d^3*e - 12*B*a*d^2*e^2 + 24*A*a*d*e^3 + (B*c*e^4*m^3 + 6*B*c*e^4*m^2 + 11*B*c*e^4*m + 6*B*c*e^4)*x^4 + (8*A*c*e^4 + (B*c*d*e^3 + A*c*e^4)*m^3 + (3*B*c*d*e^3 + 7*A*c*e^4)*m^2 + 2*(B*c*d*e^ 3 + 7*A*c*e^4)*m)*x^3 - (B*a*d^2*e^2 - 9*A*a*d*e^3)*m^2 + (12*B*a*e^4 + (A *c*d*e^3 + B*a*e^4)*m^3 - (3*B*c*d^2*e^2 - 5*A*c*d*e^3 - 8*B*a*e^4)*m^2 - (3*B*c*d^2*e^2 - 4*A*c*d*e^3 - 19*B*a*e^4)*m)*x^2 + (2*A*c*d^3*e - 7*B*a*d ^2*e^2 + 26*A*a*d*e^3)*m + (24*A*a*e^4 + (B*a*d*e^3 + A*a*e^4)*m^3 - (2*A* c*d^2*e^2 - 7*B*a*d*e^3 - 9*A*a*e^4)*m^2 + 2*(3*B*c*d^3*e - 4*A*c*d^2*e^2 + 6*B*a*d*e^3 + 13*A*a*e^4)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e ^4*m^2 + 50*e^4*m + 24*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 3958 vs. \(2 (114) = 228\).
Time = 1.05 (sec) , antiderivative size = 3958, normalized size of antiderivative = 31.41 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\text {Too large to display} \]
Piecewise((d**m*(A*a*x + A*c*x**3/3 + B*a*x**2/2 + B*c*x**4/4), Eq(e, 0)), (-2*A*a*e**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3 ) - 2*A*c*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x **3) - 6*A*c*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e **7*x**3) - 6*A*c*e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - B*a*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x** 2 + 6*e**7*x**3) - 3*B*a*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6* x**2 + 6*e**7*x**3) + 6*B*c*d**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5* x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*c*d**3/(6*d**3*e**4 + 18*d**2*e** 5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d**2*e*x*log(d/e + x)/(6*d**3 *e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 27*B*c*d**2*e*x/( 6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*c*d*e* *2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e* *7*x**3) + 18*B*c*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x* *2 + 6*e**7*x**3) + 6*B*c*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e* *5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -4)), (-A*a*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d *e**5*x + 2*e**6*x**2) + 3*A*c*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x **2) + 4*A*c*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2 ) + 4*A*c*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*c*e**...
Time = 0.20 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.89 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\frac {{\left (e^{2} {\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )} {\left (e x + d\right )}^{m} B a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \frac {{\left (e x + d\right )}^{m + 1} A a}{e {\left (m + 1\right )}} + \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} A c}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} B c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \]
(e^2*(m + 1)*x^2 + d*e*m*x - d^2)*(e*x + d)^m*B*a/((m^2 + 3*m + 2)*e^2) + (e*x + d)^(m + 1)*A*a/(e*(m + 1)) + ((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d *e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*A*c/((m^3 + 6*m^2 + 11*m + 6)* e^3) + ((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*d^4)*(e*x + d)^m*B*c/((m^4 + 10 *m^3 + 35*m^2 + 50*m + 24)*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 768 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 768, normalized size of antiderivative = 6.10 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\frac {{\left (e x + d\right )}^{m} B c e^{4} m^{3} x^{4} + {\left (e x + d\right )}^{m} B c d e^{3} m^{3} x^{3} + {\left (e x + d\right )}^{m} A c e^{4} m^{3} x^{3} + 6 \, {\left (e x + d\right )}^{m} B c e^{4} m^{2} x^{4} + {\left (e x + d\right )}^{m} A c d e^{3} m^{3} x^{2} + {\left (e x + d\right )}^{m} B a e^{4} m^{3} x^{2} + 3 \, {\left (e x + d\right )}^{m} B c d e^{3} m^{2} x^{3} + 7 \, {\left (e x + d\right )}^{m} A c e^{4} m^{2} x^{3} + 11 \, {\left (e x + d\right )}^{m} B c e^{4} m x^{4} + {\left (e x + d\right )}^{m} B a d e^{3} m^{3} x + {\left (e x + d\right )}^{m} A a e^{4} m^{3} x - 3 \, {\left (e x + d\right )}^{m} B c d^{2} e^{2} m^{2} x^{2} + 5 \, {\left (e x + d\right )}^{m} A c d e^{3} m^{2} x^{2} + 8 \, {\left (e x + d\right )}^{m} B a e^{4} m^{2} x^{2} + 2 \, {\left (e x + d\right )}^{m} B c d e^{3} m x^{3} + 14 \, {\left (e x + d\right )}^{m} A c e^{4} m x^{3} + 6 \, {\left (e x + d\right )}^{m} B c e^{4} x^{4} + {\left (e x + d\right )}^{m} A a d e^{3} m^{3} - 2 \, {\left (e x + d\right )}^{m} A c d^{2} e^{2} m^{2} x + 7 \, {\left (e x + d\right )}^{m} B a d e^{3} m^{2} x + 9 \, {\left (e x + d\right )}^{m} A a e^{4} m^{2} x - 3 \, {\left (e x + d\right )}^{m} B c d^{2} e^{2} m x^{2} + 4 \, {\left (e x + d\right )}^{m} A c d e^{3} m x^{2} + 19 \, {\left (e x + d\right )}^{m} B a e^{4} m x^{2} + 8 \, {\left (e x + d\right )}^{m} A c e^{4} x^{3} - {\left (e x + d\right )}^{m} B a d^{2} e^{2} m^{2} + 9 \, {\left (e x + d\right )}^{m} A a d e^{3} m^{2} + 6 \, {\left (e x + d\right )}^{m} B c d^{3} e m x - 8 \, {\left (e x + d\right )}^{m} A c d^{2} e^{2} m x + 12 \, {\left (e x + d\right )}^{m} B a d e^{3} m x + 26 \, {\left (e x + d\right )}^{m} A a e^{4} m x + 12 \, {\left (e x + d\right )}^{m} B a e^{4} x^{2} + 2 \, {\left (e x + d\right )}^{m} A c d^{3} e m - 7 \, {\left (e x + d\right )}^{m} B a d^{2} e^{2} m + 26 \, {\left (e x + d\right )}^{m} A a d e^{3} m + 24 \, {\left (e x + d\right )}^{m} A a e^{4} x - 6 \, {\left (e x + d\right )}^{m} B c d^{4} + 8 \, {\left (e x + d\right )}^{m} A c d^{3} e - 12 \, {\left (e x + d\right )}^{m} B a d^{2} e^{2} + 24 \, {\left (e x + d\right )}^{m} A a d e^{3}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
((e*x + d)^m*B*c*e^4*m^3*x^4 + (e*x + d)^m*B*c*d*e^3*m^3*x^3 + (e*x + d)^m *A*c*e^4*m^3*x^3 + 6*(e*x + d)^m*B*c*e^4*m^2*x^4 + (e*x + d)^m*A*c*d*e^3*m ^3*x^2 + (e*x + d)^m*B*a*e^4*m^3*x^2 + 3*(e*x + d)^m*B*c*d*e^3*m^2*x^3 + 7 *(e*x + d)^m*A*c*e^4*m^2*x^3 + 11*(e*x + d)^m*B*c*e^4*m*x^4 + (e*x + d)^m* B*a*d*e^3*m^3*x + (e*x + d)^m*A*a*e^4*m^3*x - 3*(e*x + d)^m*B*c*d^2*e^2*m^ 2*x^2 + 5*(e*x + d)^m*A*c*d*e^3*m^2*x^2 + 8*(e*x + d)^m*B*a*e^4*m^2*x^2 + 2*(e*x + d)^m*B*c*d*e^3*m*x^3 + 14*(e*x + d)^m*A*c*e^4*m*x^3 + 6*(e*x + d) ^m*B*c*e^4*x^4 + (e*x + d)^m*A*a*d*e^3*m^3 - 2*(e*x + d)^m*A*c*d^2*e^2*m^2 *x + 7*(e*x + d)^m*B*a*d*e^3*m^2*x + 9*(e*x + d)^m*A*a*e^4*m^2*x - 3*(e*x + d)^m*B*c*d^2*e^2*m*x^2 + 4*(e*x + d)^m*A*c*d*e^3*m*x^2 + 19*(e*x + d)^m* B*a*e^4*m*x^2 + 8*(e*x + d)^m*A*c*e^4*x^3 - (e*x + d)^m*B*a*d^2*e^2*m^2 + 9*(e*x + d)^m*A*a*d*e^3*m^2 + 6*(e*x + d)^m*B*c*d^3*e*m*x - 8*(e*x + d)^m* A*c*d^2*e^2*m*x + 12*(e*x + d)^m*B*a*d*e^3*m*x + 26*(e*x + d)^m*A*a*e^4*m* x + 12*(e*x + d)^m*B*a*e^4*x^2 + 2*(e*x + d)^m*A*c*d^3*e*m - 7*(e*x + d)^m *B*a*d^2*e^2*m + 26*(e*x + d)^m*A*a*d*e^3*m + 24*(e*x + d)^m*A*a*e^4*x - 6 *(e*x + d)^m*B*c*d^4 + 8*(e*x + d)^m*A*c*d^3*e - 12*(e*x + d)^m*B*a*d^2*e^ 2 + 24*(e*x + d)^m*A*a*d*e^3)/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4* m + 24*e^4)
Time = 10.93 (sec) , antiderivative size = 446, normalized size of antiderivative = 3.54 \[ \int (A+B x) (d+e x)^m \left (a+c x^2\right ) \, dx=\frac {{\left (d+e\,x\right )}^m\,\left (-6\,B\,c\,d^4+2\,A\,c\,d^3\,e\,m+8\,A\,c\,d^3\,e-B\,a\,d^2\,e^2\,m^2-7\,B\,a\,d^2\,e^2\,m-12\,B\,a\,d^2\,e^2+A\,a\,d\,e^3\,m^3+9\,A\,a\,d\,e^3\,m^2+26\,A\,a\,d\,e^3\,m+24\,A\,a\,d\,e^3\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (6\,B\,c\,d^3\,e\,m-2\,A\,c\,d^2\,e^2\,m^2-8\,A\,c\,d^2\,e^2\,m+B\,a\,d\,e^3\,m^3+7\,B\,a\,d\,e^3\,m^2+12\,B\,a\,d\,e^3\,m+A\,a\,e^4\,m^3+9\,A\,a\,e^4\,m^2+26\,A\,a\,e^4\,m+24\,A\,a\,e^4\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (-3\,B\,c\,d^2\,m+A\,c\,d\,e\,m^2+4\,A\,c\,d\,e\,m+B\,a\,e^2\,m^2+7\,B\,a\,e^2\,m+12\,B\,a\,e^2\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {B\,c\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {c\,x^3\,{\left (d+e\,x\right )}^m\,\left (4\,A\,e+A\,e\,m+B\,d\,m\right )\,\left (m^2+3\,m+2\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \]
((d + e*x)^m*(24*A*a*d*e^3 - 6*B*c*d^4 + 8*A*c*d^3*e - 12*B*a*d^2*e^2 - B* a*d^2*e^2*m^2 + 26*A*a*d*e^3*m + 2*A*c*d^3*e*m + 9*A*a*d*e^3*m^2 + A*a*d*e ^3*m^3 - 7*B*a*d^2*e^2*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x* (d + e*x)^m*(24*A*a*e^4 + 26*A*a*e^4*m + 9*A*a*e^4*m^2 + A*a*e^4*m^3 - 2*A *c*d^2*e^2*m^2 + 12*B*a*d*e^3*m + 6*B*c*d^3*e*m + 7*B*a*d*e^3*m^2 + B*a*d* e^3*m^3 - 8*A*c*d^2*e^2*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (x ^2*(m + 1)*(d + e*x)^m*(12*B*a*e^2 + 7*B*a*e^2*m - 3*B*c*d^2*m + B*a*e^2*m ^2 + 4*A*c*d*e*m + A*c*d*e*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (B*c*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35*m^2 + 10*m^3 + m^4 + 24) + (c*x^3*(d + e*x)^m*(4*A*e + A*e*m + B*d*m)*(3*m + m^2 + 2))/ (e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))