[next] [prev] [prev-tail] [tail] [up]

3.1487 \(\int (A+B x) (d+e x)^m \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=372 \[ -\frac{c (d+e x)^{m+4} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8 (m+4)}+\frac{3 c^2 (d+e x)^{m+6} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (m+6)}-\frac{c^2 (d+e x)^{m+5} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (m+5)}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) (d+e x)^{m+1}}{e^8 (m+1)}+\frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+2} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (m+2)}-\frac{3 c \left (a e^2+c d^2\right ) (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (m+3)}-\frac{c^3 (7 B d-A e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 + a
*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*
c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3
 + m))/(e^8*(3 + m)) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*
c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (c^2*(35*B*c*d^3 - 15
*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (3*c^2
*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^3*(7*B*
d - A*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(8 +
m))

_______________________________________________________________________________________

Rubi [A]  time = 0.617027, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{c (d+e x)^{m+4} \left (3 a^2 B e^4-12 a A c d e^3+30 a B c d^2 e^2-20 A c^2 d^3 e+35 B c^2 d^4\right )}{e^8 (m+4)}+\frac{3 c^2 (d+e x)^{m+6} \left (a B e^2-2 A c d e+7 B c d^2\right )}{e^8 (m+6)}-\frac{c^2 (d+e x)^{m+5} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{e^8 (m+5)}-\frac{\left (a e^2+c d^2\right )^3 (B d-A e) (d+e x)^{m+1}}{e^8 (m+1)}+\frac{\left (a e^2+c d^2\right )^2 (d+e x)^{m+2} \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^8 (m+2)}-\frac{3 c \left (a e^2+c d^2\right ) (d+e x)^{m+3} \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 (m+3)}-\frac{c^3 (7 B d-A e) (d+e x)^{m+7}}{e^8 (m+7)}+\frac{B c^3 (d+e x)^{m+8}}{e^8 (m+8)} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

-(((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^(1 + m))/(e^8*(1 + m))) + ((c*d^2 + a
*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d + e*x)^(2 + m))/(e^8*(2 + m)) - (3*
c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^(3
 + m))/(e^8*(3 + m)) + (c*(35*B*c^2*d^4 - 20*A*c^2*d^3*e + 30*a*B*c*d^2*e^2 - 12
*a*A*c*d*e^3 + 3*a^2*B*e^4)*(d + e*x)^(4 + m))/(e^8*(4 + m)) - (c^2*(35*B*c*d^3
- 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(5 + m))/(e^8*(5 + m)) + (3
*c^2*(7*B*c*d^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(6 + m))/(e^8*(6 + m)) - (c^3*(
7*B*d - A*e)*(d + e*x)^(7 + m))/(e^8*(7 + m)) + (B*c^3*(d + e*x)^(8 + m))/(e^8*(
8 + m))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 151.63, size = 371, normalized size = 1. \[ \frac{B c^{3} \left (d + e x\right )^{m + 8}}{e^{8} \left (m + 8\right )} + \frac{c^{3} \left (d + e x\right )^{m + 7} \left (A e - 7 B d\right )}{e^{8} \left (m + 7\right )} + \frac{c^{2} \left (d + e x\right )^{m + 5} \left (3 A a e^{3} + 15 A c d^{2} e - 15 B a d e^{2} - 35 B c d^{3}\right )}{e^{8} \left (m + 5\right )} + \frac{3 c^{2} \left (d + e x\right )^{m + 6} \left (- 2 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{8} \left (m + 6\right )} + \frac{3 c \left (d + e x\right )^{m + 3} \left (a e^{2} + c d^{2}\right ) \left (A a e^{3} + 5 A c d^{2} e - 3 B a d e^{2} - 7 B c d^{3}\right )}{e^{8} \left (m + 3\right )} + \frac{c \left (d + e x\right )^{m + 4} \left (- 12 A a c d e^{3} - 20 A c^{2} d^{3} e + 3 B a^{2} e^{4} + 30 B a c d^{2} e^{2} + 35 B c^{2} d^{4}\right )}{e^{8} \left (m + 4\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{e^{8} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{e^{8} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**3,x)

[Out]

B*c**3*(d + e*x)**(m + 8)/(e**8*(m + 8)) + c**3*(d + e*x)**(m + 7)*(A*e - 7*B*d)
/(e**8*(m + 7)) + c**2*(d + e*x)**(m + 5)*(3*A*a*e**3 + 15*A*c*d**2*e - 15*B*a*d
*e**2 - 35*B*c*d**3)/(e**8*(m + 5)) + 3*c**2*(d + e*x)**(m + 6)*(-2*A*c*d*e + B*
a*e**2 + 7*B*c*d**2)/(e**8*(m + 6)) + 3*c*(d + e*x)**(m + 3)*(a*e**2 + c*d**2)*(
A*a*e**3 + 5*A*c*d**2*e - 3*B*a*d*e**2 - 7*B*c*d**3)/(e**8*(m + 3)) + c*(d + e*x
)**(m + 4)*(-12*A*a*c*d*e**3 - 20*A*c**2*d**3*e + 3*B*a**2*e**4 + 30*B*a*c*d**2*
e**2 + 35*B*c**2*d**4)/(e**8*(m + 4)) + (d + e*x)**(m + 1)*(A*e - B*d)*(a*e**2 +
 c*d**2)**3/(e**8*(m + 1)) + (d + e*x)**(m + 2)*(a*e**2 + c*d**2)**2*(-6*A*c*d*e
 + B*a*e**2 + 7*B*c*d**2)/(e**8*(m + 2))

_______________________________________________________________________________________

Mathematica [B]  time = 1.82069, size = 875, normalized size = 2.35 \[ \frac{(d+e x)^{m+1} \left (A e (m+8) \left (a^3 \left (m^6+27 m^5+295 m^4+1665 m^3+5104 m^2+8028 m+5040\right ) e^6+3 a^2 c \left (m^4+22 m^3+179 m^2+638 m+840\right ) \left (2 d^2-2 e (m+1) x d+e^2 \left (m^2+3 m+2\right ) x^2\right ) e^4+3 a c^2 \left (m^2+13 m+42\right ) \left (24 d^4-24 e (m+1) x d^3+12 e^2 \left (m^2+3 m+2\right ) x^2 d^2-4 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right ) e^2+c^3 \left (720 d^6-720 e (m+1) x d^5+360 e^2 \left (m^2+3 m+2\right ) x^2 d^4-120 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^3+30 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^2-6 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d+e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6\right )\right )-B \left (a^3 \left (m^6+33 m^5+445 m^4+3135 m^3+12154 m^2+24552 m+20160\right ) (d-e (m+1) x) e^6-3 a^2 c \left (m^4+26 m^3+251 m^2+1066 m+1680\right ) \left (-6 d^3+6 e (m+1) x d^2-3 e^2 \left (m^2+3 m+2\right ) x^2 d+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right ) e^4-3 a c^2 \left (m^2+15 m+56\right ) \left (-120 d^5+120 e (m+1) x d^4-60 e^2 \left (m^2+3 m+2\right ) x^2 d^3+20 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^2-5 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d+e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right ) e^2+c^3 \left (5040 d^7-5040 e (m+1) x d^6+2520 e^2 \left (m^2+3 m+2\right ) x^2 d^5-840 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3 d^4+210 e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4 d^3-42 e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5 d^2+7 e^6 \left (m^6+21 m^5+175 m^4+735 m^3+1624 m^2+1764 m+720\right ) x^6 d-e^7 \left (m^7+28 m^6+322 m^5+1960 m^4+6769 m^3+13132 m^2+13068 m+5040\right ) x^7\right )\right )\right )}{e^8 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6) (m+7) (m+8)} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*(A*e*(8 + m)*(a^3*e^6*(5040 + 8028*m + 5104*m^2 + 1665*m^3 +
295*m^4 + 27*m^5 + m^6) + 3*a^2*c*e^4*(840 + 638*m + 179*m^2 + 22*m^3 + m^4)*(2*
d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 3*a*c^2*e^2*(42 + 13*m + m^2)
*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11
*m + 6*m^2 + m^3)*x^3 + e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4) + c^3*(720*
d^6 - 720*d^5*e*(1 + m)*x + 360*d^4*e^2*(2 + 3*m + m^2)*x^2 - 120*d^3*e^3*(6 + 1
1*m + 6*m^2 + m^3)*x^3 + 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*
d*e^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m +
1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6)*x^6)) - B*(a^3*e^6*(20160 + 24552*m
 + 12154*m^2 + 3135*m^3 + 445*m^4 + 33*m^5 + m^6)*(d - e*(1 + m)*x) - 3*a^2*c*e^
4*(1680 + 1066*m + 251*m^2 + 26*m^3 + m^4)*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2
*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)*x^3) - 3*a*c^2*e^2*(56 + 15*
m + m^2)*(-120*d^5 + 120*d^4*e*(1 + m)*x - 60*d^3*e^2*(2 + 3*m + m^2)*x^2 + 20*d
^2*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 - 5*d*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4
)*x^4 + e^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5) + c^3*(5040*d^7
 - 5040*d^6*e*(1 + m)*x + 2520*d^5*e^2*(2 + 3*m + m^2)*x^2 - 840*d^4*e^3*(6 + 11
*m + 6*m^2 + m^3)*x^3 + 210*d^3*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 42
*d^2*e^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + 7*d*e^6*(720 + 17
64*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6)*x^6 - e^7*(5040 + 13068*m +
13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7)*x^7))))/(e^8*(1 + m)*(
2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m)*(8 + m))

_______________________________________________________________________________________

Maple [B]  time = 0.028, size = 3176, normalized size = 8.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^m*(c*x^2+a)^3,x)

[Out]

(e*x+d)^(1+m)*(B*c^3*e^7*m^7*x^7+A*c^3*e^7*m^7*x^6+28*B*c^3*e^7*m^6*x^7+29*A*c^3
*e^7*m^6*x^6+3*B*a*c^2*e^7*m^7*x^5-7*B*c^3*d*e^6*m^6*x^6+322*B*c^3*e^7*m^5*x^7+3
*A*a*c^2*e^7*m^7*x^4-6*A*c^3*d*e^6*m^6*x^5+343*A*c^3*e^7*m^5*x^6+90*B*a*c^2*e^7*
m^6*x^5-147*B*c^3*d*e^6*m^5*x^6+1960*B*c^3*e^7*m^4*x^7+93*A*a*c^2*e^7*m^6*x^4-13
8*A*c^3*d*e^6*m^5*x^5+2135*A*c^3*e^7*m^4*x^6+3*B*a^2*c*e^7*m^7*x^3-15*B*a*c^2*d*
e^6*m^6*x^4+1098*B*a*c^2*e^7*m^5*x^5+42*B*c^3*d^2*e^5*m^5*x^5-1225*B*c^3*d*e^6*m
^4*x^6+6769*B*c^3*e^7*m^3*x^7+3*A*a^2*c*e^7*m^7*x^2-12*A*a*c^2*d*e^6*m^6*x^3+117
3*A*a*c^2*e^7*m^5*x^4+30*A*c^3*d^2*e^5*m^5*x^4-1230*A*c^3*d*e^6*m^4*x^5+7504*A*c
^3*e^7*m^3*x^6+96*B*a^2*c*e^7*m^6*x^3-375*B*a*c^2*d*e^6*m^5*x^4+7020*B*a*c^2*e^7
*m^4*x^5+630*B*c^3*d^2*e^5*m^4*x^5-5145*B*c^3*d*e^6*m^3*x^6+13132*B*c^3*e^7*m^2*
x^7+99*A*a^2*c*e^7*m^6*x^2-324*A*a*c^2*d*e^6*m^5*x^3+7743*A*a*c^2*e^7*m^4*x^4+54
0*A*c^3*d^2*e^5*m^4*x^4-5430*A*c^3*d*e^6*m^3*x^5+14756*A*c^3*e^7*m^2*x^6+B*a^3*e
^7*m^7*x-9*B*a^2*c*d*e^6*m^6*x^2+1254*B*a^2*c*e^7*m^5*x^3+60*B*a*c^2*d^2*e^5*m^5
*x^3-3615*B*a*c^2*d*e^6*m^4*x^4+25227*B*a*c^2*e^7*m^3*x^5-210*B*c^3*d^3*e^4*m^4*
x^4+3570*B*c^3*d^2*e^5*m^3*x^5-11368*B*c^3*d*e^6*m^2*x^6+13068*B*c^3*e^7*m*x^7+A
*a^3*e^7*m^7-6*A*a^2*c*d*e^6*m^6*x+1341*A*a^2*c*e^7*m^5*x^2+36*A*a*c^2*d^2*e^5*m
^5*x^2-3396*A*a*c^2*d*e^6*m^4*x^3+28632*A*a*c^2*e^7*m^3*x^4-120*A*c^3*d^3*e^4*m^
4*x^3+3450*A*c^3*d^2*e^5*m^3*x^4-12444*A*c^3*d*e^6*m^2*x^5+14832*A*c^3*e^7*m*x^6
+34*B*a^3*e^7*m^6*x-261*B*a^2*c*d*e^6*m^5*x^2+8592*B*a^2*c*e^7*m^4*x^3+1260*B*a*
c^2*d^2*e^5*m^4*x^3-17025*B*a*c^2*d*e^6*m^3*x^4+50490*B*a*c^2*e^7*m^2*x^5-2100*B
*c^3*d^3*e^4*m^3*x^4+9450*B*c^3*d^2*e^5*m^2*x^5-12348*B*c^3*d*e^6*m*x^6+5040*B*c
^3*e^7*x^7+35*A*a^3*e^7*m^6-186*A*a^2*c*d*e^6*m^5*x+9585*A*a^2*c*e^7*m^4*x^2+864
*A*a*c^2*d^2*e^5*m^4*x^2-17388*A*a*c^2*d*e^6*m^3*x^3+58692*A*a*c^2*e^7*m^2*x^4-1
680*A*c^3*d^3*e^4*m^3*x^3+9900*A*c^3*d^2*e^5*m^2*x^4-13872*A*c^3*d*e^6*m*x^5+576
0*A*c^3*e^7*x^6-B*a^3*d*e^6*m^6+478*B*a^3*e^7*m^5*x+18*B*a^2*c*d^2*e^5*m^5*x-297
9*B*a^2*c*d*e^6*m^4*x^2+32979*B*a^2*c*e^7*m^3*x^3-180*B*a*c^2*d^3*e^4*m^4*x^2+94
20*B*a*c^2*d^2*e^5*m^3*x^3-41010*B*a*c^2*d*e^6*m^2*x^4+51432*B*a*c^2*e^7*m*x^5+8
40*B*c^3*d^4*e^3*m^3*x^3-7350*B*c^3*d^3*e^4*m^2*x^4+11508*B*c^3*d^2*e^5*m*x^5-50
40*B*c^3*d*e^6*x^6+511*A*a^3*e^7*m^5+6*A*a^2*c*d^2*e^5*m^5-2310*A*a^2*c*d*e^6*m^
4*x+38592*A*a^2*c*e^7*m^3*x^2-72*A*a*c^2*d^3*e^4*m^4*x+7596*A*a*c^2*d^2*e^5*m^3*
x^2-44976*A*a*c^2*d*e^6*m^2*x^3+60912*A*a*c^2*e^7*m*x^4+360*A*c^3*d^4*e^3*m^3*x^
2-7080*A*c^3*d^3*e^4*m^2*x^3+12720*A*c^3*d^2*e^5*m*x^4-5760*A*c^3*d*e^6*x^5-33*B
*a^3*d*e^6*m^5+3580*B*a^3*e^7*m^4*x+486*B*a^2*c*d^2*e^5*m^4*x-16839*B*a^2*c*d*e^
6*m^3*x^2+69936*B*a^2*c*e^7*m^2*x^3-3240*B*a*c^2*d^3*e^4*m^3*x^2+30420*B*a*c^2*d
^2*e^5*m^2*x^3-47400*B*a*c^2*d*e^6*m*x^4+20160*B*a*c^2*e^7*x^5+5040*B*c^3*d^4*e^
3*m^2*x^3-10500*B*c^3*d^3*e^4*m*x^4+5040*B*c^3*d^2*e^5*x^5+4025*A*a^3*e^7*m^4+18
0*A*a^2*c*d^2*e^5*m^4-14550*A*a^2*c*d*e^6*m^3*x+86076*A*a^2*c*e^7*m^2*x^2-1584*A
*a*c^2*d^3*e^4*m^3*x+29376*A*a*c^2*d^2*e^5*m^2*x^2-54864*A*a*c^2*d*e^6*m*x^3+241
92*A*a*c^2*e^7*x^4+3960*A*c^3*d^4*e^3*m^2*x^2-11280*A*c^3*d^3*e^4*m*x^3+5760*A*c
^3*d^2*e^5*x^4-445*B*a^3*d*e^6*m^4+15289*B*a^3*e^7*m^3*x-18*B*a^2*c*d^3*e^4*m^4+
4986*B*a^2*c*d^2*e^5*m^3*x-48420*B*a^2*c*d*e^6*m^2*x^2+74628*B*a^2*c*e^7*m*x^3+3
60*B*a*c^2*d^4*e^3*m^3*x-18540*B*a*c^2*d^3*e^4*m^2*x^2+42360*B*a*c^2*d^2*e^5*m*x
^3-20160*B*a*c^2*d*e^6*x^4-2520*B*c^3*d^5*e^2*m^2*x^2+9240*B*c^3*d^4*e^3*m*x^3-5
040*B*c^3*d^3*e^4*x^4+18424*A*a^3*e^7*m^3+2130*A*a^2*c*d^2*e^5*m^3-48084*A*a^2*c
*d*e^6*m^2*x+96144*A*a^2*c*e^7*m*x^2+72*A*a*c^2*d^4*e^3*m^3-12024*A*a*c^2*d^3*e^
4*m^2*x+46800*A*a*c^2*d^2*e^5*m*x^2-24192*A*a*c^2*d*e^6*x^3-720*A*c^3*d^5*e^2*m^
2*x+9360*A*c^3*d^4*e^3*m*x^2-5760*A*c^3*d^3*e^4*x^3-3135*B*a^3*d*e^6*m^3+36706*B
*a^3*e^7*m^2*x-468*B*a^2*c*d^3*e^4*m^3+23706*B*a^2*c*d^2*e^5*m^2*x-64548*B*a^2*c
*d*e^6*m*x^2+30240*B*a^2*c*e^7*x^3+5760*B*a*c^2*d^4*e^3*m^2*x-35640*B*a*c^2*d^3*
e^4*m*x^2+20160*B*a*c^2*d^2*e^5*x^3-7560*B*c^3*d^5*e^2*m*x^2+5040*B*c^3*d^4*e^3*
x^3+48860*A*a^3*e^7*m^2+12420*A*a^2*c*d^2*e^5*m^2-75984*A*a^2*c*d*e^6*m*x+40320*
A*a^2*c*e^7*x^2+1512*A*a*c^2*d^4*e^3*m^2-34704*A*a*c^2*d^3*e^4*m*x+24192*A*a*c^2
*d^2*e^5*x^2-6480*A*c^3*d^5*e^2*m*x+5760*A*c^3*d^4*e^3*x^2-12154*B*a^3*d*e^6*m^2
+44712*B*a^3*e^7*m*x-4518*B*a^2*c*d^3*e^4*m^2+49428*B*a^2*c*d^2*e^5*m*x-30240*B*
a^2*c*d*e^6*x^2-360*B*a*c^2*d^5*e^2*m^2+25560*B*a*c^2*d^4*e^3*m*x-20160*B*a*c^2*
d^3*e^4*x^2+5040*B*c^3*d^6*e*m*x-5040*B*c^3*d^5*e^2*x^2+69264*A*a^3*e^7*m+35664*
A*a^2*c*d^2*e^5*m-40320*A*a^2*c*d*e^6*x+10512*A*a*c^2*d^4*e^3*m-24192*A*a*c^2*d^
3*e^4*x+720*A*c^3*d^6*e*m-5760*A*c^3*d^5*e^2*x-24552*B*a^3*d*e^6*m+20160*B*a^3*e
^7*x-19188*B*a^2*c*d^3*e^4*m+30240*B*a^2*c*d^2*e^5*x-5400*B*a*c^2*d^5*e^2*m+2016
0*B*a*c^2*d^4*e^3*x+5040*B*c^3*d^6*e*x+40320*A*a^3*e^7+40320*A*a^2*c*d^2*e^5+241
92*A*a*c^2*d^4*e^3+5760*A*c^3*d^6*e-20160*B*a^3*d*e^6-30240*B*a^2*c*d^3*e^4-2016
0*B*a*c^2*d^5*e^2-5040*B*c^3*d^7)/e^8/(m^8+36*m^7+546*m^6+4536*m^5+22449*m^4+672
84*m^3+118124*m^2+109584*m+40320)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.333132, size = 4207, normalized size = 11.31 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^m,x, algorithm="fricas")

[Out]

(A*a^3*d*e^7*m^7 - 5040*B*c^3*d^8 + 5760*A*c^3*d^7*e - 20160*B*a*c^2*d^6*e^2 + 2
4192*A*a*c^2*d^5*e^3 - 30240*B*a^2*c*d^4*e^4 + 40320*A*a^2*c*d^3*e^5 - 20160*B*a
^3*d^2*e^6 + 40320*A*a^3*d*e^7 + (B*c^3*e^8*m^7 + 28*B*c^3*e^8*m^6 + 322*B*c^3*e
^8*m^5 + 1960*B*c^3*e^8*m^4 + 6769*B*c^3*e^8*m^3 + 13132*B*c^3*e^8*m^2 + 13068*B
*c^3*e^8*m + 5040*B*c^3*e^8)*x^8 + (5760*A*c^3*e^8 + (B*c^3*d*e^7 + A*c^3*e^8)*m
^7 + (21*B*c^3*d*e^7 + 29*A*c^3*e^8)*m^6 + 7*(25*B*c^3*d*e^7 + 49*A*c^3*e^8)*m^5
 + 35*(21*B*c^3*d*e^7 + 61*A*c^3*e^8)*m^4 + 56*(29*B*c^3*d*e^7 + 134*A*c^3*e^8)*
m^3 + 28*(63*B*c^3*d*e^7 + 527*A*c^3*e^8)*m^2 + 144*(5*B*c^3*d*e^7 + 103*A*c^3*e
^8)*m)*x^7 - (B*a^3*d^2*e^6 - 35*A*a^3*d*e^7)*m^6 + (20160*B*a*c^2*e^8 + (A*c^3*
d*e^7 + 3*B*a*c^2*e^8)*m^7 - (7*B*c^3*d^2*e^6 - 23*A*c^3*d*e^7 - 90*B*a*c^2*e^8)
*m^6 - (105*B*c^3*d^2*e^6 - 205*A*c^3*d*e^7 - 1098*B*a*c^2*e^8)*m^5 - 5*(119*B*c
^3*d^2*e^6 - 181*A*c^3*d*e^7 - 1404*B*a*c^2*e^8)*m^4 - (1575*B*c^3*d^2*e^6 - 207
4*A*c^3*d*e^7 - 25227*B*a*c^2*e^8)*m^3 - 2*(959*B*c^3*d^2*e^6 - 1156*A*c^3*d*e^7
 - 25245*B*a*c^2*e^8)*m^2 - 24*(35*B*c^3*d^2*e^6 - 40*A*c^3*d*e^7 - 2143*B*a*c^2
*e^8)*m)*x^6 + (6*A*a^2*c*d^3*e^5 - 33*B*a^3*d^2*e^6 + 511*A*a^3*d*e^7)*m^5 + 3*
(8064*A*a*c^2*e^8 + (B*a*c^2*d*e^7 + A*a*c^2*e^8)*m^7 - (2*A*c^3*d^2*e^6 - 25*B*
a*c^2*d*e^7 - 31*A*a*c^2*e^8)*m^6 + (14*B*c^3*d^3*e^5 - 36*A*c^3*d^2*e^6 + 241*B
*a*c^2*d*e^7 + 391*A*a*c^2*e^8)*m^5 + (140*B*c^3*d^3*e^5 - 230*A*c^3*d^2*e^6 + 1
135*B*a*c^2*d*e^7 + 2581*A*a*c^2*e^8)*m^4 + 2*(245*B*c^3*d^3*e^5 - 330*A*c^3*d^2
*e^6 + 1367*B*a*c^2*d*e^7 + 4772*A*a*c^2*e^8)*m^3 + 4*(175*B*c^3*d^3*e^5 - 212*A
*c^3*d^2*e^6 + 790*B*a*c^2*d*e^7 + 4891*A*a*c^2*e^8)*m^2 + 48*(7*B*c^3*d^3*e^5 -
 8*A*c^3*d^2*e^6 + 28*B*a*c^2*d*e^7 + 423*A*a*c^2*e^8)*m)*x^5 - (18*B*a^2*c*d^4*
e^4 - 180*A*a^2*c*d^3*e^5 + 445*B*a^3*d^2*e^6 - 4025*A*a^3*d*e^7)*m^4 + 3*(10080
*B*a^2*c*e^8 + (A*a*c^2*d*e^7 + B*a^2*c*e^8)*m^7 - (5*B*a*c^2*d^2*e^6 - 27*A*a*c
^2*d*e^7 - 32*B*a^2*c*e^8)*m^6 + (10*A*c^3*d^3*e^5 - 105*B*a*c^2*d^2*e^6 + 283*A
*a*c^2*d*e^7 + 418*B*a^2*c*e^8)*m^5 - (70*B*c^3*d^4*e^4 - 140*A*c^3*d^3*e^5 + 78
5*B*a*c^2*d^2*e^6 - 1449*A*a*c^2*d*e^7 - 2864*B*a^2*c*e^8)*m^4 - (420*B*c^3*d^4*
e^4 - 590*A*c^3*d^3*e^5 + 2535*B*a*c^2*d^2*e^6 - 3748*A*a*c^2*d*e^7 - 10993*B*a^
2*c*e^8)*m^3 - 2*(385*B*c^3*d^4*e^4 - 470*A*c^3*d^3*e^5 + 1765*B*a*c^2*d^2*e^6 -
 2286*A*a*c^2*d*e^7 - 11656*B*a^2*c*e^8)*m^2 - 12*(35*B*c^3*d^4*e^4 - 40*A*c^3*d
^3*e^5 + 140*B*a*c^2*d^2*e^6 - 168*A*a*c^2*d*e^7 - 2073*B*a^2*c*e^8)*m)*x^4 + (7
2*A*a*c^2*d^5*e^3 - 468*B*a^2*c*d^4*e^4 + 2130*A*a^2*c*d^3*e^5 - 3135*B*a^3*d^2*
e^6 + 18424*A*a^3*d*e^7)*m^3 + 3*(13440*A*a^2*c*e^8 + (B*a^2*c*d*e^7 + A*a^2*c*e
^8)*m^7 - (4*A*a*c^2*d^2*e^6 - 29*B*a^2*c*d*e^7 - 33*A*a^2*c*e^8)*m^6 + (20*B*a*
c^2*d^3*e^5 - 96*A*a*c^2*d^2*e^6 + 331*B*a^2*c*d*e^7 + 447*A*a^2*c*e^8)*m^5 - (4
0*A*c^3*d^4*e^4 - 360*B*a*c^2*d^3*e^5 + 844*A*a*c^2*d^2*e^6 - 1871*B*a^2*c*d*e^7
 - 3195*A*a^2*c*e^8)*m^4 + 4*(70*B*c^3*d^5*e^3 - 110*A*c^3*d^4*e^4 + 515*B*a*c^2
*d^3*e^5 - 816*A*a*c^2*d^2*e^6 + 1345*B*a^2*c*d*e^7 + 3216*A*a^2*c*e^8)*m^3 + 4*
(210*B*c^3*d^5*e^3 - 260*A*c^3*d^4*e^4 + 990*B*a*c^2*d^3*e^5 - 1300*A*a*c^2*d^2*
e^6 + 1793*B*a^2*c*d*e^7 + 7173*A*a^2*c*e^8)*m^2 + 16*(35*B*c^3*d^5*e^3 - 40*A*c
^3*d^4*e^4 + 140*B*a*c^2*d^3*e^5 - 168*A*a*c^2*d^2*e^6 + 210*B*a^2*c*d*e^7 + 200
3*A*a^2*c*e^8)*m)*x^3 - 2*(180*B*a*c^2*d^6*e^2 - 756*A*a*c^2*d^5*e^3 + 2259*B*a^
2*c*d^4*e^4 - 6210*A*a^2*c*d^3*e^5 + 6077*B*a^3*d^2*e^6 - 24430*A*a^3*d*e^7)*m^2
 + (20160*B*a^3*e^8 + (3*A*a^2*c*d*e^7 + B*a^3*e^8)*m^7 - (9*B*a^2*c*d^2*e^6 - 9
3*A*a^2*c*d*e^7 - 34*B*a^3*e^8)*m^6 + (36*A*a*c^2*d^3*e^5 - 243*B*a^2*c*d^2*e^6
+ 1155*A*a^2*c*d*e^7 + 478*B*a^3*e^8)*m^5 - (180*B*a*c^2*d^4*e^4 - 792*A*a*c^2*d
^3*e^5 + 2493*B*a^2*c*d^2*e^6 - 7275*A*a^2*c*d*e^7 - 3580*B*a^3*e^8)*m^4 + (360*
A*c^3*d^5*e^3 - 2880*B*a*c^2*d^4*e^4 + 6012*A*a*c^2*d^3*e^5 - 11853*B*a^2*c*d^2*
e^6 + 24042*A*a^2*c*d*e^7 + 15289*B*a^3*e^8)*m^3 - 2*(1260*B*c^3*d^6*e^2 - 1620*
A*c^3*d^5*e^3 + 6390*B*a*c^2*d^4*e^4 - 8676*A*a*c^2*d^3*e^5 + 12357*B*a^2*c*d^2*
e^6 - 18996*A*a^2*c*d*e^7 - 18353*B*a^3*e^8)*m^2 - 72*(35*B*c^3*d^6*e^2 - 40*A*c
^3*d^5*e^3 + 140*B*a*c^2*d^4*e^4 - 168*A*a*c^2*d^3*e^5 + 210*B*a^2*c*d^2*e^6 - 2
80*A*a^2*c*d*e^7 - 621*B*a^3*e^8)*m)*x^2 + 12*(60*A*c^3*d^7*e - 450*B*a*c^2*d^6*
e^2 + 876*A*a*c^2*d^5*e^3 - 1599*B*a^2*c*d^4*e^4 + 2972*A*a^2*c*d^3*e^5 - 2046*B
*a^3*d^2*e^6 + 5772*A*a^3*d*e^7)*m + (40320*A*a^3*e^8 + (B*a^3*d*e^7 + A*a^3*e^8
)*m^7 - (6*A*a^2*c*d^2*e^6 - 33*B*a^3*d*e^7 - 35*A*a^3*e^8)*m^6 + (18*B*a^2*c*d^
3*e^5 - 180*A*a^2*c*d^2*e^6 + 445*B*a^3*d*e^7 + 511*A*a^3*e^8)*m^5 - (72*A*a*c^2
*d^4*e^4 - 468*B*a^2*c*d^3*e^5 + 2130*A*a^2*c*d^2*e^6 - 3135*B*a^3*d*e^7 - 4025*
A*a^3*e^8)*m^4 + 2*(180*B*a*c^2*d^5*e^3 - 756*A*a*c^2*d^4*e^4 + 2259*B*a^2*c*d^3
*e^5 - 6210*A*a^2*c*d^2*e^6 + 6077*B*a^3*d*e^7 + 9212*A*a^3*e^8)*m^3 - 4*(180*A*
c^3*d^6*e^2 - 1350*B*a*c^2*d^5*e^3 + 2628*A*a*c^2*d^4*e^4 - 4797*B*a^2*c*d^3*e^5
 + 8916*A*a^2*c*d^2*e^6 - 6138*B*a^3*d*e^7 - 12215*A*a^3*e^8)*m^2 + 144*(35*B*c^
3*d^7*e - 40*A*c^3*d^6*e^2 + 140*B*a*c^2*d^5*e^3 - 168*A*a*c^2*d^4*e^4 + 210*B*a
^2*c*d^3*e^5 - 280*A*a^2*c*d^2*e^6 + 140*B*a^3*d*e^7 + 481*A*a^3*e^8)*m)*x)*(e*x
 + d)^m/(e^8*m^8 + 36*e^8*m^7 + 546*e^8*m^6 + 4536*e^8*m^5 + 22449*e^8*m^4 + 672
84*e^8*m^3 + 118124*e^8*m^2 + 109584*e^8*m + 40320*e^8)

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**m*(c*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.300709, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^3*(B*x + A)*(e*x + d)^m,x, algorithm="giac")

[Out]

Done

[next] [prev] [prev-tail] [front] [up]