3.2640 \(\int (A+B x) (d+e x)^m \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=333 \[ -\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{(d+e x)^{m+3} \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}-\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

-(((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) - ((c
*d^2 - b*d*e + a*e^2)*(2*A*e*(2*c*d - b*e) - B*(5*c*d^2 - e*(3*b*d - a*e)))*(d +
e*x)^(2 + m))/(e^6*(2 + m)) - ((B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e
*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))*(d + e*x)^(3
+ m))/(e^6*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*
(4*b*d - a*e)))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e
)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

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Rubi [A]  time = 0.753346, antiderivative size = 330, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{(d+e x)^{m+4} \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{e^6 (m+4)}-\frac{(d+e x)^{m+3} \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{e^6 (m+3)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )^2}{e^6 (m+1)}+\frac{(d+e x)^{m+2} \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6 (m+2)}-\frac{c (d+e x)^{m+5} (-A c e-2 b B e+5 B c d)}{e^6 (m+5)}+\frac{B c^2 (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

-(((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(1 + m))/(e^6*(1 + m))) + ((c
*d^2 - b*d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e))*(d +
e*x)^(2 + m))/(e^6*(2 + m)) - ((B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e
*(2*b*d - a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))*(d + e*x)^(3
+ m))/(e^6*(3 + m)) - ((2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 + b^2*e^2 - 2*c*e*
(4*b*d - a*e)))*(d + e*x)^(4 + m))/(e^6*(4 + m)) - (c*(5*B*c*d - 2*b*B*e - A*c*e
)*(d + e*x)^(5 + m))/(e^6*(5 + m)) + (B*c^2*(d + e*x)^(6 + m))/(e^6*(6 + m))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [B]  time = 2.33842, size = 722, normalized size = 2.17 \[ \frac{(d+e x)^{m+1} \left (A e (m+6) \left (e^2 \left (m^2+9 m+20\right ) \left (a^2 e^2 \left (m^2+5 m+6\right )+2 a b e (m+3) (e (m+1) x-d)+b^2 \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+2 c e (m+5) \left (a e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )+c^2 \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )+B \left (e^2 \left (m^2+11 m+30\right ) \left (a^2 e^2 \left (m^2+7 m+12\right ) (e (m+1) x-d)+2 a b e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+b^2 \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )+2 c e (m+6) \left (a e (m+5) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+b \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )+c^2 \left (-\left (120 d^5-120 d^4 e (m+1) x+60 d^3 e^2 \left (m^2+3 m+2\right ) x^2-20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )\right )\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]

Antiderivative was successfully verified.

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

[Out]

((d + e*x)^(1 + m)*(A*e*(6 + m)*(c^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) + e^2*(20 + 9*m + m^2)*(a^2*e^2*(6 + 5*m + m^2) + 2*a*b
*e*(3 + m)*(-d + e*(1 + m)*x) + b^2*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^
2)*x^2)) + 2*c*e*(5 + m)*(a*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m +
m^2)*x^2) + b*(-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))) + B*(-(c^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)) + e^2*(30 + 11*m + m^2)*(a^2*e^2*(12 + 7*m + m^2)*(-d + e*(1 +
m)*x) + 2*a*b*e*(4 + m)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + b^
2*(-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)) + 2*c*e*(6 + m)*(a*e*(5 + m)*(-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) + b*(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)))))/(e^6*(1 + m)*(2 + m)*(3
+ m)*(4 + m)*(5 + m)*(6 + m))

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Maple [B]  time = 0.02, size = 2557, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e*x+d)^(1+m)*(B*c^2*e^5*m^5*x^5+A*c^2*e^5*m^5*x^4+2*B*b*c*e^5*m^5*x^4+15*B*c^2*
e^5*m^4*x^5+2*A*b*c*e^5*m^5*x^3+16*A*c^2*e^5*m^4*x^4+2*B*a*c*e^5*m^5*x^3+B*b^2*e
^5*m^5*x^3+32*B*b*c*e^5*m^4*x^4-5*B*c^2*d*e^4*m^4*x^4+85*B*c^2*e^5*m^3*x^5+2*A*a
*c*e^5*m^5*x^2+A*b^2*e^5*m^5*x^2+34*A*b*c*e^5*m^4*x^3-4*A*c^2*d*e^4*m^4*x^3+95*A
*c^2*e^5*m^3*x^4+2*B*a*b*e^5*m^5*x^2+34*B*a*c*e^5*m^4*x^3+17*B*b^2*e^5*m^4*x^3-8
*B*b*c*d*e^4*m^4*x^3+190*B*b*c*e^5*m^3*x^4-50*B*c^2*d*e^4*m^3*x^4+225*B*c^2*e^5*
m^2*x^5+2*A*a*b*e^5*m^5*x+36*A*a*c*e^5*m^4*x^2+18*A*b^2*e^5*m^4*x^2-6*A*b*c*d*e^
4*m^4*x^2+214*A*b*c*e^5*m^3*x^3-48*A*c^2*d*e^4*m^3*x^3+260*A*c^2*e^5*m^2*x^4+B*a
^2*e^5*m^5*x+36*B*a*b*e^5*m^4*x^2-6*B*a*c*d*e^4*m^4*x^2+214*B*a*c*e^5*m^3*x^3-3*
B*b^2*d*e^4*m^4*x^2+107*B*b^2*e^5*m^3*x^3-96*B*b*c*d*e^4*m^3*x^3+520*B*b*c*e^5*m
^2*x^4+20*B*c^2*d^2*e^3*m^3*x^3-175*B*c^2*d*e^4*m^2*x^4+274*B*c^2*e^5*m*x^5+A*a^
2*e^5*m^5+38*A*a*b*e^5*m^4*x-4*A*a*c*d*e^4*m^4*x+242*A*a*c*e^5*m^3*x^2-2*A*b^2*d
*e^4*m^4*x+121*A*b^2*e^5*m^3*x^2-84*A*b*c*d*e^4*m^3*x^2+614*A*b*c*e^5*m^2*x^3+12
*A*c^2*d^2*e^3*m^3*x^2-188*A*c^2*d*e^4*m^2*x^3+324*A*c^2*e^5*m*x^4+19*B*a^2*e^5*
m^4*x-4*B*a*b*d*e^4*m^4*x+242*B*a*b*e^5*m^3*x^2-84*B*a*c*d*e^4*m^3*x^2+614*B*a*c
*e^5*m^2*x^3-42*B*b^2*d*e^4*m^3*x^2+307*B*b^2*e^5*m^2*x^3+24*B*b*c*d^2*e^3*m^3*x
^2-376*B*b*c*d*e^4*m^2*x^3+648*B*b*c*e^5*m*x^4+120*B*c^2*d^2*e^3*m^2*x^3-250*B*c
^2*d*e^4*m*x^4+120*B*c^2*e^5*x^5+20*A*a^2*e^5*m^4-2*A*a*b*d*e^4*m^4+274*A*a*b*e^
5*m^3*x-64*A*a*c*d*e^4*m^3*x+744*A*a*c*e^5*m^2*x^2-32*A*b^2*d*e^4*m^3*x+372*A*b^
2*e^5*m^2*x^2+12*A*b*c*d^2*e^3*m^3*x-390*A*b*c*d*e^4*m^2*x^2+792*A*b*c*e^5*m*x^3
+108*A*c^2*d^2*e^3*m^2*x^2-288*A*c^2*d*e^4*m*x^3+144*A*c^2*e^5*x^4-B*a^2*d*e^4*m
^4+137*B*a^2*e^5*m^3*x-64*B*a*b*d*e^4*m^3*x+744*B*a*b*e^5*m^2*x^2+12*B*a*c*d^2*e
^3*m^3*x-390*B*a*c*d*e^4*m^2*x^2+792*B*a*c*e^5*m*x^3+6*B*b^2*d^2*e^3*m^3*x-195*B
*b^2*d*e^4*m^2*x^2+396*B*b^2*e^5*m*x^3+216*B*b*c*d^2*e^3*m^2*x^2-576*B*b*c*d*e^4
*m*x^3+288*B*b*c*e^5*x^4-60*B*c^2*d^3*e^2*m^2*x^2+220*B*c^2*d^2*e^3*m*x^3-120*B*
c^2*d*e^4*x^4+155*A*a^2*e^5*m^3-36*A*a*b*d*e^4*m^3+922*A*a*b*e^5*m^2*x+4*A*a*c*d
^2*e^3*m^3-356*A*a*c*d*e^4*m^2*x+1016*A*a*c*e^5*m*x^2+2*A*b^2*d^2*e^3*m^3-178*A*
b^2*d*e^4*m^2*x+508*A*b^2*e^5*m*x^2+144*A*b*c*d^2*e^3*m^2*x-672*A*b*c*d*e^4*m*x^
2+360*A*b*c*e^5*x^3-24*A*c^2*d^3*e^2*m^2*x+240*A*c^2*d^2*e^3*m*x^2-144*A*c^2*d*e
^4*x^3-18*B*a^2*d*e^4*m^3+461*B*a^2*e^5*m^2*x+4*B*a*b*d^2*e^3*m^3-356*B*a*b*d*e^
4*m^2*x+1016*B*a*b*e^5*m*x^2+144*B*a*c*d^2*e^3*m^2*x-672*B*a*c*d*e^4*m*x^2+360*B
*a*c*e^5*x^3+72*B*b^2*d^2*e^3*m^2*x-336*B*b^2*d*e^4*m*x^2+180*B*b^2*e^5*x^3-48*B
*b*c*d^3*e^2*m^2*x+480*B*b*c*d^2*e^3*m*x^2-288*B*b*c*d*e^4*x^3-180*B*c^2*d^3*e^2
*m*x^2+120*B*c^2*d^2*e^3*x^3+580*A*a^2*e^5*m^2-238*A*a*b*d*e^4*m^2+1404*A*a*b*e^
5*m*x+60*A*a*c*d^2*e^3*m^2-776*A*a*c*d*e^4*m*x+480*A*a*c*e^5*x^2+30*A*b^2*d^2*e^
3*m^2-388*A*b^2*d*e^4*m*x+240*A*b^2*e^5*x^2-12*A*b*c*d^3*e^2*m^2+492*A*b*c*d^2*e
^3*m*x-360*A*b*c*d*e^4*x^2-168*A*c^2*d^3*e^2*m*x+144*A*c^2*d^2*e^3*x^2-119*B*a^2
*d*e^4*m^2+702*B*a^2*e^5*m*x+60*B*a*b*d^2*e^3*m^2-776*B*a*b*d*e^4*m*x+480*B*a*b*
e^5*x^2-12*B*a*c*d^3*e^2*m^2+492*B*a*c*d^2*e^3*m*x-360*B*a*c*d*e^4*x^2-6*B*b^2*d
^3*e^2*m^2+246*B*b^2*d^2*e^3*m*x-180*B*b^2*d*e^4*x^2-336*B*b*c*d^3*e^2*m*x+288*B
*b*c*d^2*e^3*x^2+120*B*c^2*d^4*e*m*x-120*B*c^2*d^3*e^2*x^2+1044*A*a^2*e^5*m-684*
A*a*b*d*e^4*m+720*A*a*b*e^5*x+296*A*a*c*d^2*e^3*m-480*A*a*c*d*e^4*x+148*A*b^2*d^
2*e^3*m-240*A*b^2*d*e^4*x-132*A*b*c*d^3*e^2*m+360*A*b*c*d^2*e^3*x+24*A*c^2*d^4*e
*m-144*A*c^2*d^3*e^2*x-342*B*a^2*d*e^4*m+360*B*a^2*e^5*x+296*B*a*b*d^2*e^3*m-480
*B*a*b*d*e^4*x-132*B*a*c*d^3*e^2*m+360*B*a*c*d^2*e^3*x-66*B*b^2*d^3*e^2*m+180*B*
b^2*d^2*e^3*x+48*B*b*c*d^4*e*m-288*B*b*c*d^3*e^2*x+120*B*c^2*d^4*e*x+720*A*a^2*e
^5-720*A*a*b*d*e^4+480*A*a*c*d^2*e^3+240*A*b^2*d^2*e^3-360*A*b*c*d^3*e^2+144*A*c
^2*d^4*e-360*B*a^2*d*e^4+480*B*a*b*d^2*e^3-360*B*a*c*d^3*e^2-180*B*b^2*d^3*e^2+2
88*B*b*c*d^4*e-120*B*c^2*d^5)/e^6/(m^6+21*m^5+175*m^4+735*m^3+1624*m^2+1764*m+72
0)

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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 + b*x + a)^2*(B*x + A)*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312793, size = 2911, normalized size = 8.74 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*a^2*d*e^5*m^5 - 120*B*c^2*d^6 + 720*A*a^2*d*e^5 + 144*(2*B*b*c + A*c^2)*d^5*e
- 180*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e^2 + 240*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e
^3 - 360*(B*a^2 + 2*A*a*b)*d^2*e^4 + (B*c^2*e^6*m^5 + 15*B*c^2*e^6*m^4 + 85*B*c^
2*e^6*m^3 + 225*B*c^2*e^6*m^2 + 274*B*c^2*e^6*m + 120*B*c^2*e^6)*x^6 + (144*(2*B
*b*c + A*c^2)*e^6 + (B*c^2*d*e^5 + (2*B*b*c + A*c^2)*e^6)*m^5 + 2*(5*B*c^2*d*e^5
+ 8*(2*B*b*c + A*c^2)*e^6)*m^4 + 5*(7*B*c^2*d*e^5 + 19*(2*B*b*c + A*c^2)*e^6)*m
^3 + 10*(5*B*c^2*d*e^5 + 26*(2*B*b*c + A*c^2)*e^6)*m^2 + 12*(2*B*c^2*d*e^5 + 27*
(2*B*b*c + A*c^2)*e^6)*m)*x^5 + (20*A*a^2*d*e^5 - (B*a^2 + 2*A*a*b)*d^2*e^4)*m^4
+ (180*(B*b^2 + 2*(B*a + A*b)*c)*e^6 + ((2*B*b*c + A*c^2)*d*e^5 + (B*b^2 + 2*(B
*a + A*b)*c)*e^6)*m^5 - (5*B*c^2*d^2*e^4 - 12*(2*B*b*c + A*c^2)*d*e^5 - 17*(B*b^
2 + 2*(B*a + A*b)*c)*e^6)*m^4 - (30*B*c^2*d^2*e^4 - 47*(2*B*b*c + A*c^2)*d*e^5 -
107*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m^3 - (55*B*c^2*d^2*e^4 - 72*(2*B*b*c + A*c^
2)*d*e^5 - 307*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m^2 - 6*(5*B*c^2*d^2*e^4 - 6*(2*B*
b*c + A*c^2)*d*e^5 - 66*(B*b^2 + 2*(B*a + A*b)*c)*e^6)*m)*x^4 + (155*A*a^2*d*e^5
+ 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 18*(B*a^2 + 2*A*a*b)*d^2*e^4)*m^3 + (
240*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6 + ((B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + (2*B*a*
b + A*b^2 + 2*A*a*c)*e^6)*m^5 - 2*(2*(2*B*b*c + A*c^2)*d^2*e^4 - 7*(B*b^2 + 2*(B
*a + A*b)*c)*d*e^5 - 9*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m^4 + (20*B*c^2*d^3*e^3
- 36*(2*B*b*c + A*c^2)*d^2*e^4 + 65*(B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + 121*(2*B*a
*b + A*b^2 + 2*A*a*c)*e^6)*m^3 + 4*(15*B*c^2*d^3*e^3 - 20*(2*B*b*c + A*c^2)*d^2*
e^4 + 28*(B*b^2 + 2*(B*a + A*b)*c)*d*e^5 + 93*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m
^2 + 4*(10*B*c^2*d^3*e^3 - 12*(2*B*b*c + A*c^2)*d^2*e^4 + 15*(B*b^2 + 2*(B*a + A
*b)*c)*d*e^5 + 127*(2*B*a*b + A*b^2 + 2*A*a*c)*e^6)*m)*x^3 + (580*A*a^2*d*e^5 -
6*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e^2 + 30*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 1
19*(B*a^2 + 2*A*a*b)*d^2*e^4)*m^2 + (360*(B*a^2 + 2*A*a*b)*e^6 + ((2*B*a*b + A*b
^2 + 2*A*a*c)*d*e^5 + (B*a^2 + 2*A*a*b)*e^6)*m^5 - (3*(B*b^2 + 2*(B*a + A*b)*c)*
d^2*e^4 - 16*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 - 19*(B*a^2 + 2*A*a*b)*e^6)*m^4 +
(12*(2*B*b*c + A*c^2)*d^3*e^3 - 36*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^4 + 89*(2*B*
a*b + A*b^2 + 2*A*a*c)*d*e^5 + 137*(B*a^2 + 2*A*a*b)*e^6)*m^3 - (60*B*c^2*d^4*e^
2 - 84*(2*B*b*c + A*c^2)*d^3*e^3 + 123*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^4 - 194*(
2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 - 461*(B*a^2 + 2*A*a*b)*e^6)*m^2 - 6*(10*B*c^2*
d^4*e^2 - 12*(2*B*b*c + A*c^2)*d^3*e^3 + 15*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^4 -
20*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^5 - 117*(B*a^2 + 2*A*a*b)*e^6)*m)*x^2 + 2*(52
2*A*a^2*d*e^5 + 12*(2*B*b*c + A*c^2)*d^5*e - 33*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e^
2 + 74*(2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^3 - 171*(B*a^2 + 2*A*a*b)*d^2*e^4)*m +
(720*A*a^2*e^6 + (A*a^2*e^6 + (B*a^2 + 2*A*a*b)*d*e^5)*m^5 + 2*(10*A*a^2*e^6 - (
2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^4 + 9*(B*a^2 + 2*A*a*b)*d*e^5)*m^4 + (155*A*a^2
*e^6 + 6*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 30*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*
e^4 + 119*(B*a^2 + 2*A*a*b)*d*e^5)*m^3 + 2*(290*A*a^2*e^6 - 12*(2*B*b*c + A*c^2)
*d^4*e^2 + 33*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 74*(2*B*a*b + A*b^2 + 2*A*a*c)
*d^2*e^4 + 171*(B*a^2 + 2*A*a*b)*d*e^5)*m^2 + 12*(10*B*c^2*d^5*e + 87*A*a^2*e^6
- 12*(2*B*b*c + A*c^2)*d^4*e^2 + 15*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^3 - 20*(2*B*
a*b + A*b^2 + 2*A*a*c)*d^2*e^4 + 30*(B*a^2 + 2*A*a*b)*d*e^5)*m)*x)*(e*x + d)^m/(
e^6*m^6 + 21*e^6*m^5 + 175*e^6*m^4 + 735*e^6*m^3 + 1624*e^6*m^2 + 1764*e^6*m + 7
20*e^6)

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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+b*x+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.317099, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done