3.3152 \(\int (a+b x)^4 (A+B x) (d+e x)^m \, dx\)

Optimal. Leaf size=234 \[ -\frac{b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}+\frac{2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac{(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac{2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac{b^4 B (d+e x)^{m+6}}{e^6 (m+6)} \]

[Out]

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

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Rubi [A]  time = 0.461223, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ -\frac{b^3 (d+e x)^{m+5} (-4 a B e-A b e+5 b B d)}{e^6 (m+5)}+\frac{2 b^2 (b d-a e) (d+e x)^{m+4} (-3 a B e-2 A b e+5 b B d)}{e^6 (m+4)}-\frac{(b d-a e)^4 (B d-A e) (d+e x)^{m+1}}{e^6 (m+1)}+\frac{(b d-a e)^3 (d+e x)^{m+2} (-a B e-4 A b e+5 b B d)}{e^6 (m+2)}-\frac{2 b (b d-a e)^2 (d+e x)^{m+3} (-2 a B e-3 A b e+5 b B d)}{e^6 (m+3)}+\frac{b^4 B (d+e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4*(A + B*x)*(d + e*x)^m,x]

[Out]

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

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Rubi in Sympy [A]  time = 86.8218, size = 224, normalized size = 0.96 \[ \frac{B b^{4} \left (d + e x\right )^{m + 6}}{e^{6} \left (m + 6\right )} + \frac{b^{3} \left (d + e x\right )^{m + 5} \left (A b e + 4 B a e - 5 B b d\right )}{e^{6} \left (m + 5\right )} + \frac{2 b^{2} \left (d + e x\right )^{m + 4} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{e^{6} \left (m + 4\right )} + \frac{2 b \left (d + e x\right )^{m + 3} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{e^{6} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{e^{6} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{e^{6} \left (m + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4*(B*x+A)*(e*x+d)**m,x)

[Out]

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

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Mathematica [B]  time = 1.3319, size = 635, normalized size = 2.71 \[ \frac{(d+e x)^{m+1} \left (a^4 e^4 \left (m^4+18 m^3+119 m^2+342 m+360\right ) (A e (m+2)-B d+B e (m+1) x)+4 a^3 b e^3 \left (m^3+15 m^2+74 m+120\right ) \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+6 a^2 b^2 e^2 \left (m^2+11 m+30\right ) \left (A e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )+B \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )+4 a b^3 e (m+6) \left (A e (m+5) \left (-6 d^3+6 d^2 e (m+1) x-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )+B \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )+b^4 \left (-\left (B \left (120 d^5-120 d^4 e (m+1) x+60 d^3 e^2 \left (m^2+3 m+2\right ) x^2-20 d^2 e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+5 d e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4-e^5 \left (m^5+15 m^4+85 m^3+225 m^2+274 m+120\right ) x^5\right )-A e (m+6) \left (24 d^4-24 d^3 e (m+1) x+12 d^2 e^2 \left (m^2+3 m+2\right ) x^2-4 d e^3 \left (m^3+6 m^2+11 m+6\right ) x^3+e^4 \left (m^4+10 m^3+35 m^2+50 m+24\right ) x^4\right )\right )\right )\right )}{e^6 (m+1) (m+2) (m+3) (m+4) (m+5) (m+6)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4*(A + B*x)*(d + e*x)^m,x]

[Out]

((d + e*x)^(1 + m)*(a^4*e^4*(360 + 342*m + 119*m^2 + 18*m^3 + m^4)*(-(B*d) + A*e
*(2 + m) + B*e*(1 + m)*x) + 4*a^3*b*e^3*(120 + 74*m + 15*m^2 + m^3)*(A*e*(3 + m)
*(-d + e*(1 + m)*x) + B*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2)) + 6
*a^2*b^2*e^2*(30 + 11*m + m^2)*(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)) + 4*a*b^3*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)) - b^4
*(-(A*e*(6 + m)*(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)) + B*(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^6*(1 + m)*
(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m))

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Maple [B]  time = 0.019, size = 2355, normalized size = 10.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4*(B*x+A)*(e*x+d)^m,x)

[Out]

(e*x+d)^(1+m)*(B*b^4*e^5*m^5*x^5+A*b^4*e^5*m^5*x^4+4*B*a*b^3*e^5*m^5*x^4+15*B*b^
4*e^5*m^4*x^5+4*A*a*b^3*e^5*m^5*x^3+16*A*b^4*e^5*m^4*x^4+6*B*a^2*b^2*e^5*m^5*x^3
+64*B*a*b^3*e^5*m^4*x^4-5*B*b^4*d*e^4*m^4*x^4+85*B*b^4*e^5*m^3*x^5+6*A*a^2*b^2*e
^5*m^5*x^2+68*A*a*b^3*e^5*m^4*x^3-4*A*b^4*d*e^4*m^4*x^3+95*A*b^4*e^5*m^3*x^4+4*B
*a^3*b*e^5*m^5*x^2+102*B*a^2*b^2*e^5*m^4*x^3-16*B*a*b^3*d*e^4*m^4*x^3+380*B*a*b^
3*e^5*m^3*x^4-50*B*b^4*d*e^4*m^3*x^4+225*B*b^4*e^5*m^2*x^5+4*A*a^3*b*e^5*m^5*x+1
08*A*a^2*b^2*e^5*m^4*x^2-12*A*a*b^3*d*e^4*m^4*x^2+428*A*a*b^3*e^5*m^3*x^3-48*A*b
^4*d*e^4*m^3*x^3+260*A*b^4*e^5*m^2*x^4+B*a^4*e^5*m^5*x+72*B*a^3*b*e^5*m^4*x^2-18
*B*a^2*b^2*d*e^4*m^4*x^2+642*B*a^2*b^2*e^5*m^3*x^3-192*B*a*b^3*d*e^4*m^3*x^3+104
0*B*a*b^3*e^5*m^2*x^4+20*B*b^4*d^2*e^3*m^3*x^3-175*B*b^4*d*e^4*m^2*x^4+274*B*b^4
*e^5*m*x^5+A*a^4*e^5*m^5+76*A*a^3*b*e^5*m^4*x-12*A*a^2*b^2*d*e^4*m^4*x+726*A*a^2
*b^2*e^5*m^3*x^2-168*A*a*b^3*d*e^4*m^3*x^2+1228*A*a*b^3*e^5*m^2*x^3+12*A*b^4*d^2
*e^3*m^3*x^2-188*A*b^4*d*e^4*m^2*x^3+324*A*b^4*e^5*m*x^4+19*B*a^4*e^5*m^4*x-8*B*
a^3*b*d*e^4*m^4*x+484*B*a^3*b*e^5*m^3*x^2-252*B*a^2*b^2*d*e^4*m^3*x^2+1842*B*a^2
*b^2*e^5*m^2*x^3+48*B*a*b^3*d^2*e^3*m^3*x^2-752*B*a*b^3*d*e^4*m^2*x^3+1296*B*a*b
^3*e^5*m*x^4+120*B*b^4*d^2*e^3*m^2*x^3-250*B*b^4*d*e^4*m*x^4+120*B*b^4*e^5*x^5+2
0*A*a^4*e^5*m^4-4*A*a^3*b*d*e^4*m^4+548*A*a^3*b*e^5*m^3*x-192*A*a^2*b^2*d*e^4*m^
3*x+2232*A*a^2*b^2*e^5*m^2*x^2+24*A*a*b^3*d^2*e^3*m^3*x-780*A*a*b^3*d*e^4*m^2*x^
2+1584*A*a*b^3*e^5*m*x^3+108*A*b^4*d^2*e^3*m^2*x^2-288*A*b^4*d*e^4*m*x^3+144*A*b
^4*e^5*x^4-B*a^4*d*e^4*m^4+137*B*a^4*e^5*m^3*x-128*B*a^3*b*d*e^4*m^3*x+1488*B*a^
3*b*e^5*m^2*x^2+36*B*a^2*b^2*d^2*e^3*m^3*x-1170*B*a^2*b^2*d*e^4*m^2*x^2+2376*B*a
^2*b^2*e^5*m*x^3+432*B*a*b^3*d^2*e^3*m^2*x^2-1152*B*a*b^3*d*e^4*m*x^3+576*B*a*b^
3*e^5*x^4-60*B*b^4*d^3*e^2*m^2*x^2+220*B*b^4*d^2*e^3*m*x^3-120*B*b^4*d*e^4*x^4+1
55*A*a^4*e^5*m^3-72*A*a^3*b*d*e^4*m^3+1844*A*a^3*b*e^5*m^2*x+12*A*a^2*b^2*d^2*e^
3*m^3-1068*A*a^2*b^2*d*e^4*m^2*x+3048*A*a^2*b^2*e^5*m*x^2+288*A*a*b^3*d^2*e^3*m^
2*x-1344*A*a*b^3*d*e^4*m*x^2+720*A*a*b^3*e^5*x^3-24*A*b^4*d^3*e^2*m^2*x+240*A*b^
4*d^2*e^3*m*x^2-144*A*b^4*d*e^4*x^3-18*B*a^4*d*e^4*m^3+461*B*a^4*e^5*m^2*x+8*B*a
^3*b*d^2*e^3*m^3-712*B*a^3*b*d*e^4*m^2*x+2032*B*a^3*b*e^5*m*x^2+432*B*a^2*b^2*d^
2*e^3*m^2*x-2016*B*a^2*b^2*d*e^4*m*x^2+1080*B*a^2*b^2*e^5*x^3-96*B*a*b^3*d^3*e^2
*m^2*x+960*B*a*b^3*d^2*e^3*m*x^2-576*B*a*b^3*d*e^4*x^3-180*B*b^4*d^3*e^2*m*x^2+1
20*B*b^4*d^2*e^3*x^3+580*A*a^4*e^5*m^2-476*A*a^3*b*d*e^4*m^2+2808*A*a^3*b*e^5*m*
x+180*A*a^2*b^2*d^2*e^3*m^2-2328*A*a^2*b^2*d*e^4*m*x+1440*A*a^2*b^2*e^5*x^2-24*A
*a*b^3*d^3*e^2*m^2+984*A*a*b^3*d^2*e^3*m*x-720*A*a*b^3*d*e^4*x^2-168*A*b^4*d^3*e
^2*m*x+144*A*b^4*d^2*e^3*x^2-119*B*a^4*d*e^4*m^2+702*B*a^4*e^5*m*x+120*B*a^3*b*d
^2*e^3*m^2-1552*B*a^3*b*d*e^4*m*x+960*B*a^3*b*e^5*x^2-36*B*a^2*b^2*d^3*e^2*m^2+1
476*B*a^2*b^2*d^2*e^3*m*x-1080*B*a^2*b^2*d*e^4*x^2-672*B*a*b^3*d^3*e^2*m*x+576*B
*a*b^3*d^2*e^3*x^2+120*B*b^4*d^4*e*m*x-120*B*b^4*d^3*e^2*x^2+1044*A*a^4*e^5*m-13
68*A*a^3*b*d*e^4*m+1440*A*a^3*b*e^5*x+888*A*a^2*b^2*d^2*e^3*m-1440*A*a^2*b^2*d*e
^4*x-264*A*a*b^3*d^3*e^2*m+720*A*a*b^3*d^2*e^3*x+24*A*b^4*d^4*e*m-144*A*b^4*d^3*
e^2*x-342*B*a^4*d*e^4*m+360*B*a^4*e^5*x+592*B*a^3*b*d^2*e^3*m-960*B*a^3*b*d*e^4*
x-396*B*a^2*b^2*d^3*e^2*m+1080*B*a^2*b^2*d^2*e^3*x+96*B*a*b^3*d^4*e*m-576*B*a*b^
3*d^3*e^2*x+120*B*b^4*d^4*e*x+720*A*a^4*e^5-1440*A*a^3*b*d*e^4+1440*A*a^2*b^2*d^
2*e^3-720*A*a*b^3*d^3*e^2+144*A*b^4*d^4*e-360*B*a^4*d*e^4+960*B*a^3*b*d^2*e^3-10
80*B*a^2*b^2*d^3*e^2+576*B*a*b^3*d^4*e-120*B*b^4*d^5)/e^6/(m^6+21*m^5+175*m^4+73
5*m^3+1624*m^2+1764*m+720)

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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((B*x + A)*(b*x + a)^4*(e*x + d)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.332078, size = 3070, normalized size = 13.12 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^4*(e*x + d)^m,x, algorithm="fricas")

[Out]

(A*a^4*d*e^5*m^5 - 120*B*b^4*d^6 + 720*A*a^4*d*e^5 + 144*(4*B*a*b^3 + A*b^4)*d^5
*e - 360*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 480*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e
^3 - 360*(B*a^4 + 4*A*a^3*b)*d^2*e^4 + (B*b^4*e^6*m^5 + 15*B*b^4*e^6*m^4 + 85*B*
b^4*e^6*m^3 + 225*B*b^4*e^6*m^2 + 274*B*b^4*e^6*m + 120*B*b^4*e^6)*x^6 + (144*(4
*B*a*b^3 + A*b^4)*e^6 + (B*b^4*d*e^5 + (4*B*a*b^3 + A*b^4)*e^6)*m^5 + 2*(5*B*b^4
*d*e^5 + 8*(4*B*a*b^3 + A*b^4)*e^6)*m^4 + 5*(7*B*b^4*d*e^5 + 19*(4*B*a*b^3 + A*b
^4)*e^6)*m^3 + 10*(5*B*b^4*d*e^5 + 26*(4*B*a*b^3 + A*b^4)*e^6)*m^2 + 12*(2*B*b^4
*d*e^5 + 27*(4*B*a*b^3 + A*b^4)*e^6)*m)*x^5 + (20*A*a^4*d*e^5 - (B*a^4 + 4*A*a^3
*b)*d^2*e^4)*m^4 + (360*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6 + ((4*B*a*b^3 + A*b^4)*d*e
^5 + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^5 - (5*B*b^4*d^2*e^4 - 12*(4*B*a*b^3 + A
*b^4)*d*e^5 - 34*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^4 - (30*B*b^4*d^2*e^4 - 47*(4*
B*a*b^3 + A*b^4)*d*e^5 - 214*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^3 - (55*B*b^4*d^2*
e^4 - 72*(4*B*a*b^3 + A*b^4)*d*e^5 - 614*(3*B*a^2*b^2 + 2*A*a*b^3)*e^6)*m^2 - 6*
(5*B*b^4*d^2*e^4 - 6*(4*B*a*b^3 + A*b^4)*d*e^5 - 132*(3*B*a^2*b^2 + 2*A*a*b^3)*e
^6)*m)*x^4 + (155*A*a^4*d*e^5 + 4*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3 - 18*(B*a^4
+ 4*A*a^3*b)*d^2*e^4)*m^3 + 2*(240*(2*B*a^3*b + 3*A*a^2*b^2)*e^6 + ((3*B*a^2*b^2
+ 2*A*a*b^3)*d*e^5 + (2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m^5 - 2*((4*B*a*b^3 + A*b^4
)*d^2*e^4 - 7*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 - 9*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)
*m^4 + (10*B*b^4*d^3*e^3 - 18*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 65*(3*B*a^2*b^2 + 2*
A*a*b^3)*d*e^5 + 121*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m^3 + 2*(15*B*b^4*d^3*e^3 -
20*(4*B*a*b^3 + A*b^4)*d^2*e^4 + 56*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 186*(2*B*a
^3*b + 3*A*a^2*b^2)*e^6)*m^2 + 4*(5*B*b^4*d^3*e^3 - 6*(4*B*a*b^3 + A*b^4)*d^2*e^
4 + 15*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^5 + 127*(2*B*a^3*b + 3*A*a^2*b^2)*e^6)*m)*x
^3 + (580*A*a^4*d*e^5 - 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 60*(2*B*a^3*b + 3
*A*a^2*b^2)*d^3*e^3 - 119*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*m^2 + (360*(B*a^4 + 4*A*a
^3*b)*e^6 + (2*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 + (B*a^4 + 4*A*a^3*b)*e^6)*m^5 -
(6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 - 32*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 19*(
B*a^4 + 4*A*a^3*b)*e^6)*m^4 + (12*(4*B*a*b^3 + A*b^4)*d^3*e^3 - 72*(3*B*a^2*b^2
+ 2*A*a*b^3)*d^2*e^4 + 178*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 + 137*(B*a^4 + 4*A*a^
3*b)*e^6)*m^3 - (60*B*b^4*d^4*e^2 - 84*(4*B*a*b^3 + A*b^4)*d^3*e^3 + 246*(3*B*a^
2*b^2 + 2*A*a*b^3)*d^2*e^4 - 388*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 461*(B*a^4 +
4*A*a^3*b)*e^6)*m^2 - 6*(10*B*b^4*d^4*e^2 - 12*(4*B*a*b^3 + A*b^4)*d^3*e^3 + 30*
(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^4 - 40*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^5 - 117*(B*
a^4 + 4*A*a^3*b)*e^6)*m)*x^2 + 2*(522*A*a^4*d*e^5 + 12*(4*B*a*b^3 + A*b^4)*d^5*e
- 66*(3*B*a^2*b^2 + 2*A*a*b^3)*d^4*e^2 + 148*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^3
- 171*(B*a^4 + 4*A*a^3*b)*d^2*e^4)*m + (720*A*a^4*e^6 + (A*a^4*e^6 + (B*a^4 + 4*
A*a^3*b)*d*e^5)*m^5 + 2*(10*A*a^4*e^6 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 9*
(B*a^4 + 4*A*a^3*b)*d*e^5)*m^4 + (155*A*a^4*e^6 + 12*(3*B*a^2*b^2 + 2*A*a*b^3)*d
^3*e^3 - 60*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 119*(B*a^4 + 4*A*a^3*b)*d*e^5)*m
^3 + 2*(290*A*a^4*e^6 - 12*(4*B*a*b^3 + A*b^4)*d^4*e^2 + 66*(3*B*a^2*b^2 + 2*A*a
*b^3)*d^3*e^3 - 148*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 + 171*(B*a^4 + 4*A*a^3*b)*
d*e^5)*m^2 + 12*(10*B*b^4*d^5*e + 87*A*a^4*e^6 - 12*(4*B*a*b^3 + A*b^4)*d^4*e^2
+ 30*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^3 - 40*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^4 +
30*(B*a^4 + 4*A*a^3*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 + 720*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)**4*(B*x+A)*(e*x+d)**m,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.259358, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^4*(e*x + d)^m,x, algorithm="giac")

[Out]

Done