3.2641 \(\int (A+B x) (d+e x)^m \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=153 \[ -\frac{(d+e x)^{m+2} \left (A e (2 c d-b e)-B \left (3 c d^2-e (2 b d-a e)\right )\right )}{e^4 (m+2)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.252049, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{(d+e x)^{m+2} \left (-B e (2 b d-a e)-A e (2 c d-b e)+3 B c d^2\right )}{e^4 (m+2)}-\frac{(B d-A e) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+3} (-A c e-b B e+3 B c d)}{e^4 (m+3)}+\frac{B c (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 53.7157, size = 141, normalized size = 0.92 \[ \frac{B c \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{e^{4} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (A b e^{2} - 2 A c d e + B a e^{2} - 2 B b d e + 3 B c d^{2}\right )}{e^{4} \left (m + 2\right )} + \frac{\left (d + e x\right )^{m + 3} \left (A c e + B b e - 3 B c d\right )}{e^{4} \left (m + 3\right )} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.394351, size = 207, normalized size = 1.35 \[ \frac{(d+e x)^{m+1} \left (A e (m+4) \left (e (m+3) (a e (m+2)-b d+b e (m+1) x)+c \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+B e (m+4) \left (a e (m+3) (e (m+1) x-d)+b \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )-B c \left (6 d^3-6 d^2 e (m+1) x+3 d e^2 \left (m^2+3 m+2\right ) x^2-e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.01, size = 498, normalized size = 3.3 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( Bc{e}^{3}{m}^{3}{x}^{3}+Ac{e}^{3}{m}^{3}{x}^{2}+Bb{e}^{3}{m}^{3}{x}^{2}+6\,Bc{e}^{3}{m}^{2}{x}^{3}+Ab{e}^{3}{m}^{3}x+7\,Ac{e}^{3}{m}^{2}{x}^{2}+Ba{e}^{3}{m}^{3}x+7\,Bb{e}^{3}{m}^{2}{x}^{2}-3\,Bcd{e}^{2}{m}^{2}{x}^{2}+11\,Bc{e}^{3}m{x}^{3}+Aa{e}^{3}{m}^{3}+8\,Ab{e}^{3}{m}^{2}x-2\,Acd{e}^{2}{m}^{2}x+14\,Ac{e}^{3}m{x}^{2}+8\,Ba{e}^{3}{m}^{2}x-2\,Bbd{e}^{2}{m}^{2}x+14\,Bb{e}^{3}m{x}^{2}-9\,Bcd{e}^{2}m{x}^{2}+6\,Bc{x}^{3}{e}^{3}+9\,Aa{e}^{3}{m}^{2}-Abd{e}^{2}{m}^{2}+19\,Ab{e}^{3}mx-10\,Acd{e}^{2}mx+8\,Ac{e}^{3}{x}^{2}-Bad{e}^{2}{m}^{2}+19\,Ba{e}^{3}mx-10\,Bbd{e}^{2}mx+8\,Bb{e}^{3}{x}^{2}+6\,Bc{d}^{2}emx-6\,Bcd{e}^{2}{x}^{2}+26\,Aa{e}^{3}m-7\,Abd{e}^{2}m+12\,Ab{e}^{3}x+2\,Ac{d}^{2}em-8\,Acd{e}^{2}x-7\,Bad{e}^{2}m+12\,Ba{e}^{3}x+2\,Bb{d}^{2}em-8\,Bbd{e}^{2}x+6\,Bc{d}^{2}ex+24\,aA{e}^{3}-12\,Abd{e}^{2}+8\,Ac{d}^{2}e-12\,aBd{e}^{2}+8\,Bb{d}^{2}e-6\,Bc{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \]

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),x)

[Out]

(e*x+d)^(1+m)*(B*c*e^3*m^3*x^3+A*c*e^3*m^3*x^2+B*b*e^3*m^3*x^2+6*B*c*e^3*m^2*x^3
+A*b*e^3*m^3*x+7*A*c*e^3*m^2*x^2+B*a*e^3*m^3*x+7*B*b*e^3*m^2*x^2-3*B*c*d*e^2*m^2
*x^2+11*B*c*e^3*m*x^3+A*a*e^3*m^3+8*A*b*e^3*m^2*x-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-2*B*b*d*e^2*m^2*x+14*B*b*e^3*m*x^2-9*B*c*d*e^2*m*x^2+6*B*c*
e^3*x^3+9*A*a*e^3*m^2-A*b*d*e^2*m^2+19*A*b*e^3*m*x-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-10*B*b*d*e^2*m*x+8*B*b*e^3*x^2+6*B*c*d^2*e*m*x-6*
B*c*d*e^2*x^2+26*A*a*e^3*m-7*A*b*d*e^2*m+12*A*b*e^3*x+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+2*B*b*d^2*e*m-8*B*b*d*e^2*x+6*B*c*d^2*e*x+24*A*a*e^
3-12*A*b*d*e^2+8*A*c*d^2*e-12*B*a*d*e^2+8*B*b*d^2*e-6*B*c*d^3)/e^4/(m^4+10*m^3+3
5*m^2+50*m+24)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.295608, size = 726, normalized size = 4.75 \[ \frac{{\left (A a d e^{3} m^{3} - 6 \, B c d^{4} + 24 \, A a d e^{3} + 8 \,{\left (B b + A c\right )} d^{3} e - 12 \,{\left (B a + A b\right )} d^{2} e^{2} +{\left (B c e^{4} m^{3} + 6 \, B c e^{4} m^{2} + 11 \, B c e^{4} m + 6 \, B c e^{4}\right )} x^{4} +{\left (8 \,{\left (B b + A c\right )} e^{4} +{\left (B c d e^{3} +{\left (B b + A c\right )} e^{4}\right )} m^{3} +{\left (3 \, B c d e^{3} + 7 \,{\left (B b + A c\right )} e^{4}\right )} m^{2} + 2 \,{\left (B c d e^{3} + 7 \,{\left (B b + A c\right )} e^{4}\right )} m\right )} x^{3} +{\left (9 \, A a d e^{3} -{\left (B a + A b\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (B a + A b\right )} e^{4} +{\left ({\left (B b + A c\right )} d e^{3} +{\left (B a + A b\right )} e^{4}\right )} m^{3} -{\left (3 \, B c d^{2} e^{2} - 5 \,{\left (B b + A c\right )} d e^{3} - 8 \,{\left (B a + A b\right )} e^{4}\right )} m^{2} -{\left (3 \, B c d^{2} e^{2} - 4 \,{\left (B b + A c\right )} d e^{3} - 19 \,{\left (B a + A b\right )} e^{4}\right )} m\right )} x^{2} +{\left (26 \, A a d e^{3} + 2 \,{\left (B b + A c\right )} d^{3} e - 7 \,{\left (B a + A b\right )} d^{2} e^{2}\right )} m +{\left (24 \, A a e^{4} +{\left (A a e^{4} +{\left (B a + A b\right )} d e^{3}\right )} m^{3} +{\left (9 \, A a e^{4} - 2 \,{\left (B b + A c\right )} d^{2} e^{2} + 7 \,{\left (B a + A b\right )} d e^{3}\right )} m^{2} + 2 \,{\left (3 \, B c d^{3} e + 13 \, A a e^{4} - 4 \,{\left (B b + A c\right )} d^{2} e^{2} + 6 \,{\left (B a + A b\right )} d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(A*a*d*e^3*m^3 - 6*B*c*d^4 + 24*A*a*d*e^3 + 8*(B*b + A*c)*d^3*e - 12*(B*a + A*b)
*d^2*e^2 + (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*(B*
b + A*c)*e^4 + (B*c*d*e^3 + (B*b + A*c)*e^4)*m^3 + (3*B*c*d*e^3 + 7*(B*b + A*c)*
e^4)*m^2 + 2*(B*c*d*e^3 + 7*(B*b + A*c)*e^4)*m)*x^3 + (9*A*a*d*e^3 - (B*a + A*b)
*d^2*e^2)*m^2 + (12*(B*a + A*b)*e^4 + ((B*b + A*c)*d*e^3 + (B*a + A*b)*e^4)*m^3
- (3*B*c*d^2*e^2 - 5*(B*b + A*c)*d*e^3 - 8*(B*a + A*b)*e^4)*m^2 - (3*B*c*d^2*e^2
- 4*(B*b + A*c)*d*e^3 - 19*(B*a + A*b)*e^4)*m)*x^2 + (26*A*a*d*e^3 + 2*(B*b + A
*c)*d^3*e - 7*(B*a + A*b)*d^2*e^2)*m + (24*A*a*e^4 + (A*a*e^4 + (B*a + A*b)*d*e^
3)*m^3 + (9*A*a*e^4 - 2*(B*b + A*c)*d^2*e^2 + 7*(B*a + A*b)*d*e^3)*m^2 + 2*(3*B*
c*d^3*e + 13*A*a*e^4 - 4*(B*b + A*c)*d^2*e^2 + 6*(B*a + A*b)*d*e^3)*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)

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Sympy [A]  time = 20.9965, size = 5846, normalized size = 38.21 \[ \text{result too large to display} \]

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),x)

[Out]

Piecewise((d**m*(A*a*x + A*b*x**2/2 + A*c*x**3/3 + B*a*x**2/2 + B*b*x**3/3 + B*c
*x**4/4), Eq(e, 0)), (-2*A*a*d**2*e**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e
**6*x**2 + 6*d**2*e**7*x**3) + 3*A*b*d*e**4*x**2/(6*d**5*e**4 + 18*d**4*e**5*x +
18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + A*b*e**5*x**3/(6*d**5*e**4 + 18*d**4*e*
*5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 2*A*c*d*e**4*x**3/(6*d**5*e**4 +
18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 3*B*a*d*e**4*x**2/(6*d*
*5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + B*a*e**5*x**3
/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 2*B*b*d
*e**4*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3)
+ 6*B*c*d**5*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6
*d**2*e**7*x**3) + 2*B*c*d**5/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2
+ 6*d**2*e**7*x**3) + 18*B*c*d**4*e*x*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x
+ 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 18*B*c*d**3*e**2*x**2*log(d/e + x)/(6
*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3) - 9*B*c*d**3
*e**2*x**2/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3)
+ 6*B*c*d**2*e**3*x**3*log(d/e + x)/(6*d**5*e**4 + 18*d**4*e**5*x + 18*d**3*e**
6*x**2 + 6*d**2*e**7*x**3) - 9*B*c*d**2*e**3*x**3/(6*d**5*e**4 + 18*d**4*e**5*x
+ 18*d**3*e**6*x**2 + 6*d**2*e**7*x**3), Eq(m, -4)), (-A*a*d*e**3/(2*d**3*e**4 +
4*d**2*e**5*x + 2*d*e**6*x**2) + A*b*e**4*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2
*d*e**6*x**2) + 2*A*c*d**3*e*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**
6*x**2) + A*c*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*A*c*d**2*
e**2*x*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*A*c*d*e**3
*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 2*A*c*d*e**3*
x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + B*a*e**4*x**2/(2*d**3*e**4
+ 4*d**2*e**5*x + 2*d*e**6*x**2) + 2*B*b*d**3*e*log(d/e + x)/(2*d**3*e**4 + 4*d*
*2*e**5*x + 2*d*e**6*x**2) + B*b*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*
x**2) + 4*B*b*d**2*e**2*x*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x
**2) + 2*B*b*d*e**3*x**2*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x*
*2) - 2*B*b*d*e**3*x**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*B*c*d*
*4*log(d/e + x)/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 3*B*c*d**4/(2*d*
*3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 12*B*c*d**3*e*x*log(d/e + x)/(2*d**3*
e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 6*B*c*d**2*e**2*x**2*log(d/e + x)/(2*d**
3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 6*B*c*d**2*e**2*x**2/(2*d**3*e**4 + 4*
d**2*e**5*x + 2*d*e**6*x**2) + 2*B*c*d*e**3*x**3/(2*d**3*e**4 + 4*d**2*e**5*x +
2*d*e**6*x**2), Eq(m, -3)), (-2*A*a*e**3/(2*d*e**4 + 2*e**5*x) + 2*A*b*d*e**2*lo
g(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*A*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*A*b*e*
*3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*c*d**2*e*log(d/e + x)/(2*d*e**4 +
2*e**5*x) - 4*A*c*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*A*c*d*e**2*x*log(d/e + x)/(2*
d*e**4 + 2*e**5*x) + 2*A*c*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*d*e**2*log(d/
e + x)/(2*d*e**4 + 2*e**5*x) + 2*B*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a*e**3*x
*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*B*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e*
*5*x) - 4*B*b*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*B*b*d*e**2*x*log(d/e + x)/(2*d*e*
*4 + 2*e**5*x) + 2*B*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3*log(d/e + x)
/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*c*d**2*e*x*log(d
/e + x)/(2*d*e**4 + 2*e**5*x) - 3*B*c*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*c*e*
*3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (A*a*log(d/e + x)/e - A*b*d*log(d/e +
x)/e**2 + A*b*x/e + A*c*d**2*log(d/e + x)/e**3 - A*c*d*x/e**2 + A*c*x**2/(2*e)
- B*a*d*log(d/e + x)/e**2 + B*a*x/e + B*b*d**2*log(d/e + x)/e**3 - B*b*d*x/e**2
+ B*b*x**2/(2*e) - B*c*d**3*log(d/e + x)/e**4 + B*c*d**2*x/e**3 - B*c*d*x**2/(2*
e**2) + B*c*x**3/(3*e), Eq(m, -1)), (A*a*d*e**3*m**3*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a*d*e**3*m**2*(d + e*x)*
*m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a*d*e*
*3*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 24*A*a*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + A*a*e**4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a*e**4*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e
**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a*e**4*m*x*(d + e*x)**m/(e
**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 24*A*a*e**4*x*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - A*b
*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*
m + 24*e**4) - 7*A*b*d**2*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**
4*m**2 + 50*e**4*m + 24*e**4) - 12*A*b*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e*
*4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*b*d*e**3*m**3*x*(d + e*x)**m/(
e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*A*b*d*e**3*m*
*2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4
) + 12*A*b*d*e**3*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50
*e**4*m + 24*e**4) + A*b*e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*b*e**4*m**2*x**2*(d + e*x)**m/(e**4*m
**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 19*A*b*e**4*m*x**2*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*
A*b*e**4*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) + 2*A*c*d**3*e*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 8*A*c*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) - 2*A*c*d**2*e**2*m**2*x*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*A*c*d**2*e**2*m*x
*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
A*c*d*e**3*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*
e**4*m + 24*e**4) + 5*A*c*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**
3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 4*A*c*d*e**3*m*x**2*(d + e*x)**m/(e**4
*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*c*e**4*m**3*x**3*
(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7
*A*c*e**4*m**2*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + 14*A*c*e**4*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) + 8*A*c*e**4*x**3*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - B*a*d**2*e**2*m**2*(d + e*x)
**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 7*B*a*d**2
*e**2*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e
**4) - 12*B*a*d**2*e**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 +
50*e**4*m + 24*e**4) + B*a*d*e**3*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3
+ 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*B*a*d*e**3*m**2*x*(d + e*x)**m/(e**4*m
**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a*d*e**3*m*x*(d
+ e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*a*
e**4*m**3*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) + 8*B*a*e**4*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e
**4*m**2 + 50*e**4*m + 24*e**4) + 19*B*a*e**4*m*x**2*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 12*B*a*e**4*x**2*(d + e*x)**
m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*B*b*d**3*e
*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4)
+ 8*B*b*d**3*e*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m
+ 24*e**4) - 2*B*b*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*B*b*d**2*e**2*m*x*(d + e*x)**m/(e**4*m**4
+ 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + B*b*d*e**3*m**3*x**2*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 5*B*b
*d*e**3*m**2*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + 4*B*b*d*e**3*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35
*e**4*m**2 + 50*e**4*m + 24*e**4) + B*b*e**4*m**3*x**3*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 7*B*b*e**4*m**2*x**3*(d +
e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 14*B*b
*e**4*m*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m +
24*e**4) + 8*B*b*e**4*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) - 6*B*c*d**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*d**3*e*m*x*(d + e*x)**m/(e**4*m**4 +
10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 3*B*c*d**2*e**2*m**2*x**2*(
d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 3*
B*c*d**2*e**2*m*x**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*
e**4*m + 24*e**4) + B*c*d*e**3*m**3*x**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3
+ 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 3*B*c*d*e**3*m**2*x**3*(d + e*x)**m/(e**
4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 2*B*c*d*e**3*m*x**
3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) +
B*c*e**4*m**3*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e
**4*m + 24*e**4) + 6*B*c*e**4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 +
35*e**4*m**2 + 50*e**4*m + 24*e**4) + 11*B*c*e**4*m*x**4*(d + e*x)**m/(e**4*m**
4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*B*c*e**4*x**4*(d + e*
x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.317545, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done