3.3154 \(\int (a+b x)^2 (A+B x) (d+e x)^m \, dx\)
Optimal. Leaf size=138 \[ -\frac{(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac{b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac{b^2 B (d+e x)^{m+4}}{e^4 (m+4)} \]
[Out]
-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a*e)*(3
*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b*B*d - A*b*e
- 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e*x)^(4 + m))/(e^4*(4
+ m))
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Rubi [A] time = 0.248719, antiderivative size = 138, 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
\[ -\frac{(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac{(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac{b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac{b^2 B (d+e x)^{m+4}}{e^4 (m+4)} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
[Out]
-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a*e)*(3
*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b*B*d - A*b*e
- 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e*x)^(4 + m))/(e^4*(4
+ m))
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Rubi in Sympy [A] time = 42.1774, size = 126, normalized size = 0.91 \[ \frac{B b^{2} \left (d + e x\right )^{m + 4}}{e^{4} \left (m + 4\right )} + \frac{b \left (d + e x\right )^{m + 3} \left (A b e + 2 B a e - 3 B b d\right )}{e^{4} \left (m + 3\right )} + \frac{\left (d + e x\right )^{m + 1} \left (A e - B d\right ) \left (a e - b d\right )^{2}}{e^{4} \left (m + 1\right )} + \frac{\left (d + e x\right )^{m + 2} \left (a e - b d\right ) \left (2 A b e + B a e - 3 B b d\right )}{e^{4} \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**2*(B*x+A)*(e*x+d)**m,x)
[Out]
B*b**2*(d + e*x)**(m + 4)/(e**4*(m + 4)) + b*(d + e*x)**(m + 3)*(A*b*e + 2*B*a*e
- 3*B*b*d)/(e**4*(m + 3)) + (d + e*x)**(m + 1)*(A*e - B*d)*(a*e - b*d)**2/(e**4
*(m + 1)) + (d + e*x)**(m + 2)*(a*e - b*d)*(2*A*b*e + B*a*e - 3*B*b*d)/(e**4*(m
+ 2))
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Mathematica [A] time = 0.344285, size = 221, normalized size = 1.6 \[ \frac{(d+e x)^{m+1} \left (a^2 e^2 \left (m^2+7 m+12\right ) (A e (m+2)-B d+B e (m+1) x)+2 a b e (m+4) \left (A e (m+3) (e (m+1) x-d)+B \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+b^2 \left (-\left (B \left (6 d^3-6 d^2 e (m+1) x+3 d e^2 \left (m^2+3 m+2\right ) x^2-e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )-A e (m+4) \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]
[Out]
((d + e*x)^(1 + m)*(a^2*e^2*(12 + 7*m + m^2)*(-(B*d) + A*e*(2 + m) + B*e*(1 + m)
*x) + 2*a*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)) - b^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))))/(e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m
))
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Maple [B] time = 0.014, size = 576, normalized size = 4.2 \[{\frac{ \left ( ex+d \right ) ^{1+m} \left ( B{b}^{2}{e}^{3}{m}^{3}{x}^{3}+A{b}^{2}{e}^{3}{m}^{3}{x}^{2}+2\,Bab{e}^{3}{m}^{3}{x}^{2}+6\,B{b}^{2}{e}^{3}{m}^{2}{x}^{3}+2\,Aab{e}^{3}{m}^{3}x+7\,A{b}^{2}{e}^{3}{m}^{2}{x}^{2}+B{a}^{2}{e}^{3}{m}^{3}x+14\,Bab{e}^{3}{m}^{2}{x}^{2}-3\,B{b}^{2}d{e}^{2}{m}^{2}{x}^{2}+11\,B{b}^{2}{e}^{3}m{x}^{3}+A{a}^{2}{e}^{3}{m}^{3}+16\,Aab{e}^{3}{m}^{2}x-2\,A{b}^{2}d{e}^{2}{m}^{2}x+14\,A{b}^{2}{e}^{3}m{x}^{2}+8\,B{a}^{2}{e}^{3}{m}^{2}x-4\,Babd{e}^{2}{m}^{2}x+28\,Bab{e}^{3}m{x}^{2}-9\,B{b}^{2}d{e}^{2}m{x}^{2}+6\,{b}^{2}B{x}^{3}{e}^{3}+9\,A{a}^{2}{e}^{3}{m}^{2}-2\,Aabd{e}^{2}{m}^{2}+38\,Aab{e}^{3}mx-10\,A{b}^{2}d{e}^{2}mx+8\,A{b}^{2}{e}^{3}{x}^{2}-B{a}^{2}d{e}^{2}{m}^{2}+19\,B{a}^{2}{e}^{3}mx-20\,Babd{e}^{2}mx+16\,Bab{e}^{3}{x}^{2}+6\,B{b}^{2}{d}^{2}emx-6\,B{b}^{2}d{e}^{2}{x}^{2}+26\,A{a}^{2}{e}^{3}m-14\,Aabd{e}^{2}m+24\,Aab{e}^{3}x+2\,A{b}^{2}{d}^{2}em-8\,A{b}^{2}d{e}^{2}x-7\,B{a}^{2}d{e}^{2}m+12\,B{a}^{2}{e}^{3}x+4\,Bab{d}^{2}em-16\,Babd{e}^{2}x+6\,B{b}^{2}{d}^{2}ex+24\,A{a}^{2}{e}^{3}-24\,Aabd{e}^{2}+8\,A{b}^{2}{d}^{2}e-12\,B{a}^{2}d{e}^{2}+16\,Bab{d}^{2}e-6\,B{b}^{2}{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^2*(B*x+A)*(e*x+d)^m,x)
[Out]
(e*x+d)^(1+m)*(B*b^2*e^3*m^3*x^3+A*b^2*e^3*m^3*x^2+2*B*a*b*e^3*m^3*x^2+6*B*b^2*e
^3*m^2*x^3+2*A*a*b*e^3*m^3*x+7*A*b^2*e^3*m^2*x^2+B*a^2*e^3*m^3*x+14*B*a*b*e^3*m^
2*x^2-3*B*b^2*d*e^2*m^2*x^2+11*B*b^2*e^3*m*x^3+A*a^2*e^3*m^3+16*A*a*b*e^3*m^2*x-
2*A*b^2*d*e^2*m^2*x+14*A*b^2*e^3*m*x^2+8*B*a^2*e^3*m^2*x-4*B*a*b*d*e^2*m^2*x+28*
B*a*b*e^3*m*x^2-9*B*b^2*d*e^2*m*x^2+6*B*b^2*e^3*x^3+9*A*a^2*e^3*m^2-2*A*a*b*d*e^
2*m^2+38*A*a*b*e^3*m*x-10*A*b^2*d*e^2*m*x+8*A*b^2*e^3*x^2-B*a^2*d*e^2*m^2+19*B*a
^2*e^3*m*x-20*B*a*b*d*e^2*m*x+16*B*a*b*e^3*x^2+6*B*b^2*d^2*e*m*x-6*B*b^2*d*e^2*x
^2+26*A*a^2*e^3*m-14*A*a*b*d*e^2*m+24*A*a*b*e^3*x+2*A*b^2*d^2*e*m-8*A*b^2*d*e^2*
x-7*B*a^2*d*e^2*m+12*B*a^2*e^3*x+4*B*a*b*d^2*e*m-16*B*a*b*d*e^2*x+6*B*b^2*d^2*e*
x+24*A*a^2*e^3-24*A*a*b*d*e^2+8*A*b^2*d^2*e-12*B*a^2*d*e^2+16*B*a*b*d^2*e-6*B*b^
2*d^3)/e^4/(m^4+10*m^3+35*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((B*x + A)*(b*x + a)^2*(e*x + d)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.288519, size = 891, normalized size = 6.46 \[ \frac{{\left (A a^{2} d e^{3} m^{3} - 6 \, B b^{2} d^{4} + 24 \, A a^{2} d e^{3} + 8 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 12 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2} +{\left (B b^{2} e^{4} m^{3} + 6 \, B b^{2} e^{4} m^{2} + 11 \, B b^{2} e^{4} m + 6 \, B b^{2} e^{4}\right )} x^{4} +{\left (8 \,{\left (2 \, B a b + A b^{2}\right )} e^{4} +{\left (B b^{2} d e^{3} +{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{3} +{\left (3 \, B b^{2} d e^{3} + 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m^{2} + 2 \,{\left (B b^{2} d e^{3} + 7 \,{\left (2 \, B a b + A b^{2}\right )} e^{4}\right )} m\right )} x^{3} +{\left (9 \, A a^{2} d e^{3} -{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m^{2} +{\left (12 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4} +{\left ({\left (2 \, B a b + A b^{2}\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{3} -{\left (3 \, B b^{2} d^{2} e^{2} - 5 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 8 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m^{2} -{\left (3 \, B b^{2} d^{2} e^{2} - 4 \,{\left (2 \, B a b + A b^{2}\right )} d e^{3} - 19 \,{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} m\right )} x^{2} +{\left (26 \, A a^{2} d e^{3} + 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{3} e - 7 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} m +{\left (24 \, A a^{2} e^{4} +{\left (A a^{2} e^{4} +{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{3} +{\left (9 \, A a^{2} e^{4} - 2 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 7 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m^{2} + 2 \,{\left (3 \, B b^{2} d^{3} e + 13 \, A a^{2} e^{4} - 4 \,{\left (2 \, B a b + A b^{2}\right )} d^{2} e^{2} + 6 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^m,x, algorithm="fricas")
[Out]
(A*a^2*d*e^3*m^3 - 6*B*b^2*d^4 + 24*A*a^2*d*e^3 + 8*(2*B*a*b + A*b^2)*d^3*e - 12
*(B*a^2 + 2*A*a*b)*d^2*e^2 + (B*b^2*e^4*m^3 + 6*B*b^2*e^4*m^2 + 11*B*b^2*e^4*m +
6*B*b^2*e^4)*x^4 + (8*(2*B*a*b + A*b^2)*e^4 + (B*b^2*d*e^3 + (2*B*a*b + A*b^2)*
e^4)*m^3 + (3*B*b^2*d*e^3 + 7*(2*B*a*b + A*b^2)*e^4)*m^2 + 2*(B*b^2*d*e^3 + 7*(2
*B*a*b + A*b^2)*e^4)*m)*x^3 + (9*A*a^2*d*e^3 - (B*a^2 + 2*A*a*b)*d^2*e^2)*m^2 +
(12*(B*a^2 + 2*A*a*b)*e^4 + ((2*B*a*b + A*b^2)*d*e^3 + (B*a^2 + 2*A*a*b)*e^4)*m^
3 - (3*B*b^2*d^2*e^2 - 5*(2*B*a*b + A*b^2)*d*e^3 - 8*(B*a^2 + 2*A*a*b)*e^4)*m^2
- (3*B*b^2*d^2*e^2 - 4*(2*B*a*b + A*b^2)*d*e^3 - 19*(B*a^2 + 2*A*a*b)*e^4)*m)*x^
2 + (26*A*a^2*d*e^3 + 2*(2*B*a*b + A*b^2)*d^3*e - 7*(B*a^2 + 2*A*a*b)*d^2*e^2)*m
+ (24*A*a^2*e^4 + (A*a^2*e^4 + (B*a^2 + 2*A*a*b)*d*e^3)*m^3 + (9*A*a^2*e^4 - 2*
(2*B*a*b + A*b^2)*d^2*e^2 + 7*(B*a^2 + 2*A*a*b)*d*e^3)*m^2 + 2*(3*B*b^2*d^3*e +
13*A*a^2*e^4 - 4*(2*B*a*b + A*b^2)*d^2*e^2 + 6*(B*a^2 + 2*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 = 10.0918, size = 6094, normalized size = 44.16 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)**2*(B*x+A)*(e*x+d)**m,x)
[Out]
Piecewise((d**m*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b
*x**3/3 + B*b**2*x**4/4), Eq(e, 0)), (-2*A*a**2*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) + 6*A*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) + 2*A*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*b**2
*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**2*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**2*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) + 4*B*a*b*d*e**4*x**3/(6*d**5*e**4 + 18*d**4*e**5*x + 1
8*d**3*e**6*x**2 + 6*d**2*e**7*x**3) + 6*B*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*d**2*e**3*x**3*lo
g(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*b**2*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**2*d*e**3/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d
*e**6*x**2) + 2*A*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*b**2*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*b
**2*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*A*b**2*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*b**2*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*b**2*d*e**3*x
**2/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + B*a**2*e**4*x**2/(2*d**3*e**
4 + 4*d**2*e**5*x + 2*d*e**6*x**2) + 4*B*a*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) + 2*B*a*b*d**3*e/(2*d**3*e**4 + 4*d**2*e**5*x + 2
*d*e**6*x**2) + 8*B*a*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) + 4*B*a*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) - 4*B*a*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*b**2*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*b**2*d**4/(2*d**3*e**4 + 4*d**2*e**5*x + 2*d*e**6*x**2) - 12*B*b**2*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*b**2*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*b**2*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*b**2*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**2*e**3/(2*d
*e**4 + 2*e**5*x) + 4*A*a*b*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*
d*e**2/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x)
- 4*A*b**2*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*b**2*d**2*e/(2*d*e**
4 + 2*e**5*x) - 4*A*b**2*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*A*b**2*
e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*d*e**2*log(d/e + x)/(2*d*e**4 + 2*e**
5*x) + 2*B*a**2*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*e**3*x*log(d/e + x)/(2*d
*e**4 + 2*e**5*x) - 8*B*a*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 8*B*a*b*
d**2*e/(2*d*e**4 + 2*e**5*x) - 8*B*a*b*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*
x) + 4*B*a*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**3*log(d/e + x)/(2*d*e
**4 + 2*e**5*x) + 6*B*b**2*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**2*e*x*log(d/
e + x)/(2*d*e**4 + 2*e**5*x) - 3*B*b**2*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*b*
*2*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (A*a**2*log(d/e + x)/e - 2*A*a*b
*d*log(d/e + x)/e**2 + 2*A*a*b*x/e + A*b**2*d**2*log(d/e + x)/e**3 - A*b**2*d*x/
e**2 + A*b**2*x**2/(2*e) - B*a**2*d*log(d/e + x)/e**2 + B*a**2*x/e + 2*B*a*b*d**
2*log(d/e + x)/e**3 - 2*B*a*b*d*x/e**2 + B*a*b*x**2/e - B*b**2*d**3*log(d/e + x)
/e**4 + B*b**2*d**2*x/e**3 - B*b**2*d*x**2/(2*e**2) + B*b**2*x**3/(3*e), Eq(m, -
1)), (A*a**2*d*e**3*m**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) + 9*A*a**2*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**2*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**2*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**2*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**2*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**2*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**2*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) - 2*A*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) - 14*A*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) - 24*A*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) + 2*A*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) + 14*A*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) + 24*A*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) + 2*A*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) + 16*A*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) + 38*
A*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) + 24*A*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*b**2*d**3*e*m*(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) + 8*A*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*A*b**2*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) - B*a**2*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**2*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**2*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**2*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**2*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**2*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**2*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 + 2
4*e**4) + 8*B*a**2*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**2*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*B*a**2*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) + 4*B*a*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) + 16*B*a*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) - 4*B*a*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) - 16*B*a*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) + 2*B*a*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) + 10*B*a*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) + 8*B*a*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) + 2*B*a*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)
+ 14*B*a*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) + 28*B*a*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) + 16*B*a*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*b**2*d**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*
b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*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*b**2*e**4*m**2*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + 11*B*b**2*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*b**2*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 = 0.23825, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)*(b*x + a)^2*(e*x + d)^m,x, algorithm="giac")
[Out]
Done