3.2317 \(\int (A+B x) (d+e x)^5 \left (a+b x+c x^2\right )^2 \, dx\)
Optimal. Leaf size=304 \[ -\frac{(d+e x)^9 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{9 e^6}-\frac{(d+e x)^8 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{8 e^6}-\frac{(d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{7 e^6}-\frac{(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6}-\frac{c (d+e x)^{10} (-A c e-2 b B e+5 B c d)}{10 e^6}+\frac{B c^2 (d+e x)^{11}}{11 e^6} \]
[Out]
-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^6)/(6*e^6) - ((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)^7)/(7*e^
6) - ((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)^8)/(8*e^6) - ((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)^9)/(9*e^6) -
(c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^10)/(10*e^6) + (B*c^2*(d + e*x)^11)/(1
1*e^6)
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Rubi [A] time = 2.43245, antiderivative size = 302, 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
\[ -\frac{(d+e x)^9 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{9 e^6}-\frac{(d+e x)^8 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{8 e^6}+\frac{(d+e x)^7 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{7 e^6}-\frac{(d+e x)^6 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{6 e^6}-\frac{c (d+e x)^{10} (-A c e-2 b B e+5 B c d)}{10 e^6}+\frac{B c^2 (d+e x)^{11}}{11 e^6} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)*(d + e*x)^5*(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)^6)/(6*e^6) + ((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)^7)/(7*e^
6) - ((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)^8)/(8*e^6) - ((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)^9)/(9*e^6) -
(c*(5*B*c*d - 2*b*B*e - A*c*e)*(d + e*x)^10)/(10*e^6) + (B*c^2*(d + e*x)^11)/(1
1*e^6)
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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)**5*(c*x**2+b*x+a)**2,x)
[Out]
Timed out
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Mathematica [B] time = 0.875961, size = 665, normalized size = 2.19 \[ \frac{1}{7} e x^7 \left (B \left (e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+20 c d^2 e (a e+b d)+5 c^2 d^4\right )+A e \left (10 c d e (a e+2 b d)+b e^2 (2 a e+5 b d)+10 c^2 d^3\right )\right )+\frac{1}{5} d x^5 \left (A \left (5 a^2 e^4+20 a c d^2 e^2+c^2 d^4\right )+2 b d \left (10 a A e^3+10 a B d e^2+5 A c d^2 e+B c d^3\right )+10 a B d e \left (a e^2+c d^2\right )+5 b^2 d^2 e (2 A e+B d)\right )+\frac{1}{6} x^6 \left (A e \left (e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )+20 c d^2 e (a e+b d)+5 c^2 d^4\right )+B \left (5 d e^2 \left (a^2 e^2+4 a b d e+2 b^2 d^2\right )+10 c d^3 e (2 a e+b d)+c^2 d^5\right )\right )+a^2 A d^5 x+\frac{1}{9} e^3 x^9 \left (B \left (2 c e (a e+5 b d)+b^2 e^2+10 c^2 d^2\right )+A c e (2 b e+5 c d)\right )+\frac{1}{8} e^2 x^8 \left (A e \left (2 c e (a e+5 b d)+b^2 e^2+10 c^2 d^2\right )+B \left (10 c d e (a e+2 b d)+b e^2 (2 a e+5 b d)+10 c^2 d^3\right )\right )+\frac{1}{3} d^3 x^3 \left (A \left (10 a b d e+2 a \left (5 a e^2+c d^2\right )+b^2 d^2\right )+a B d (5 a e+2 b d)\right )+\frac{1}{4} d^2 x^4 \left (2 b d \left (10 a A e^2+5 a B d e+A c d^2\right )+2 a \left (5 a A e^3+5 a B d e^2+5 A c d^2 e+B c d^3\right )+b^2 d^2 (5 A e+B d)\right )+\frac{1}{2} a d^4 x^2 (5 a A e+a B d+2 A b d)+\frac{1}{10} c e^4 x^{10} (A c e+2 b B e+5 B c d)+\frac{1}{11} B c^2 e^5 x^{11} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)*(d + e*x)^5*(a + b*x + c*x^2)^2,x]
[Out]
a^2*A*d^5*x + (a*d^4*(2*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (d^3*(a*B*d*(2*b*d + 5
*a*e) + A*(b^2*d^2 + 10*a*b*d*e + 2*a*(c*d^2 + 5*a*e^2)))*x^3)/3 + (d^2*(b^2*d^2
*(B*d + 5*A*e) + 2*b*d*(A*c*d^2 + 5*a*B*d*e + 10*a*A*e^2) + 2*a*(B*c*d^3 + 5*A*c
*d^2*e + 5*a*B*d*e^2 + 5*a*A*e^3))*x^4)/4 + (d*(5*b^2*d^2*e*(B*d + 2*A*e) + 10*a
*B*d*e*(c*d^2 + a*e^2) + 2*b*d*(B*c*d^3 + 5*A*c*d^2*e + 10*a*B*d*e^2 + 10*a*A*e^
3) + A*(c^2*d^4 + 20*a*c*d^2*e^2 + 5*a^2*e^4))*x^5)/5 + ((B*(c^2*d^5 + 10*c*d^3*
e*(b*d + 2*a*e) + 5*d*e^2*(2*b^2*d^2 + 4*a*b*d*e + a^2*e^2)) + A*e*(5*c^2*d^4 +
20*c*d^2*e*(b*d + a*e) + e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2)))*x^6)/6 + (e*(
A*e*(10*c^2*d^3 + 10*c*d*e*(2*b*d + a*e) + b*e^2*(5*b*d + 2*a*e)) + B*(5*c^2*d^4
+ 20*c*d^2*e*(b*d + a*e) + e^2*(10*b^2*d^2 + 10*a*b*d*e + a^2*e^2)))*x^7)/7 + (
e^2*(A*e*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(5*b*d + a*e)) + B*(10*c^2*d^3 + 10*c*d*e
*(2*b*d + a*e) + b*e^2*(5*b*d + 2*a*e)))*x^8)/8 + (e^3*(A*c*e*(5*c*d + 2*b*e) +
B*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(5*b*d + a*e)))*x^9)/9 + (c*e^4*(5*B*c*d + 2*b*B
*e + A*c*e)*x^10)/10 + (B*c^2*e^5*x^11)/11
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Maple [B] time = 0.004, size = 671, normalized size = 2.2 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(e*x+d)^5*(c*x^2+b*x+a)^2,x)
[Out]
1/11*B*e^5*c^2*x^11+1/10*((A*e^5+5*B*d*e^4)*c^2+2*B*e^5*b*c)*x^10+1/9*((5*A*d*e^
4+10*B*d^2*e^3)*c^2+2*(A*e^5+5*B*d*e^4)*b*c+B*e^5*(2*a*c+b^2))*x^9+1/8*((10*A*d^
2*e^3+10*B*d^3*e^2)*c^2+2*(5*A*d*e^4+10*B*d^2*e^3)*b*c+(A*e^5+5*B*d*e^4)*(2*a*c+
b^2)+2*B*e^5*a*b)*x^8+1/7*((10*A*d^3*e^2+5*B*d^4*e)*c^2+2*(10*A*d^2*e^3+10*B*d^3
*e^2)*b*c+(5*A*d*e^4+10*B*d^2*e^3)*(2*a*c+b^2)+2*(A*e^5+5*B*d*e^4)*a*b+B*e^5*a^2
)*x^7+1/6*((5*A*d^4*e+B*d^5)*c^2+2*(10*A*d^3*e^2+5*B*d^4*e)*b*c+(10*A*d^2*e^3+10
*B*d^3*e^2)*(2*a*c+b^2)+2*(5*A*d*e^4+10*B*d^2*e^3)*a*b+(A*e^5+5*B*d*e^4)*a^2)*x^
6+1/5*(A*d^5*c^2+2*(5*A*d^4*e+B*d^5)*b*c+(10*A*d^3*e^2+5*B*d^4*e)*(2*a*c+b^2)+2*
(10*A*d^2*e^3+10*B*d^3*e^2)*a*b+(5*A*d*e^4+10*B*d^2*e^3)*a^2)*x^5+1/4*(2*A*d^5*b
*c+(5*A*d^4*e+B*d^5)*(2*a*c+b^2)+2*(10*A*d^3*e^2+5*B*d^4*e)*a*b+(10*A*d^2*e^3+10
*B*d^3*e^2)*a^2)*x^4+1/3*(A*d^5*(2*a*c+b^2)+2*(5*A*d^4*e+B*d^5)*a*b+(10*A*d^3*e^
2+5*B*d^4*e)*a^2)*x^3+1/2*(2*A*d^5*a*b+(5*A*d^4*e+B*d^5)*a^2)*x^2+A*d^5*a^2*x
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Maxima [A] time = 0.710719, size = 875, normalized size = 2.88 \[ \text{result too large to display} \]
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)^5,x, algorithm="maxima")
[Out]
1/11*B*c^2*e^5*x^11 + 1/10*(5*B*c^2*d*e^4 + (2*B*b*c + A*c^2)*e^5)*x^10 + 1/9*(1
0*B*c^2*d^2*e^3 + 5*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*(B*a + A*b)*c)*e^5)*x^9
+ A*a^2*d^5*x + 1/8*(10*B*c^2*d^3*e^2 + 10*(2*B*b*c + A*c^2)*d^2*e^3 + 5*(B*b^2
+ 2*(B*a + A*b)*c)*d*e^4 + (2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^8 + 1/7*(5*B*c^2*
d^4*e + 10*(2*B*b*c + A*c^2)*d^3*e^2 + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 + 5*
(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a*b)*e^5)*x^7 + 1/6*(B*c^2*d^5
+ A*a^2*e^5 + 5*(2*B*b*c + A*c^2)*d^4*e + 10*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 +
10*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 5*(B*a^2 + 2*A*a*b)*d*e^4)*x^6 + 1/5*(
5*A*a^2*d*e^4 + (2*B*b*c + A*c^2)*d^5 + 5*(B*b^2 + 2*(B*a + A*b)*c)*d^4*e + 10*(
2*B*a*b + A*b^2 + 2*A*a*c)*d^3*e^2 + 10*(B*a^2 + 2*A*a*b)*d^2*e^3)*x^5 + 1/4*(10
*A*a^2*d^2*e^3 + (B*b^2 + 2*(B*a + A*b)*c)*d^5 + 5*(2*B*a*b + A*b^2 + 2*A*a*c)*d
^4*e + 10*(B*a^2 + 2*A*a*b)*d^3*e^2)*x^4 + 1/3*(10*A*a^2*d^3*e^2 + (2*B*a*b + A*
b^2 + 2*A*a*c)*d^5 + 5*(B*a^2 + 2*A*a*b)*d^4*e)*x^3 + 1/2*(5*A*a^2*d^4*e + (B*a^
2 + 2*A*a*b)*d^5)*x^2
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Fricas [A] time = 0.239975, size = 1, normalized size = 0. \[ \text{result too large to display} \]
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)^5,x, algorithm="fricas")
[Out]
1/11*x^11*e^5*c^2*B + 1/2*x^10*e^4*d*c^2*B + 1/5*x^10*e^5*c*b*B + 1/10*x^10*e^5*
c^2*A + 10/9*x^9*e^3*d^2*c^2*B + 10/9*x^9*e^4*d*c*b*B + 1/9*x^9*e^5*b^2*B + 2/9*
x^9*e^5*c*a*B + 5/9*x^9*e^4*d*c^2*A + 2/9*x^9*e^5*c*b*A + 5/4*x^8*e^2*d^3*c^2*B
+ 5/2*x^8*e^3*d^2*c*b*B + 5/8*x^8*e^4*d*b^2*B + 5/4*x^8*e^4*d*c*a*B + 1/4*x^8*e^
5*b*a*B + 5/4*x^8*e^3*d^2*c^2*A + 5/4*x^8*e^4*d*c*b*A + 1/8*x^8*e^5*b^2*A + 1/4*
x^8*e^5*c*a*A + 5/7*x^7*e*d^4*c^2*B + 20/7*x^7*e^2*d^3*c*b*B + 10/7*x^7*e^3*d^2*
b^2*B + 20/7*x^7*e^3*d^2*c*a*B + 10/7*x^7*e^4*d*b*a*B + 1/7*x^7*e^5*a^2*B + 10/7
*x^7*e^2*d^3*c^2*A + 20/7*x^7*e^3*d^2*c*b*A + 5/7*x^7*e^4*d*b^2*A + 10/7*x^7*e^4
*d*c*a*A + 2/7*x^7*e^5*b*a*A + 1/6*x^6*d^5*c^2*B + 5/3*x^6*e*d^4*c*b*B + 5/3*x^6
*e^2*d^3*b^2*B + 10/3*x^6*e^2*d^3*c*a*B + 10/3*x^6*e^3*d^2*b*a*B + 5/6*x^6*e^4*d
*a^2*B + 5/6*x^6*e*d^4*c^2*A + 10/3*x^6*e^2*d^3*c*b*A + 5/3*x^6*e^3*d^2*b^2*A +
10/3*x^6*e^3*d^2*c*a*A + 5/3*x^6*e^4*d*b*a*A + 1/6*x^6*e^5*a^2*A + 2/5*x^5*d^5*c
*b*B + x^5*e*d^4*b^2*B + 2*x^5*e*d^4*c*a*B + 4*x^5*e^2*d^3*b*a*B + 2*x^5*e^3*d^2
*a^2*B + 1/5*x^5*d^5*c^2*A + 2*x^5*e*d^4*c*b*A + 2*x^5*e^2*d^3*b^2*A + 4*x^5*e^2
*d^3*c*a*A + 4*x^5*e^3*d^2*b*a*A + x^5*e^4*d*a^2*A + 1/4*x^4*d^5*b^2*B + 1/2*x^4
*d^5*c*a*B + 5/2*x^4*e*d^4*b*a*B + 5/2*x^4*e^2*d^3*a^2*B + 1/2*x^4*d^5*c*b*A + 5
/4*x^4*e*d^4*b^2*A + 5/2*x^4*e*d^4*c*a*A + 5*x^4*e^2*d^3*b*a*A + 5/2*x^4*e^3*d^2
*a^2*A + 2/3*x^3*d^5*b*a*B + 5/3*x^3*e*d^4*a^2*B + 1/3*x^3*d^5*b^2*A + 2/3*x^3*d
^5*c*a*A + 10/3*x^3*e*d^4*b*a*A + 10/3*x^3*e^2*d^3*a^2*A + 1/2*x^2*d^5*a^2*B + x
^2*d^5*b*a*A + 5/2*x^2*e*d^4*a^2*A + x*d^5*a^2*A
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Sympy [A] time = 0.476728, size = 957, normalized size = 3.15 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x+A)*(e*x+d)**5*(c*x**2+b*x+a)**2,x)
[Out]
A*a**2*d**5*x + B*c**2*e**5*x**11/11 + x**10*(A*c**2*e**5/10 + B*b*c*e**5/5 + B*
c**2*d*e**4/2) + x**9*(2*A*b*c*e**5/9 + 5*A*c**2*d*e**4/9 + 2*B*a*c*e**5/9 + B*b
**2*e**5/9 + 10*B*b*c*d*e**4/9 + 10*B*c**2*d**2*e**3/9) + x**8*(A*a*c*e**5/4 + A
*b**2*e**5/8 + 5*A*b*c*d*e**4/4 + 5*A*c**2*d**2*e**3/4 + B*a*b*e**5/4 + 5*B*a*c*
d*e**4/4 + 5*B*b**2*d*e**4/8 + 5*B*b*c*d**2*e**3/2 + 5*B*c**2*d**3*e**2/4) + x**
7*(2*A*a*b*e**5/7 + 10*A*a*c*d*e**4/7 + 5*A*b**2*d*e**4/7 + 20*A*b*c*d**2*e**3/7
+ 10*A*c**2*d**3*e**2/7 + B*a**2*e**5/7 + 10*B*a*b*d*e**4/7 + 20*B*a*c*d**2*e**
3/7 + 10*B*b**2*d**2*e**3/7 + 20*B*b*c*d**3*e**2/7 + 5*B*c**2*d**4*e/7) + x**6*(
A*a**2*e**5/6 + 5*A*a*b*d*e**4/3 + 10*A*a*c*d**2*e**3/3 + 5*A*b**2*d**2*e**3/3 +
10*A*b*c*d**3*e**2/3 + 5*A*c**2*d**4*e/6 + 5*B*a**2*d*e**4/6 + 10*B*a*b*d**2*e*
*3/3 + 10*B*a*c*d**3*e**2/3 + 5*B*b**2*d**3*e**2/3 + 5*B*b*c*d**4*e/3 + B*c**2*d
**5/6) + x**5*(A*a**2*d*e**4 + 4*A*a*b*d**2*e**3 + 4*A*a*c*d**3*e**2 + 2*A*b**2*
d**3*e**2 + 2*A*b*c*d**4*e + A*c**2*d**5/5 + 2*B*a**2*d**2*e**3 + 4*B*a*b*d**3*e
**2 + 2*B*a*c*d**4*e + B*b**2*d**4*e + 2*B*b*c*d**5/5) + x**4*(5*A*a**2*d**2*e**
3/2 + 5*A*a*b*d**3*e**2 + 5*A*a*c*d**4*e/2 + 5*A*b**2*d**4*e/4 + A*b*c*d**5/2 +
5*B*a**2*d**3*e**2/2 + 5*B*a*b*d**4*e/2 + B*a*c*d**5/2 + B*b**2*d**5/4) + x**3*(
10*A*a**2*d**3*e**2/3 + 10*A*a*b*d**4*e/3 + 2*A*a*c*d**5/3 + A*b**2*d**5/3 + 5*B
*a**2*d**4*e/3 + 2*B*a*b*d**5/3) + x**2*(5*A*a**2*d**4*e/2 + A*a*b*d**5 + B*a**2
*d**5/2)
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.254827, 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)^5,x, algorithm="giac")
[Out]
Done