3.1675 \(\int (A+B x) (d+e x)^5 \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=206 \[ -\frac{b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac{2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac{(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{b^4 B (d+e x)^{11}}{11 e^6} \]

[Out]

-((b*d - a*e)^4*(B*d - A*e)*(d + e*x)^6)/(6*e^6) + ((b*d - a*e)^3*(5*b*B*d - 4*A
*b*e - a*B*e)*(d + e*x)^7)/(7*e^6) - (b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B
*e)*(d + e*x)^8)/(4*e^6) + (2*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d +
e*x)^9)/(9*e^6) - (b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^10)/(10*e^6) + (b^
4*B*(d + e*x)^11)/(11*e^6)

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Rubi [A]  time = 1.48232, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{b^3 (d+e x)^{10} (-4 a B e-A b e+5 b B d)}{10 e^6}+\frac{2 b^2 (d+e x)^9 (b d-a e) (-3 a B e-2 A b e+5 b B d)}{9 e^6}-\frac{b (d+e x)^8 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{4 e^6}+\frac{(d+e x)^7 (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{7 e^6}-\frac{(d+e x)^6 (b d-a e)^4 (B d-A e)}{6 e^6}+\frac{b^4 B (d+e x)^{11}}{11 e^6} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

-((b*d - a*e)^4*(B*d - A*e)*(d + e*x)^6)/(6*e^6) + ((b*d - a*e)^3*(5*b*B*d - 4*A
*b*e - a*B*e)*(d + e*x)^7)/(7*e^6) - (b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B
*e)*(d + e*x)^8)/(4*e^6) + (2*b^2*(b*d - a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d +
e*x)^9)/(9*e^6) - (b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^10)/(10*e^6) + (b^
4*B*(d + e*x)^11)/(11*e^6)

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Rubi in Sympy [A]  time = 177.702, size = 202, normalized size = 0.98 \[ \frac{B b^{4} \left (d + e x\right )^{11}}{11 e^{6}} + \frac{b^{3} \left (d + e x\right )^{10} \left (A b e + 4 B a e - 5 B b d\right )}{10 e^{6}} + \frac{2 b^{2} \left (d + e x\right )^{9} \left (a e - b d\right ) \left (2 A b e + 3 B a e - 5 B b d\right )}{9 e^{6}} + \frac{b \left (d + e x\right )^{8} \left (a e - b d\right )^{2} \left (3 A b e + 2 B a e - 5 B b d\right )}{4 e^{6}} + \frac{\left (d + e x\right )^{7} \left (a e - b d\right )^{3} \left (4 A b e + B a e - 5 B b d\right )}{7 e^{6}} + \frac{\left (d + e x\right )^{6} \left (A e - B d\right ) \left (a e - b d\right )^{4}}{6 e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

B*b**4*(d + e*x)**11/(11*e**6) + b**3*(d + e*x)**10*(A*b*e + 4*B*a*e - 5*B*b*d)/
(10*e**6) + 2*b**2*(d + e*x)**9*(a*e - b*d)*(2*A*b*e + 3*B*a*e - 5*B*b*d)/(9*e**
6) + b*(d + e*x)**8*(a*e - b*d)**2*(3*A*b*e + 2*B*a*e - 5*B*b*d)/(4*e**6) + (d +
e*x)**7*(a*e - b*d)**3*(4*A*b*e + B*a*e - 5*B*b*d)/(7*e**6) + (d + e*x)**6*(A*e
- B*d)*(a*e - b*d)**4/(6*e**6)

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Mathematica [B]  time = 0.424906, size = 615, normalized size = 2.99 \[ a^4 A d^5 x+\frac{1}{2} a^3 d^4 x^2 (5 a A e+a B d+4 A b d)+\frac{1}{3} a^2 d^3 x^3 \left (2 A \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+a B d (5 a e+4 b d)\right )+\frac{1}{9} b^2 e^3 x^9 \left (6 a^2 B e^2+4 a b e (A e+5 B d)+5 b^2 d (A e+2 B d)\right )+\frac{1}{4} b e^2 x^8 \left (2 a^3 B e^3+3 a^2 b e^2 (A e+5 B d)+10 a b^2 d e (A e+2 B d)+5 b^3 d^2 (A e+B d)\right )+\frac{1}{2} a d^2 x^4 \left (a B d \left (5 a^2 e^2+10 a b d e+3 b^2 d^2\right )+A \left (5 a^3 e^3+20 a^2 b d e^2+15 a b^2 d^2 e+2 b^3 d^3\right )\right )+\frac{1}{7} e x^7 \left (a^4 B e^4+4 a^3 b e^3 (A e+5 B d)+30 a^2 b^2 d e^2 (A e+2 B d)+40 a b^3 d^2 e (A e+B d)+5 b^4 d^3 (2 A e+B d)\right )+\frac{1}{6} x^6 \left (a^4 e^4 (A e+5 B d)+20 a^3 b d e^3 (A e+2 B d)+60 a^2 b^2 d^2 e^2 (A e+B d)+20 a b^3 d^3 e (2 A e+B d)+b^4 d^4 (5 A e+B d)\right )+\frac{1}{5} d x^5 \left (2 a B d \left (5 a^3 e^3+20 a^2 b d e^2+15 a b^2 d^2 e+2 b^3 d^3\right )+A \left (5 a^4 e^4+40 a^3 b d e^3+60 a^2 b^2 d^2 e^2+20 a b^3 d^3 e+b^4 d^4\right )\right )+\frac{1}{10} b^3 e^4 x^{10} (4 a B e+A b e+5 b B d)+\frac{1}{11} b^4 B e^5 x^{11} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)*(d + e*x)^5*(a^2 + 2*a*b*x + b^2*x^2)^2,x]

[Out]

a^4*A*d^5*x + (a^3*d^4*(4*A*b*d + a*B*d + 5*a*A*e)*x^2)/2 + (a^2*d^3*(a*B*d*(4*b
*d + 5*a*e) + 2*A*(3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2))*x^3)/3 + (a*d^2*(a*B*d*(
3*b^2*d^2 + 10*a*b*d*e + 5*a^2*e^2) + A*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d
*e^2 + 5*a^3*e^3))*x^4)/2 + (d*(2*a*B*d*(2*b^3*d^3 + 15*a*b^2*d^2*e + 20*a^2*b*d
*e^2 + 5*a^3*e^3) + A*(b^4*d^4 + 20*a*b^3*d^3*e + 60*a^2*b^2*d^2*e^2 + 40*a^3*b*
d*e^3 + 5*a^4*e^4))*x^5)/5 + ((60*a^2*b^2*d^2*e^2*(B*d + A*e) + 20*a^3*b*d*e^3*(
2*B*d + A*e) + a^4*e^4*(5*B*d + A*e) + 20*a*b^3*d^3*e*(B*d + 2*A*e) + b^4*d^4*(B
*d + 5*A*e))*x^6)/6 + (e*(a^4*B*e^4 + 40*a*b^3*d^2*e*(B*d + A*e) + 30*a^2*b^2*d*
e^2*(2*B*d + A*e) + 4*a^3*b*e^3*(5*B*d + A*e) + 5*b^4*d^3*(B*d + 2*A*e))*x^7)/7
+ (b*e^2*(2*a^3*B*e^3 + 5*b^3*d^2*(B*d + A*e) + 10*a*b^2*d*e*(2*B*d + A*e) + 3*a
^2*b*e^2*(5*B*d + A*e))*x^8)/4 + (b^2*e^3*(6*a^2*B*e^2 + 5*b^2*d*(2*B*d + A*e) +
4*a*b*e*(5*B*d + A*e))*x^9)/9 + (b^3*e^4*(5*b*B*d + A*b*e + 4*a*B*e)*x^10)/10 +
(b^4*B*e^5*x^11)/11

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Maple [B]  time = 0.003, size = 692, normalized size = 3.4 \[{\frac{B{e}^{5}{b}^{4}{x}^{11}}{11}}+{\frac{ \left ( \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){b}^{4}+4\,B{e}^{5}a{b}^{3} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){b}^{4}+4\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ) a{b}^{3}+6\,B{e}^{5}{a}^{2}{b}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){b}^{4}+4\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ) a{b}^{3}+6\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{2}{b}^{2}+4\,B{e}^{5}{a}^{3}b \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){b}^{4}+4\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ) a{b}^{3}+6\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{2}{b}^{2}+4\, \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{3}b+B{e}^{5}{a}^{4} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){b}^{4}+4\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ) a{b}^{3}+6\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{2}{b}^{2}+4\, \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{3}b+ \left ( A{e}^{5}+5\,Bd{e}^{4} \right ){a}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{5}{b}^{4}+4\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ) a{b}^{3}+6\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{2}{b}^{2}+4\, \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{3}b+ \left ( 5\,Ad{e}^{4}+10\,B{d}^{2}{e}^{3} \right ){a}^{4} \right ){x}^{5}}{5}}+{\frac{ \left ( 4\,A{d}^{5}a{b}^{3}+6\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{2}{b}^{2}+4\, \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{3}b+ \left ( 10\,A{d}^{2}{e}^{3}+10\,B{d}^{3}{e}^{2} \right ){a}^{4} \right ){x}^{4}}{4}}+{\frac{ \left ( 6\,A{d}^{5}{a}^{2}{b}^{2}+4\, \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{3}b+ \left ( 10\,A{d}^{3}{e}^{2}+5\,B{d}^{4}e \right ){a}^{4} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{5}{a}^{3}b+ \left ( 5\,A{d}^{4}e+B{d}^{5} \right ){a}^{4} \right ){x}^{2}}{2}}+A{d}^{5}{a}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(e*x+d)^5*(b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

1/11*B*e^5*b^4*x^11+1/10*((A*e^5+5*B*d*e^4)*b^4+4*B*e^5*a*b^3)*x^10+1/9*((5*A*d*
e^4+10*B*d^2*e^3)*b^4+4*(A*e^5+5*B*d*e^4)*a*b^3+6*B*e^5*a^2*b^2)*x^9+1/8*((10*A*
d^2*e^3+10*B*d^3*e^2)*b^4+4*(5*A*d*e^4+10*B*d^2*e^3)*a*b^3+6*(A*e^5+5*B*d*e^4)*a
^2*b^2+4*B*e^5*a^3*b)*x^8+1/7*((10*A*d^3*e^2+5*B*d^4*e)*b^4+4*(10*A*d^2*e^3+10*B
*d^3*e^2)*a*b^3+6*(5*A*d*e^4+10*B*d^2*e^3)*a^2*b^2+4*(A*e^5+5*B*d*e^4)*a^3*b+B*e
^5*a^4)*x^7+1/6*((5*A*d^4*e+B*d^5)*b^4+4*(10*A*d^3*e^2+5*B*d^4*e)*a*b^3+6*(10*A*
d^2*e^3+10*B*d^3*e^2)*a^2*b^2+4*(5*A*d*e^4+10*B*d^2*e^3)*a^3*b+(A*e^5+5*B*d*e^4)
*a^4)*x^6+1/5*(A*d^5*b^4+4*(5*A*d^4*e+B*d^5)*a*b^3+6*(10*A*d^3*e^2+5*B*d^4*e)*a^
2*b^2+4*(10*A*d^2*e^3+10*B*d^3*e^2)*a^3*b+(5*A*d*e^4+10*B*d^2*e^3)*a^4)*x^5+1/4*
(4*A*d^5*a*b^3+6*(5*A*d^4*e+B*d^5)*a^2*b^2+4*(10*A*d^3*e^2+5*B*d^4*e)*a^3*b+(10*
A*d^2*e^3+10*B*d^3*e^2)*a^4)*x^4+1/3*(6*A*d^5*a^2*b^2+4*(5*A*d^4*e+B*d^5)*a^3*b+
(10*A*d^3*e^2+5*B*d^4*e)*a^4)*x^3+1/2*(4*A*d^5*a^3*b+(5*A*d^4*e+B*d^5)*a^4)*x^2+
A*d^5*a^4*x

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Maxima [A]  time = 0.698145, size = 926, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^5,x, algorithm="maxima")

[Out]

1/11*B*b^4*e^5*x^11 + A*a^4*d^5*x + 1/10*(5*B*b^4*d*e^4 + (4*B*a*b^3 + A*b^4)*e^
5)*x^10 + 1/9*(10*B*b^4*d^2*e^3 + 5*(4*B*a*b^3 + A*b^4)*d*e^4 + 2*(3*B*a^2*b^2 +
2*A*a*b^3)*e^5)*x^9 + 1/4*(5*B*b^4*d^3*e^2 + 5*(4*B*a*b^3 + A*b^4)*d^2*e^3 + 5*
(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^4 + (2*B*a^3*b + 3*A*a^2*b^2)*e^5)*x^8 + 1/7*(5*B*
b^4*d^4*e + 10*(4*B*a*b^3 + A*b^4)*d^3*e^2 + 20*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^
3 + 10*(2*B*a^3*b + 3*A*a^2*b^2)*d*e^4 + (B*a^4 + 4*A*a^3*b)*e^5)*x^7 + 1/6*(B*b
^4*d^5 + A*a^4*e^5 + 5*(4*B*a*b^3 + A*b^4)*d^4*e + 20*(3*B*a^2*b^2 + 2*A*a*b^3)*
d^3*e^2 + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + 5*(B*a^4 + 4*A*a^3*b)*d*e^4)*x^
6 + 1/5*(5*A*a^4*d*e^4 + (4*B*a*b^3 + A*b^4)*d^5 + 10*(3*B*a^2*b^2 + 2*A*a*b^3)*
d^4*e + 20*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^2 + 10*(B*a^4 + 4*A*a^3*b)*d^2*e^3)*x
^5 + 1/2*(5*A*a^4*d^2*e^3 + (3*B*a^2*b^2 + 2*A*a*b^3)*d^5 + 5*(2*B*a^3*b + 3*A*a
^2*b^2)*d^4*e + 5*(B*a^4 + 4*A*a^3*b)*d^3*e^2)*x^4 + 1/3*(10*A*a^4*d^3*e^2 + 2*(
2*B*a^3*b + 3*A*a^2*b^2)*d^5 + 5*(B*a^4 + 4*A*a^3*b)*d^4*e)*x^3 + 1/2*(5*A*a^4*d
^4*e + (B*a^4 + 4*A*a^3*b)*d^5)*x^2

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Fricas [A]  time = 0.2433, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^5,x, algorithm="fricas")

[Out]

1/11*x^11*e^5*b^4*B + 1/2*x^10*e^4*d*b^4*B + 2/5*x^10*e^5*b^3*a*B + 1/10*x^10*e^
5*b^4*A + 10/9*x^9*e^3*d^2*b^4*B + 20/9*x^9*e^4*d*b^3*a*B + 2/3*x^9*e^5*b^2*a^2*
B + 5/9*x^9*e^4*d*b^4*A + 4/9*x^9*e^5*b^3*a*A + 5/4*x^8*e^2*d^3*b^4*B + 5*x^8*e^
3*d^2*b^3*a*B + 15/4*x^8*e^4*d*b^2*a^2*B + 1/2*x^8*e^5*b*a^3*B + 5/4*x^8*e^3*d^2
*b^4*A + 5/2*x^8*e^4*d*b^3*a*A + 3/4*x^8*e^5*b^2*a^2*A + 5/7*x^7*e*d^4*b^4*B + 4
0/7*x^7*e^2*d^3*b^3*a*B + 60/7*x^7*e^3*d^2*b^2*a^2*B + 20/7*x^7*e^4*d*b*a^3*B +
1/7*x^7*e^5*a^4*B + 10/7*x^7*e^2*d^3*b^4*A + 40/7*x^7*e^3*d^2*b^3*a*A + 30/7*x^7
*e^4*d*b^2*a^2*A + 4/7*x^7*e^5*b*a^3*A + 1/6*x^6*d^5*b^4*B + 10/3*x^6*e*d^4*b^3*
a*B + 10*x^6*e^2*d^3*b^2*a^2*B + 20/3*x^6*e^3*d^2*b*a^3*B + 5/6*x^6*e^4*d*a^4*B
+ 5/6*x^6*e*d^4*b^4*A + 20/3*x^6*e^2*d^3*b^3*a*A + 10*x^6*e^3*d^2*b^2*a^2*A + 10
/3*x^6*e^4*d*b*a^3*A + 1/6*x^6*e^5*a^4*A + 4/5*x^5*d^5*b^3*a*B + 6*x^5*e*d^4*b^2
*a^2*B + 8*x^5*e^2*d^3*b*a^3*B + 2*x^5*e^3*d^2*a^4*B + 1/5*x^5*d^5*b^4*A + 4*x^5
*e*d^4*b^3*a*A + 12*x^5*e^2*d^3*b^2*a^2*A + 8*x^5*e^3*d^2*b*a^3*A + x^5*e^4*d*a^
4*A + 3/2*x^4*d^5*b^2*a^2*B + 5*x^4*e*d^4*b*a^3*B + 5/2*x^4*e^2*d^3*a^4*B + x^4*
d^5*b^3*a*A + 15/2*x^4*e*d^4*b^2*a^2*A + 10*x^4*e^2*d^3*b*a^3*A + 5/2*x^4*e^3*d^
2*a^4*A + 4/3*x^3*d^5*b*a^3*B + 5/3*x^3*e*d^4*a^4*B + 2*x^3*d^5*b^2*a^2*A + 20/3
*x^3*e*d^4*b*a^3*A + 10/3*x^3*e^2*d^3*a^4*A + 1/2*x^2*d^5*a^4*B + 2*x^2*d^5*b*a^
3*A + 5/2*x^2*e*d^4*a^4*A + x*d^5*a^4*A

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Sympy [A]  time = 0.453262, size = 884, normalized size = 4.29 \[ A a^{4} d^{5} x + \frac{B b^{4} e^{5} x^{11}}{11} + x^{10} \left (\frac{A b^{4} e^{5}}{10} + \frac{2 B a b^{3} e^{5}}{5} + \frac{B b^{4} d e^{4}}{2}\right ) + x^{9} \left (\frac{4 A a b^{3} e^{5}}{9} + \frac{5 A b^{4} d e^{4}}{9} + \frac{2 B a^{2} b^{2} e^{5}}{3} + \frac{20 B a b^{3} d e^{4}}{9} + \frac{10 B b^{4} d^{2} e^{3}}{9}\right ) + x^{8} \left (\frac{3 A a^{2} b^{2} e^{5}}{4} + \frac{5 A a b^{3} d e^{4}}{2} + \frac{5 A b^{4} d^{2} e^{3}}{4} + \frac{B a^{3} b e^{5}}{2} + \frac{15 B a^{2} b^{2} d e^{4}}{4} + 5 B a b^{3} d^{2} e^{3} + \frac{5 B b^{4} d^{3} e^{2}}{4}\right ) + x^{7} \left (\frac{4 A a^{3} b e^{5}}{7} + \frac{30 A a^{2} b^{2} d e^{4}}{7} + \frac{40 A a b^{3} d^{2} e^{3}}{7} + \frac{10 A b^{4} d^{3} e^{2}}{7} + \frac{B a^{4} e^{5}}{7} + \frac{20 B a^{3} b d e^{4}}{7} + \frac{60 B a^{2} b^{2} d^{2} e^{3}}{7} + \frac{40 B a b^{3} d^{3} e^{2}}{7} + \frac{5 B b^{4} d^{4} e}{7}\right ) + x^{6} \left (\frac{A a^{4} e^{5}}{6} + \frac{10 A a^{3} b d e^{4}}{3} + 10 A a^{2} b^{2} d^{2} e^{3} + \frac{20 A a b^{3} d^{3} e^{2}}{3} + \frac{5 A b^{4} d^{4} e}{6} + \frac{5 B a^{4} d e^{4}}{6} + \frac{20 B a^{3} b d^{2} e^{3}}{3} + 10 B a^{2} b^{2} d^{3} e^{2} + \frac{10 B a b^{3} d^{4} e}{3} + \frac{B b^{4} d^{5}}{6}\right ) + x^{5} \left (A a^{4} d e^{4} + 8 A a^{3} b d^{2} e^{3} + 12 A a^{2} b^{2} d^{3} e^{2} + 4 A a b^{3} d^{4} e + \frac{A b^{4} d^{5}}{5} + 2 B a^{4} d^{2} e^{3} + 8 B a^{3} b d^{3} e^{2} + 6 B a^{2} b^{2} d^{4} e + \frac{4 B a b^{3} d^{5}}{5}\right ) + x^{4} \left (\frac{5 A a^{4} d^{2} e^{3}}{2} + 10 A a^{3} b d^{3} e^{2} + \frac{15 A a^{2} b^{2} d^{4} e}{2} + A a b^{3} d^{5} + \frac{5 B a^{4} d^{3} e^{2}}{2} + 5 B a^{3} b d^{4} e + \frac{3 B a^{2} b^{2} d^{5}}{2}\right ) + x^{3} \left (\frac{10 A a^{4} d^{3} e^{2}}{3} + \frac{20 A a^{3} b d^{4} e}{3} + 2 A a^{2} b^{2} d^{5} + \frac{5 B a^{4} d^{4} e}{3} + \frac{4 B a^{3} b d^{5}}{3}\right ) + x^{2} \left (\frac{5 A a^{4} d^{4} e}{2} + 2 A a^{3} b d^{5} + \frac{B a^{4} d^{5}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)**5*(b**2*x**2+2*a*b*x+a**2)**2,x)

[Out]

A*a**4*d**5*x + B*b**4*e**5*x**11/11 + x**10*(A*b**4*e**5/10 + 2*B*a*b**3*e**5/5
+ B*b**4*d*e**4/2) + x**9*(4*A*a*b**3*e**5/9 + 5*A*b**4*d*e**4/9 + 2*B*a**2*b**
2*e**5/3 + 20*B*a*b**3*d*e**4/9 + 10*B*b**4*d**2*e**3/9) + x**8*(3*A*a**2*b**2*e
**5/4 + 5*A*a*b**3*d*e**4/2 + 5*A*b**4*d**2*e**3/4 + B*a**3*b*e**5/2 + 15*B*a**2
*b**2*d*e**4/4 + 5*B*a*b**3*d**2*e**3 + 5*B*b**4*d**3*e**2/4) + x**7*(4*A*a**3*b
*e**5/7 + 30*A*a**2*b**2*d*e**4/7 + 40*A*a*b**3*d**2*e**3/7 + 10*A*b**4*d**3*e**
2/7 + B*a**4*e**5/7 + 20*B*a**3*b*d*e**4/7 + 60*B*a**2*b**2*d**2*e**3/7 + 40*B*a
*b**3*d**3*e**2/7 + 5*B*b**4*d**4*e/7) + x**6*(A*a**4*e**5/6 + 10*A*a**3*b*d*e**
4/3 + 10*A*a**2*b**2*d**2*e**3 + 20*A*a*b**3*d**3*e**2/3 + 5*A*b**4*d**4*e/6 + 5
*B*a**4*d*e**4/6 + 20*B*a**3*b*d**2*e**3/3 + 10*B*a**2*b**2*d**3*e**2 + 10*B*a*b
**3*d**4*e/3 + B*b**4*d**5/6) + x**5*(A*a**4*d*e**4 + 8*A*a**3*b*d**2*e**3 + 12*
A*a**2*b**2*d**3*e**2 + 4*A*a*b**3*d**4*e + A*b**4*d**5/5 + 2*B*a**4*d**2*e**3 +
8*B*a**3*b*d**3*e**2 + 6*B*a**2*b**2*d**4*e + 4*B*a*b**3*d**5/5) + x**4*(5*A*a*
*4*d**2*e**3/2 + 10*A*a**3*b*d**3*e**2 + 15*A*a**2*b**2*d**4*e/2 + A*a*b**3*d**5
+ 5*B*a**4*d**3*e**2/2 + 5*B*a**3*b*d**4*e + 3*B*a**2*b**2*d**5/2) + x**3*(10*A
*a**4*d**3*e**2/3 + 20*A*a**3*b*d**4*e/3 + 2*A*a**2*b**2*d**5 + 5*B*a**4*d**4*e/
3 + 4*B*a**3*b*d**5/3) + x**2*(5*A*a**4*d**4*e/2 + 2*A*a**3*b*d**5 + B*a**4*d**5
/2)

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GIAC/XCAS [A]  time = 0.277589, size = 1115, normalized size = 5.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2*(B*x + A)*(e*x + d)^5,x, algorithm="giac")

[Out]

1/11*B*b^4*x^11*e^5 + 1/2*B*b^4*d*x^10*e^4 + 10/9*B*b^4*d^2*x^9*e^3 + 5/4*B*b^4*
d^3*x^8*e^2 + 5/7*B*b^4*d^4*x^7*e + 1/6*B*b^4*d^5*x^6 + 2/5*B*a*b^3*x^10*e^5 + 1
/10*A*b^4*x^10*e^5 + 20/9*B*a*b^3*d*x^9*e^4 + 5/9*A*b^4*d*x^9*e^4 + 5*B*a*b^3*d^
2*x^8*e^3 + 5/4*A*b^4*d^2*x^8*e^3 + 40/7*B*a*b^3*d^3*x^7*e^2 + 10/7*A*b^4*d^3*x^
7*e^2 + 10/3*B*a*b^3*d^4*x^6*e + 5/6*A*b^4*d^4*x^6*e + 4/5*B*a*b^3*d^5*x^5 + 1/5
*A*b^4*d^5*x^5 + 2/3*B*a^2*b^2*x^9*e^5 + 4/9*A*a*b^3*x^9*e^5 + 15/4*B*a^2*b^2*d*
x^8*e^4 + 5/2*A*a*b^3*d*x^8*e^4 + 60/7*B*a^2*b^2*d^2*x^7*e^3 + 40/7*A*a*b^3*d^2*
x^7*e^3 + 10*B*a^2*b^2*d^3*x^6*e^2 + 20/3*A*a*b^3*d^3*x^6*e^2 + 6*B*a^2*b^2*d^4*
x^5*e + 4*A*a*b^3*d^4*x^5*e + 3/2*B*a^2*b^2*d^5*x^4 + A*a*b^3*d^5*x^4 + 1/2*B*a^
3*b*x^8*e^5 + 3/4*A*a^2*b^2*x^8*e^5 + 20/7*B*a^3*b*d*x^7*e^4 + 30/7*A*a^2*b^2*d*
x^7*e^4 + 20/3*B*a^3*b*d^2*x^6*e^3 + 10*A*a^2*b^2*d^2*x^6*e^3 + 8*B*a^3*b*d^3*x^
5*e^2 + 12*A*a^2*b^2*d^3*x^5*e^2 + 5*B*a^3*b*d^4*x^4*e + 15/2*A*a^2*b^2*d^4*x^4*
e + 4/3*B*a^3*b*d^5*x^3 + 2*A*a^2*b^2*d^5*x^3 + 1/7*B*a^4*x^7*e^5 + 4/7*A*a^3*b*
x^7*e^5 + 5/6*B*a^4*d*x^6*e^4 + 10/3*A*a^3*b*d*x^6*e^4 + 2*B*a^4*d^2*x^5*e^3 + 8
*A*a^3*b*d^2*x^5*e^3 + 5/2*B*a^4*d^3*x^4*e^2 + 10*A*a^3*b*d^3*x^4*e^2 + 5/3*B*a^
4*d^4*x^3*e + 20/3*A*a^3*b*d^4*x^3*e + 1/2*B*a^4*d^5*x^2 + 2*A*a^3*b*d^5*x^2 + 1
/6*A*a^4*x^6*e^5 + A*a^4*d*x^5*e^4 + 5/2*A*a^4*d^2*x^4*e^3 + 10/3*A*a^4*d^3*x^3*
e^2 + 5/2*A*a^4*d^4*x^2*e + A*a^4*d^5*x