3.1040 \(\int (a+b x)^6 (A+B x) (d+e x)^2 \, dx\)

Optimal. Leaf size=118 \[ \frac{e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac{(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac{B e^2 (a+b x)^{10}}{10 b^4} \]

[Out]

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

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Rubi [A]  time = 0.869071, antiderivative size = 118, 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 \[ \frac{e (a+b x)^9 (-3 a B e+A b e+2 b B d)}{9 b^4}+\frac{(a+b x)^8 (b d-a e) (-3 a B e+2 A b e+b B d)}{8 b^4}+\frac{(a+b x)^7 (A b-a B) (b d-a e)^2}{7 b^4}+\frac{B e^2 (a+b x)^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6*(A + B*x)*(d + e*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 56.3559, size = 112, normalized size = 0.95 \[ \frac{B e^{2} \left (a + b x\right )^{10}}{10 b^{4}} + \frac{e \left (a + b x\right )^{9} \left (A b e - 3 B a e + 2 B b d\right )}{9 b^{4}} - \frac{\left (a + b x\right )^{8} \left (a e - b d\right ) \left (2 A b e - 3 B a e + B b d\right )}{8 b^{4}} + \frac{\left (a + b x\right )^{7} \left (A b - B a\right ) \left (a e - b d\right )^{2}}{7 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6*(B*x+A)*(e*x+d)**2,x)

[Out]

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

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Mathematica [B]  time = 0.561149, size = 386, normalized size = 3.27 \[ \frac{x \left (210 a^6 \left (4 A \left (3 d^2+3 d e x+e^2 x^2\right )+B x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+252 a^5 b x \left (5 A \left (6 d^2+8 d e x+3 e^2 x^2\right )+2 B x \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )+630 a^4 b^2 x^2 \left (2 A \left (10 d^2+15 d e x+6 e^2 x^2\right )+B x \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )+120 a^3 b^3 x^3 \left (7 A \left (15 d^2+24 d e x+10 e^2 x^2\right )+4 B x \left (21 d^2+35 d e x+15 e^2 x^2\right )\right )+45 a^2 b^4 x^4 \left (8 A \left (21 d^2+35 d e x+15 e^2 x^2\right )+5 B x \left (28 d^2+48 d e x+21 e^2 x^2\right )\right )+30 a b^5 x^5 \left (3 A \left (28 d^2+48 d e x+21 e^2 x^2\right )+2 B x \left (36 d^2+63 d e x+28 e^2 x^2\right )\right )+b^6 x^6 \left (10 A \left (36 d^2+63 d e x+28 e^2 x^2\right )+7 B x \left (45 d^2+80 d e x+36 e^2 x^2\right )\right )\right )}{2520} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6*(A + B*x)*(d + e*x)^2,x]

[Out]

(x*(210*a^6*(4*A*(3*d^2 + 3*d*e*x + e^2*x^2) + B*x*(6*d^2 + 8*d*e*x + 3*e^2*x^2)
) + 252*a^5*b*x*(5*A*(6*d^2 + 8*d*e*x + 3*e^2*x^2) + 2*B*x*(10*d^2 + 15*d*e*x +
6*e^2*x^2)) + 630*a^4*b^2*x^2*(2*A*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + B*x*(15*d^2
+ 24*d*e*x + 10*e^2*x^2)) + 120*a^3*b^3*x^3*(7*A*(15*d^2 + 24*d*e*x + 10*e^2*x^
2) + 4*B*x*(21*d^2 + 35*d*e*x + 15*e^2*x^2)) + 45*a^2*b^4*x^4*(8*A*(21*d^2 + 35*
d*e*x + 15*e^2*x^2) + 5*B*x*(28*d^2 + 48*d*e*x + 21*e^2*x^2)) + 30*a*b^5*x^5*(3*
A*(28*d^2 + 48*d*e*x + 21*e^2*x^2) + 2*B*x*(36*d^2 + 63*d*e*x + 28*e^2*x^2)) + b
^6*x^6*(10*A*(36*d^2 + 63*d*e*x + 28*e^2*x^2) + 7*B*x*(45*d^2 + 80*d*e*x + 36*e^
2*x^2))))/2520

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Maple [B]  time = 0.003, size = 469, normalized size = 4. \[{\frac{{b}^{6}B{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ({b}^{6}A+6\,a{b}^{5}B \right ){e}^{2}+2\,{b}^{6}Bde \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ){e}^{2}+2\, \left ({b}^{6}A+6\,a{b}^{5}B \right ) de+{b}^{6}B{d}^{2} \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ){e}^{2}+2\, \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ) de+ \left ({b}^{6}A+6\,a{b}^{5}B \right ){d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ){e}^{2}+2\, \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ) de+ \left ( 6\,a{b}^{5}A+15\,{a}^{2}{b}^{4}B \right ){d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ){e}^{2}+2\, \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ) de+ \left ( 15\,{a}^{2}{b}^{4}A+20\,{a}^{3}{b}^{3}B \right ){d}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( \left ( 6\,{a}^{5}bA+{a}^{6}B \right ){e}^{2}+2\, \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ) de+ \left ( 20\,{a}^{3}{b}^{3}A+15\,{a}^{4}{b}^{2}B \right ){d}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{6}A{e}^{2}+2\, \left ( 6\,{a}^{5}bA+{a}^{6}B \right ) de+ \left ( 15\,{a}^{4}{b}^{2}A+6\,{a}^{5}bB \right ){d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{6}Ade+ \left ( 6\,{a}^{5}bA+{a}^{6}B \right ){d}^{2} \right ){x}^{2}}{2}}+{a}^{6}A{d}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6*(B*x+A)*(e*x+d)^2,x)

[Out]

1/10*b^6*B*e^2*x^10+1/9*((A*b^6+6*B*a*b^5)*e^2+2*b^6*B*d*e)*x^9+1/8*((6*A*a*b^5+
15*B*a^2*b^4)*e^2+2*(A*b^6+6*B*a*b^5)*d*e+b^6*B*d^2)*x^8+1/7*((15*A*a^2*b^4+20*B
*a^3*b^3)*e^2+2*(6*A*a*b^5+15*B*a^2*b^4)*d*e+(A*b^6+6*B*a*b^5)*d^2)*x^7+1/6*((20
*A*a^3*b^3+15*B*a^4*b^2)*e^2+2*(15*A*a^2*b^4+20*B*a^3*b^3)*d*e+(6*A*a*b^5+15*B*a
^2*b^4)*d^2)*x^6+1/5*((15*A*a^4*b^2+6*B*a^5*b)*e^2+2*(20*A*a^3*b^3+15*B*a^4*b^2)
*d*e+(15*A*a^2*b^4+20*B*a^3*b^3)*d^2)*x^5+1/4*((6*A*a^5*b+B*a^6)*e^2+2*(15*A*a^4
*b^2+6*B*a^5*b)*d*e+(20*A*a^3*b^3+15*B*a^4*b^2)*d^2)*x^4+1/3*(a^6*A*e^2+2*(6*A*a
^5*b+B*a^6)*d*e+(15*A*a^4*b^2+6*B*a^5*b)*d^2)*x^3+1/2*(2*a^6*A*d*e+(6*A*a^5*b+B*
a^6)*d^2)*x^2+a^6*A*d^2*x

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Maxima [A]  time = 1.33687, size = 643, normalized size = 5.45 \[ \frac{1}{10} \, B b^{6} e^{2} x^{10} + A a^{6} d^{2} x + \frac{1}{9} \,{\left (2 \, B b^{6} d e +{\left (6 \, B a b^{5} + A b^{6}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B b^{6} d^{2} + 2 \,{\left (6 \, B a b^{5} + A b^{6}\right )} d e + 3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} e^{2}\right )} x^{8} + \frac{1}{7} \,{\left ({\left (6 \, B a b^{5} + A b^{6}\right )} d^{2} + 6 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d e + 5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (3 \,{\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} d^{2} + 10 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d e + 5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left (5 \,{\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} d^{2} + 10 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d e + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (5 \,{\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} d^{2} + 6 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d e +{\left (B a^{6} + 6 \, A a^{5} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{6} e^{2} + 3 \,{\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} d^{2} + 2 \,{\left (B a^{6} + 6 \, A a^{5} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{6} d e +{\left (B a^{6} + 6 \, A a^{5} b\right )} d^{2}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.190751, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} e^{2} b^{6} B + \frac{2}{9} x^{9} e d b^{6} B + \frac{2}{3} x^{9} e^{2} b^{5} a B + \frac{1}{9} x^{9} e^{2} b^{6} A + \frac{1}{8} x^{8} d^{2} b^{6} B + \frac{3}{2} x^{8} e d b^{5} a B + \frac{15}{8} x^{8} e^{2} b^{4} a^{2} B + \frac{1}{4} x^{8} e d b^{6} A + \frac{3}{4} x^{8} e^{2} b^{5} a A + \frac{6}{7} x^{7} d^{2} b^{5} a B + \frac{30}{7} x^{7} e d b^{4} a^{2} B + \frac{20}{7} x^{7} e^{2} b^{3} a^{3} B + \frac{1}{7} x^{7} d^{2} b^{6} A + \frac{12}{7} x^{7} e d b^{5} a A + \frac{15}{7} x^{7} e^{2} b^{4} a^{2} A + \frac{5}{2} x^{6} d^{2} b^{4} a^{2} B + \frac{20}{3} x^{6} e d b^{3} a^{3} B + \frac{5}{2} x^{6} e^{2} b^{2} a^{4} B + x^{6} d^{2} b^{5} a A + 5 x^{6} e d b^{4} a^{2} A + \frac{10}{3} x^{6} e^{2} b^{3} a^{3} A + 4 x^{5} d^{2} b^{3} a^{3} B + 6 x^{5} e d b^{2} a^{4} B + \frac{6}{5} x^{5} e^{2} b a^{5} B + 3 x^{5} d^{2} b^{4} a^{2} A + 8 x^{5} e d b^{3} a^{3} A + 3 x^{5} e^{2} b^{2} a^{4} A + \frac{15}{4} x^{4} d^{2} b^{2} a^{4} B + 3 x^{4} e d b a^{5} B + \frac{1}{4} x^{4} e^{2} a^{6} B + 5 x^{4} d^{2} b^{3} a^{3} A + \frac{15}{2} x^{4} e d b^{2} a^{4} A + \frac{3}{2} x^{4} e^{2} b a^{5} A + 2 x^{3} d^{2} b a^{5} B + \frac{2}{3} x^{3} e d a^{6} B + 5 x^{3} d^{2} b^{2} a^{4} A + 4 x^{3} e d b a^{5} A + \frac{1}{3} x^{3} e^{2} a^{6} A + \frac{1}{2} x^{2} d^{2} a^{6} B + 3 x^{2} d^{2} b a^{5} A + x^{2} e d a^{6} A + x d^{2} a^{6} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.339248, size = 568, normalized size = 4.81 \[ A a^{6} d^{2} x + \frac{B b^{6} e^{2} x^{10}}{10} + x^{9} \left (\frac{A b^{6} e^{2}}{9} + \frac{2 B a b^{5} e^{2}}{3} + \frac{2 B b^{6} d e}{9}\right ) + x^{8} \left (\frac{3 A a b^{5} e^{2}}{4} + \frac{A b^{6} d e}{4} + \frac{15 B a^{2} b^{4} e^{2}}{8} + \frac{3 B a b^{5} d e}{2} + \frac{B b^{6} d^{2}}{8}\right ) + x^{7} \left (\frac{15 A a^{2} b^{4} e^{2}}{7} + \frac{12 A a b^{5} d e}{7} + \frac{A b^{6} d^{2}}{7} + \frac{20 B a^{3} b^{3} e^{2}}{7} + \frac{30 B a^{2} b^{4} d e}{7} + \frac{6 B a b^{5} d^{2}}{7}\right ) + x^{6} \left (\frac{10 A a^{3} b^{3} e^{2}}{3} + 5 A a^{2} b^{4} d e + A a b^{5} d^{2} + \frac{5 B a^{4} b^{2} e^{2}}{2} + \frac{20 B a^{3} b^{3} d e}{3} + \frac{5 B a^{2} b^{4} d^{2}}{2}\right ) + x^{5} \left (3 A a^{4} b^{2} e^{2} + 8 A a^{3} b^{3} d e + 3 A a^{2} b^{4} d^{2} + \frac{6 B a^{5} b e^{2}}{5} + 6 B a^{4} b^{2} d e + 4 B a^{3} b^{3} d^{2}\right ) + x^{4} \left (\frac{3 A a^{5} b e^{2}}{2} + \frac{15 A a^{4} b^{2} d e}{2} + 5 A a^{3} b^{3} d^{2} + \frac{B a^{6} e^{2}}{4} + 3 B a^{5} b d e + \frac{15 B a^{4} b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{6} e^{2}}{3} + 4 A a^{5} b d e + 5 A a^{4} b^{2} d^{2} + \frac{2 B a^{6} d e}{3} + 2 B a^{5} b d^{2}\right ) + x^{2} \left (A a^{6} d e + 3 A a^{5} b d^{2} + \frac{B a^{6} d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6*(B*x+A)*(e*x+d)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.227954, size = 745, normalized size = 6.31 \[ \frac{1}{10} \, B b^{6} x^{10} e^{2} + \frac{2}{9} \, B b^{6} d x^{9} e + \frac{1}{8} \, B b^{6} d^{2} x^{8} + \frac{2}{3} \, B a b^{5} x^{9} e^{2} + \frac{1}{9} \, A b^{6} x^{9} e^{2} + \frac{3}{2} \, B a b^{5} d x^{8} e + \frac{1}{4} \, A b^{6} d x^{8} e + \frac{6}{7} \, B a b^{5} d^{2} x^{7} + \frac{1}{7} \, A b^{6} d^{2} x^{7} + \frac{15}{8} \, B a^{2} b^{4} x^{8} e^{2} + \frac{3}{4} \, A a b^{5} x^{8} e^{2} + \frac{30}{7} \, B a^{2} b^{4} d x^{7} e + \frac{12}{7} \, A a b^{5} d x^{7} e + \frac{5}{2} \, B a^{2} b^{4} d^{2} x^{6} + A a b^{5} d^{2} x^{6} + \frac{20}{7} \, B a^{3} b^{3} x^{7} e^{2} + \frac{15}{7} \, A a^{2} b^{4} x^{7} e^{2} + \frac{20}{3} \, B a^{3} b^{3} d x^{6} e + 5 \, A a^{2} b^{4} d x^{6} e + 4 \, B a^{3} b^{3} d^{2} x^{5} + 3 \, A a^{2} b^{4} d^{2} x^{5} + \frac{5}{2} \, B a^{4} b^{2} x^{6} e^{2} + \frac{10}{3} \, A a^{3} b^{3} x^{6} e^{2} + 6 \, B a^{4} b^{2} d x^{5} e + 8 \, A a^{3} b^{3} d x^{5} e + \frac{15}{4} \, B a^{4} b^{2} d^{2} x^{4} + 5 \, A a^{3} b^{3} d^{2} x^{4} + \frac{6}{5} \, B a^{5} b x^{5} e^{2} + 3 \, A a^{4} b^{2} x^{5} e^{2} + 3 \, B a^{5} b d x^{4} e + \frac{15}{2} \, A a^{4} b^{2} d x^{4} e + 2 \, B a^{5} b d^{2} x^{3} + 5 \, A a^{4} b^{2} d^{2} x^{3} + \frac{1}{4} \, B a^{6} x^{4} e^{2} + \frac{3}{2} \, A a^{5} b x^{4} e^{2} + \frac{2}{3} \, B a^{6} d x^{3} e + 4 \, A a^{5} b d x^{3} e + \frac{1}{2} \, B a^{6} d^{2} x^{2} + 3 \, A a^{5} b d^{2} x^{2} + \frac{1}{3} \, A a^{6} x^{3} e^{2} + A a^{6} d x^{2} e + A a^{6} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^6*(e*x + d)^2,x, algorithm="giac")

[Out]

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