3.1020 \(\int (a+b x)^3 (A+B x) (d+e x)^3 \, dx\)

Optimal. Leaf size=159 \[ \frac{e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 (a+b x)^8}{8 b^5} \]

[Out]

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

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Rubi [A]  time = 0.581042, antiderivative size = 159, 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 \[ \frac{e^2 (a+b x)^7 (-4 a B e+A b e+3 b B d)}{7 b^5}+\frac{e (a+b x)^6 (b d-a e) (-2 a B e+A b e+b B d)}{2 b^5}+\frac{(a+b x)^5 (b d-a e)^2 (-4 a B e+3 A b e+b B d)}{5 b^5}+\frac{(a+b x)^4 (A b-a B) (b d-a e)^3}{4 b^5}+\frac{B e^3 (a+b x)^8}{8 b^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(A + B*x)*(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 72.6691, size = 153, normalized size = 0.96 \[ \frac{B e^{3} \left (a + b x\right )^{8}}{8 b^{5}} + \frac{e^{2} \left (a + b x\right )^{7} \left (A b e - 4 B a e + 3 B b d\right )}{7 b^{5}} - \frac{e \left (a + b x\right )^{6} \left (a e - b d\right ) \left (A b e - 2 B a e + B b d\right )}{2 b^{5}} + \frac{\left (a + b x\right )^{5} \left (a e - b d\right )^{2} \left (3 A b e - 4 B a e + B b d\right )}{5 b^{5}} - \frac{\left (a + b x\right )^{4} \left (A b - B a\right ) \left (a e - b d\right )^{3}}{4 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)*(e*x+d)**3,x)

[Out]

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

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Mathematica [A]  time = 0.181243, size = 297, normalized size = 1.87 \[ a^3 A d^3 x+a d x^3 \left (A \left (a^2 e^2+3 a b d e+b^2 d^2\right )+a B d (a e+b d)\right )+\frac{1}{2} b e x^6 \left (a^2 B e^2+a b e (A e+3 B d)+b^2 d (A e+B d)\right )+\frac{1}{2} a^2 d^2 x^2 (3 A (a e+b d)+a B d)+\frac{1}{5} x^5 \left (a^3 B e^3+3 a^2 b e^2 (A e+3 B d)+9 a b^2 d e (A e+B d)+b^3 d^2 (3 A e+B d)\right )+\frac{1}{4} x^4 \left (3 a B d \left (a^2 e^2+3 a b d e+b^2 d^2\right )+A \left (a^3 e^3+9 a^2 b d e^2+9 a b^2 d^2 e+b^3 d^3\right )\right )+\frac{1}{7} b^2 e^2 x^7 (3 a B e+A b e+3 b B d)+\frac{1}{8} b^3 B e^3 x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^3,x]

[Out]

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

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Maple [B]  time = 0.001, size = 339, normalized size = 2.1 \[{\frac{{b}^{3}B{e}^{3}{x}^{8}}{8}}+{\frac{ \left ( \left ({b}^{3}A+3\,a{b}^{2}B \right ){e}^{3}+3\,{b}^{3}Bd{e}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){e}^{3}+3\, \left ({b}^{3}A+3\,a{b}^{2}B \right ) d{e}^{2}+3\,{b}^{3}B{d}^{2}e \right ){x}^{6}}{6}}+{\frac{ \left ( \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){e}^{3}+3\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ) d{e}^{2}+3\, \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{2}e+{b}^{3}B{d}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}A{e}^{3}+3\, \left ( 3\,A{a}^{2}b+B{a}^{3} \right ) d{e}^{2}+3\, \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{2}e+ \left ({b}^{3}A+3\,a{b}^{2}B \right ){d}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}Ad{e}^{2}+3\, \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){d}^{2}e+ \left ( 3\,a{b}^{2}A+3\,{a}^{2}bB \right ){d}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}A{d}^{2}e+ \left ( 3\,A{a}^{2}b+B{a}^{3} \right ){d}^{3} \right ){x}^{2}}{2}}+{a}^{3}A{d}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)*(e*x+d)^3,x)

[Out]

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

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Maxima [A]  time = 1.35037, size = 439, normalized size = 2.76 \[ \frac{1}{8} \, B b^{3} e^{3} x^{8} + A a^{3} d^{3} x + \frac{1}{7} \,{\left (3 \, B b^{3} d e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{3}\right )} x^{7} + \frac{1}{2} \,{\left (B b^{3} d^{2} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} e^{3}\right )} x^{6} + \frac{1}{5} \,{\left (B b^{3} d^{3} + 3 \,{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{2} +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (A a^{3} e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} + 9 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e + 3 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{2}\right )} x^{4} +{\left (A a^{3} d e^{2} +{\left (B a^{2} b + A a b^{2}\right )} d^{3} +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e\right )} x^{3} + \frac{1}{2} \,{\left (3 \, A a^{3} d^{2} e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.1921, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} e^{3} b^{3} B + \frac{3}{7} x^{7} e^{2} d b^{3} B + \frac{3}{7} x^{7} e^{3} b^{2} a B + \frac{1}{7} x^{7} e^{3} b^{3} A + \frac{1}{2} x^{6} e d^{2} b^{3} B + \frac{3}{2} x^{6} e^{2} d b^{2} a B + \frac{1}{2} x^{6} e^{3} b a^{2} B + \frac{1}{2} x^{6} e^{2} d b^{3} A + \frac{1}{2} x^{6} e^{3} b^{2} a A + \frac{1}{5} x^{5} d^{3} b^{3} B + \frac{9}{5} x^{5} e d^{2} b^{2} a B + \frac{9}{5} x^{5} e^{2} d b a^{2} B + \frac{1}{5} x^{5} e^{3} a^{3} B + \frac{3}{5} x^{5} e d^{2} b^{3} A + \frac{9}{5} x^{5} e^{2} d b^{2} a A + \frac{3}{5} x^{5} e^{3} b a^{2} A + \frac{3}{4} x^{4} d^{3} b^{2} a B + \frac{9}{4} x^{4} e d^{2} b a^{2} B + \frac{3}{4} x^{4} e^{2} d a^{3} B + \frac{1}{4} x^{4} d^{3} b^{3} A + \frac{9}{4} x^{4} e d^{2} b^{2} a A + \frac{9}{4} x^{4} e^{2} d b a^{2} A + \frac{1}{4} x^{4} e^{3} a^{3} A + x^{3} d^{3} b a^{2} B + x^{3} e d^{2} a^{3} B + x^{3} d^{3} b^{2} a A + 3 x^{3} e d^{2} b a^{2} A + x^{3} e^{2} d a^{3} A + \frac{1}{2} x^{2} d^{3} a^{3} B + \frac{3}{2} x^{2} d^{3} b a^{2} A + \frac{3}{2} x^{2} e d^{2} a^{3} A + x d^{3} a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.280518, size = 422, normalized size = 2.65 \[ A a^{3} d^{3} x + \frac{B b^{3} e^{3} x^{8}}{8} + x^{7} \left (\frac{A b^{3} e^{3}}{7} + \frac{3 B a b^{2} e^{3}}{7} + \frac{3 B b^{3} d e^{2}}{7}\right ) + x^{6} \left (\frac{A a b^{2} e^{3}}{2} + \frac{A b^{3} d e^{2}}{2} + \frac{B a^{2} b e^{3}}{2} + \frac{3 B a b^{2} d e^{2}}{2} + \frac{B b^{3} d^{2} e}{2}\right ) + x^{5} \left (\frac{3 A a^{2} b e^{3}}{5} + \frac{9 A a b^{2} d e^{2}}{5} + \frac{3 A b^{3} d^{2} e}{5} + \frac{B a^{3} e^{3}}{5} + \frac{9 B a^{2} b d e^{2}}{5} + \frac{9 B a b^{2} d^{2} e}{5} + \frac{B b^{3} d^{3}}{5}\right ) + x^{4} \left (\frac{A a^{3} e^{3}}{4} + \frac{9 A a^{2} b d e^{2}}{4} + \frac{9 A a b^{2} d^{2} e}{4} + \frac{A b^{3} d^{3}}{4} + \frac{3 B a^{3} d e^{2}}{4} + \frac{9 B a^{2} b d^{2} e}{4} + \frac{3 B a b^{2} d^{3}}{4}\right ) + x^{3} \left (A a^{3} d e^{2} + 3 A a^{2} b d^{2} e + A a b^{2} d^{3} + B a^{3} d^{2} e + B a^{2} b d^{3}\right ) + x^{2} \left (\frac{3 A a^{3} d^{2} e}{2} + \frac{3 A a^{2} b d^{3}}{2} + \frac{B a^{3} d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)*(e*x+d)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218806, size = 543, normalized size = 3.42 \[ \frac{1}{8} \, B b^{3} x^{8} e^{3} + \frac{3}{7} \, B b^{3} d x^{7} e^{2} + \frac{1}{2} \, B b^{3} d^{2} x^{6} e + \frac{1}{5} \, B b^{3} d^{3} x^{5} + \frac{3}{7} \, B a b^{2} x^{7} e^{3} + \frac{1}{7} \, A b^{3} x^{7} e^{3} + \frac{3}{2} \, B a b^{2} d x^{6} e^{2} + \frac{1}{2} \, A b^{3} d x^{6} e^{2} + \frac{9}{5} \, B a b^{2} d^{2} x^{5} e + \frac{3}{5} \, A b^{3} d^{2} x^{5} e + \frac{3}{4} \, B a b^{2} d^{3} x^{4} + \frac{1}{4} \, A b^{3} d^{3} x^{4} + \frac{1}{2} \, B a^{2} b x^{6} e^{3} + \frac{1}{2} \, A a b^{2} x^{6} e^{3} + \frac{9}{5} \, B a^{2} b d x^{5} e^{2} + \frac{9}{5} \, A a b^{2} d x^{5} e^{2} + \frac{9}{4} \, B a^{2} b d^{2} x^{4} e + \frac{9}{4} \, A a b^{2} d^{2} x^{4} e + B a^{2} b d^{3} x^{3} + A a b^{2} d^{3} x^{3} + \frac{1}{5} \, B a^{3} x^{5} e^{3} + \frac{3}{5} \, A a^{2} b x^{5} e^{3} + \frac{3}{4} \, B a^{3} d x^{4} e^{2} + \frac{9}{4} \, A a^{2} b d x^{4} e^{2} + B a^{3} d^{2} x^{3} e + 3 \, A a^{2} b d^{2} x^{3} e + \frac{1}{2} \, B a^{3} d^{3} x^{2} + \frac{3}{2} \, A a^{2} b d^{3} x^{2} + \frac{1}{4} \, A a^{3} x^{4} e^{3} + A a^{3} d x^{3} e^{2} + \frac{3}{2} \, A a^{3} d^{2} x^{2} e + A a^{3} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3*(e*x + d)^3,x, algorithm="giac")

[Out]

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