3.1772 \(\int (a+b x)^3 \left (a c+(b c+a d) x+b d x^2\right )^3 \, dx\)

Optimal. Leaf size=92 \[ \frac{d^2 (a+b x)^9 (b c-a d)}{3 b^4}+\frac{3 d (a+b x)^8 (b c-a d)^2}{8 b^4}+\frac{(a+b x)^7 (b c-a d)^3}{7 b^4}+\frac{d^3 (a+b x)^{10}}{10 b^4} \]

[Out]

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

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Rubi [A]  time = 0.493537, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{d^2 (a+b x)^9 (b c-a d)}{3 b^4}+\frac{3 d (a+b x)^8 (b c-a d)^2}{8 b^4}+\frac{(a+b x)^7 (b c-a d)^3}{7 b^4}+\frac{d^3 (a+b x)^{10}}{10 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 57.9183, size = 80, normalized size = 0.87 \[ \frac{d^{3} \left (a + b x\right )^{10}}{10 b^{4}} - \frac{d^{2} \left (a + b x\right )^{9} \left (a d - b c\right )}{3 b^{4}} + \frac{3 d \left (a + b x\right )^{8} \left (a d - b c\right )^{2}}{8 b^{4}} - \frac{\left (a + b x\right )^{7} \left (a d - b c\right )^{3}}{7 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

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

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Mathematica [B]  time = 0.157285, size = 276, normalized size = 3. \[ \frac{1}{840} x \left (210 a^6 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+252 a^5 b x \left (10 c^3+20 c^2 d x+15 c d^2 x^2+4 d^3 x^3\right )+210 a^4 b^2 x^2 \left (20 c^3+45 c^2 d x+36 c d^2 x^2+10 d^3 x^3\right )+120 a^3 b^3 x^3 \left (35 c^3+84 c^2 d x+70 c d^2 x^2+20 d^3 x^3\right )+45 a^2 b^4 x^4 \left (56 c^3+140 c^2 d x+120 c d^2 x^2+35 d^3 x^3\right )+10 a b^5 x^5 \left (84 c^3+216 c^2 d x+189 c d^2 x^2+56 d^3 x^3\right )+b^6 x^6 \left (120 c^3+315 c^2 d x+280 c d^2 x^2+84 d^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^3*(a*c + (b*c + a*d)*x + b*d*x^2)^3,x]

[Out]

(x*(210*a^6*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + 252*a^5*b*x*(10*c^3 +
20*c^2*d*x + 15*c*d^2*x^2 + 4*d^3*x^3) + 210*a^4*b^2*x^2*(20*c^3 + 45*c^2*d*x +
36*c*d^2*x^2 + 10*d^3*x^3) + 120*a^3*b^3*x^3*(35*c^3 + 84*c^2*d*x + 70*c*d^2*x^2
+ 20*d^3*x^3) + 45*a^2*b^4*x^4*(56*c^3 + 140*c^2*d*x + 120*c*d^2*x^2 + 35*d^3*x
^3) + 10*a*b^5*x^5*(84*c^3 + 216*c^2*d*x + 189*c*d^2*x^2 + 56*d^3*x^3) + b^6*x^6
*(120*c^3 + 315*c^2*d*x + 280*c*d^2*x^2 + 84*d^3*x^3)))/840

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Maple [B]  time = 0.003, size = 811, normalized size = 8.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(a*c+(a*d+b*c)*x+x^2*b*d)^3,x)

[Out]

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

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Maxima [A]  time = 0.741085, size = 441, normalized size = 4.79 \[ \frac{1}{10} \, b^{6} d^{3} x^{10} + a^{6} c^{3} x + \frac{1}{3} \,{\left (b^{6} c d^{2} + 2 \, a b^{5} d^{3}\right )} x^{9} + \frac{3}{8} \,{\left (b^{6} c^{2} d + 6 \, a b^{5} c d^{2} + 5 \, a^{2} b^{4} d^{3}\right )} x^{8} + \frac{1}{7} \,{\left (b^{6} c^{3} + 18 \, a b^{5} c^{2} d + 45 \, a^{2} b^{4} c d^{2} + 20 \, a^{3} b^{3} d^{3}\right )} x^{7} + \frac{1}{2} \,{\left (2 \, a b^{5} c^{3} + 15 \, a^{2} b^{4} c^{2} d + 20 \, a^{3} b^{3} c d^{2} + 5 \, a^{4} b^{2} d^{3}\right )} x^{6} + \frac{3}{5} \,{\left (5 \, a^{2} b^{4} c^{3} + 20 \, a^{3} b^{3} c^{2} d + 15 \, a^{4} b^{2} c d^{2} + 2 \, a^{5} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (20 \, a^{3} b^{3} c^{3} + 45 \, a^{4} b^{2} c^{2} d + 18 \, a^{5} b c d^{2} + a^{6} d^{3}\right )} x^{4} +{\left (5 \, a^{4} b^{2} c^{3} + 6 \, a^{5} b c^{2} d + a^{6} c d^{2}\right )} x^{3} + \frac{3}{2} \,{\left (2 \, a^{5} b c^{3} + a^{6} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a)^3,x, algorithm="maxima")

[Out]

1/10*b^6*d^3*x^10 + a^6*c^3*x + 1/3*(b^6*c*d^2 + 2*a*b^5*d^3)*x^9 + 3/8*(b^6*c^2
*d + 6*a*b^5*c*d^2 + 5*a^2*b^4*d^3)*x^8 + 1/7*(b^6*c^3 + 18*a*b^5*c^2*d + 45*a^2
*b^4*c*d^2 + 20*a^3*b^3*d^3)*x^7 + 1/2*(2*a*b^5*c^3 + 15*a^2*b^4*c^2*d + 20*a^3*
b^3*c*d^2 + 5*a^4*b^2*d^3)*x^6 + 3/5*(5*a^2*b^4*c^3 + 20*a^3*b^3*c^2*d + 15*a^4*
b^2*c*d^2 + 2*a^5*b*d^3)*x^5 + 1/4*(20*a^3*b^3*c^3 + 45*a^4*b^2*c^2*d + 18*a^5*b
*c*d^2 + a^6*d^3)*x^4 + (5*a^4*b^2*c^3 + 6*a^5*b*c^2*d + a^6*c*d^2)*x^3 + 3/2*(2
*a^5*b*c^3 + a^6*c^2*d)*x^2

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Fricas [A]  time = 0.180223, size = 1, normalized size = 0.01 \[ \frac{1}{10} x^{10} d^{3} b^{6} + \frac{1}{3} x^{9} d^{2} c b^{6} + \frac{2}{3} x^{9} d^{3} b^{5} a + \frac{3}{8} x^{8} d c^{2} b^{6} + \frac{9}{4} x^{8} d^{2} c b^{5} a + \frac{15}{8} x^{8} d^{3} b^{4} a^{2} + \frac{1}{7} x^{7} c^{3} b^{6} + \frac{18}{7} x^{7} d c^{2} b^{5} a + \frac{45}{7} x^{7} d^{2} c b^{4} a^{2} + \frac{20}{7} x^{7} d^{3} b^{3} a^{3} + x^{6} c^{3} b^{5} a + \frac{15}{2} x^{6} d c^{2} b^{4} a^{2} + 10 x^{6} d^{2} c b^{3} a^{3} + \frac{5}{2} x^{6} d^{3} b^{2} a^{4} + 3 x^{5} c^{3} b^{4} a^{2} + 12 x^{5} d c^{2} b^{3} a^{3} + 9 x^{5} d^{2} c b^{2} a^{4} + \frac{6}{5} x^{5} d^{3} b a^{5} + 5 x^{4} c^{3} b^{3} a^{3} + \frac{45}{4} x^{4} d c^{2} b^{2} a^{4} + \frac{9}{2} x^{4} d^{2} c b a^{5} + \frac{1}{4} x^{4} d^{3} a^{6} + 5 x^{3} c^{3} b^{2} a^{4} + 6 x^{3} d c^{2} b a^{5} + x^{3} d^{2} c a^{6} + 3 x^{2} c^{3} b a^{5} + \frac{3}{2} x^{2} d c^{2} a^{6} + x c^{3} a^{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a)^3,x, algorithm="fricas")

[Out]

1/10*x^10*d^3*b^6 + 1/3*x^9*d^2*c*b^6 + 2/3*x^9*d^3*b^5*a + 3/8*x^8*d*c^2*b^6 +
9/4*x^8*d^2*c*b^5*a + 15/8*x^8*d^3*b^4*a^2 + 1/7*x^7*c^3*b^6 + 18/7*x^7*d*c^2*b^
5*a + 45/7*x^7*d^2*c*b^4*a^2 + 20/7*x^7*d^3*b^3*a^3 + x^6*c^3*b^5*a + 15/2*x^6*d
*c^2*b^4*a^2 + 10*x^6*d^2*c*b^3*a^3 + 5/2*x^6*d^3*b^2*a^4 + 3*x^5*c^3*b^4*a^2 +
12*x^5*d*c^2*b^3*a^3 + 9*x^5*d^2*c*b^2*a^4 + 6/5*x^5*d^3*b*a^5 + 5*x^4*c^3*b^3*a
^3 + 45/4*x^4*d*c^2*b^2*a^4 + 9/2*x^4*d^2*c*b*a^5 + 1/4*x^4*d^3*a^6 + 5*x^3*c^3*
b^2*a^4 + 6*x^3*d*c^2*b*a^5 + x^3*d^2*c*a^6 + 3*x^2*c^3*b*a^5 + 3/2*x^2*d*c^2*a^
6 + x*c^3*a^6

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Sympy [A]  time = 0.36601, size = 364, normalized size = 3.96 \[ a^{6} c^{3} x + \frac{b^{6} d^{3} x^{10}}{10} + x^{9} \left (\frac{2 a b^{5} d^{3}}{3} + \frac{b^{6} c d^{2}}{3}\right ) + x^{8} \left (\frac{15 a^{2} b^{4} d^{3}}{8} + \frac{9 a b^{5} c d^{2}}{4} + \frac{3 b^{6} c^{2} d}{8}\right ) + x^{7} \left (\frac{20 a^{3} b^{3} d^{3}}{7} + \frac{45 a^{2} b^{4} c d^{2}}{7} + \frac{18 a b^{5} c^{2} d}{7} + \frac{b^{6} c^{3}}{7}\right ) + x^{6} \left (\frac{5 a^{4} b^{2} d^{3}}{2} + 10 a^{3} b^{3} c d^{2} + \frac{15 a^{2} b^{4} c^{2} d}{2} + a b^{5} c^{3}\right ) + x^{5} \left (\frac{6 a^{5} b d^{3}}{5} + 9 a^{4} b^{2} c d^{2} + 12 a^{3} b^{3} c^{2} d + 3 a^{2} b^{4} c^{3}\right ) + x^{4} \left (\frac{a^{6} d^{3}}{4} + \frac{9 a^{5} b c d^{2}}{2} + \frac{45 a^{4} b^{2} c^{2} d}{4} + 5 a^{3} b^{3} c^{3}\right ) + x^{3} \left (a^{6} c d^{2} + 6 a^{5} b c^{2} d + 5 a^{4} b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{6} c^{2} d}{2} + 3 a^{5} b c^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(a*c+(a*d+b*c)*x+b*d*x**2)**3,x)

[Out]

a**6*c**3*x + b**6*d**3*x**10/10 + x**9*(2*a*b**5*d**3/3 + b**6*c*d**2/3) + x**8
*(15*a**2*b**4*d**3/8 + 9*a*b**5*c*d**2/4 + 3*b**6*c**2*d/8) + x**7*(20*a**3*b**
3*d**3/7 + 45*a**2*b**4*c*d**2/7 + 18*a*b**5*c**2*d/7 + b**6*c**3/7) + x**6*(5*a
**4*b**2*d**3/2 + 10*a**3*b**3*c*d**2 + 15*a**2*b**4*c**2*d/2 + a*b**5*c**3) + x
**5*(6*a**5*b*d**3/5 + 9*a**4*b**2*c*d**2 + 12*a**3*b**3*c**2*d + 3*a**2*b**4*c*
*3) + x**4*(a**6*d**3/4 + 9*a**5*b*c*d**2/2 + 45*a**4*b**2*c**2*d/4 + 5*a**3*b**
3*c**3) + x**3*(a**6*c*d**2 + 6*a**5*b*c**2*d + 5*a**4*b**2*c**3) + x**2*(3*a**6
*c**2*d/2 + 3*a**5*b*c**3)

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GIAC/XCAS [A]  time = 0.209842, size = 489, normalized size = 5.32 \[ \frac{1}{10} \, b^{6} d^{3} x^{10} + \frac{1}{3} \, b^{6} c d^{2} x^{9} + \frac{2}{3} \, a b^{5} d^{3} x^{9} + \frac{3}{8} \, b^{6} c^{2} d x^{8} + \frac{9}{4} \, a b^{5} c d^{2} x^{8} + \frac{15}{8} \, a^{2} b^{4} d^{3} x^{8} + \frac{1}{7} \, b^{6} c^{3} x^{7} + \frac{18}{7} \, a b^{5} c^{2} d x^{7} + \frac{45}{7} \, a^{2} b^{4} c d^{2} x^{7} + \frac{20}{7} \, a^{3} b^{3} d^{3} x^{7} + a b^{5} c^{3} x^{6} + \frac{15}{2} \, a^{2} b^{4} c^{2} d x^{6} + 10 \, a^{3} b^{3} c d^{2} x^{6} + \frac{5}{2} \, a^{4} b^{2} d^{3} x^{6} + 3 \, a^{2} b^{4} c^{3} x^{5} + 12 \, a^{3} b^{3} c^{2} d x^{5} + 9 \, a^{4} b^{2} c d^{2} x^{5} + \frac{6}{5} \, a^{5} b d^{3} x^{5} + 5 \, a^{3} b^{3} c^{3} x^{4} + \frac{45}{4} \, a^{4} b^{2} c^{2} d x^{4} + \frac{9}{2} \, a^{5} b c d^{2} x^{4} + \frac{1}{4} \, a^{6} d^{3} x^{4} + 5 \, a^{4} b^{2} c^{3} x^{3} + 6 \, a^{5} b c^{2} d x^{3} + a^{6} c d^{2} x^{3} + 3 \, a^{5} b c^{3} x^{2} + \frac{3}{2} \, a^{6} c^{2} d x^{2} + a^{6} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*d*x^2 + a*c + (b*c + a*d)*x)^3*(b*x + a)^3,x, algorithm="giac")

[Out]

1/10*b^6*d^3*x^10 + 1/3*b^6*c*d^2*x^9 + 2/3*a*b^5*d^3*x^9 + 3/8*b^6*c^2*d*x^8 +
9/4*a*b^5*c*d^2*x^8 + 15/8*a^2*b^4*d^3*x^8 + 1/7*b^6*c^3*x^7 + 18/7*a*b^5*c^2*d*
x^7 + 45/7*a^2*b^4*c*d^2*x^7 + 20/7*a^3*b^3*d^3*x^7 + a*b^5*c^3*x^6 + 15/2*a^2*b
^4*c^2*d*x^6 + 10*a^3*b^3*c*d^2*x^6 + 5/2*a^4*b^2*d^3*x^6 + 3*a^2*b^4*c^3*x^5 +
12*a^3*b^3*c^2*d*x^5 + 9*a^4*b^2*c*d^2*x^5 + 6/5*a^5*b*d^3*x^5 + 5*a^3*b^3*c^3*x
^4 + 45/4*a^4*b^2*c^2*d*x^4 + 9/2*a^5*b*c*d^2*x^4 + 1/4*a^6*d^3*x^4 + 5*a^4*b^2*
c^3*x^3 + 6*a^5*b*c^2*d*x^3 + a^6*c*d^2*x^3 + 3*a^5*b*c^3*x^2 + 3/2*a^6*c^2*d*x^
2 + a^6*c^3*x