3.1260 \(\int (a+b x)^3 (c+d x)^3 \, dx\)

Optimal. Leaf size=92 \[ \frac{d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac{3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac{(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac{d^3 (a+b x)^7}{7 b^4} \]

[Out]

((b*c - a*d)^3*(a + b*x)^4)/(4*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^5)/(5*b^4) +
(d^2*(b*c - a*d)*(a + b*x)^6)/(2*b^4) + (d^3*(a + b*x)^7)/(7*b^4)

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Rubi [A]  time = 0.179506, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{d^2 (a+b x)^6 (b c-a d)}{2 b^4}+\frac{3 d (a+b x)^5 (b c-a d)^2}{5 b^4}+\frac{(a+b x)^4 (b c-a d)^3}{4 b^4}+\frac{d^3 (a+b x)^7}{7 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(c + d*x)^3,x]

[Out]

((b*c - a*d)^3*(a + b*x)^4)/(4*b^4) + (3*d*(b*c - a*d)^2*(a + b*x)^5)/(5*b^4) +
(d^2*(b*c - a*d)*(a + b*x)^6)/(2*b^4) + (d^3*(a + b*x)^7)/(7*b^4)

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Rubi in Sympy [A]  time = 29.1398, size = 80, normalized size = 0.87 \[ \frac{d^{3} \left (a + b x\right )^{7}}{7 b^{4}} - \frac{d^{2} \left (a + b x\right )^{6} \left (a d - b c\right )}{2 b^{4}} + \frac{3 d \left (a + b x\right )^{5} \left (a d - b c\right )^{2}}{5 b^{4}} - \frac{\left (a + b x\right )^{4} \left (a d - b c\right )^{3}}{4 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*(a + b*x)**7/(7*b**4) - d**2*(a + b*x)**6*(a*d - b*c)/(2*b**4) + 3*d*(a + b
*x)**5*(a*d - b*c)**2/(5*b**4) - (a + b*x)**4*(a*d - b*c)**3/(4*b**4)

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Mathematica [A]  time = 0.0351396, size = 161, normalized size = 1.75 \[ a^3 c^3 x+\frac{3}{5} b d x^5 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+a c x^3 \left (a^2 d^2+3 a b c d+b^2 c^2\right )+\frac{3}{2} a^2 c^2 x^2 (a d+b c)+\frac{1}{4} x^4 \left (a^3 d^3+9 a^2 b c d^2+9 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{2} b^2 d^2 x^6 (a d+b c)+\frac{1}{7} b^3 d^3 x^7 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.001, size = 177, normalized size = 1.9 \[{\frac{{b}^{3}{d}^{3}{x}^{7}}{7}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{3}+3\,{b}^{3}c{d}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{3}+9\,a{b}^{2}c{d}^{2}+3\,{b}^{3}{c}^{2}d \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{3}{d}^{3}+9\,{a}^{2}bc{d}^{2}+9\,a{b}^{2}{c}^{2}d+{b}^{3}{c}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{3}c{d}^{2}+9\,{a}^{2}b{c}^{2}d+3\,a{b}^{2}{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{3}{c}^{2}d+3\,{a}^{2}b{c}^{3} \right ){x}^{2}}{2}}+{a}^{3}{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.36666, size = 225, normalized size = 2.45 \[ \frac{1}{7} \, b^{3} d^{3} x^{7} + a^{3} c^{3} x + \frac{1}{2} \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{6} + \frac{3}{5} \,{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{4} +{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{3} + \frac{3}{2} \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.181809, size = 1, normalized size = 0.01 \[ \frac{1}{7} x^{7} d^{3} b^{3} + \frac{1}{2} x^{6} d^{2} c b^{3} + \frac{1}{2} x^{6} d^{3} b^{2} a + \frac{3}{5} x^{5} d c^{2} b^{3} + \frac{9}{5} x^{5} d^{2} c b^{2} a + \frac{3}{5} x^{5} d^{3} b a^{2} + \frac{1}{4} x^{4} c^{3} b^{3} + \frac{9}{4} x^{4} d c^{2} b^{2} a + \frac{9}{4} x^{4} d^{2} c b a^{2} + \frac{1}{4} x^{4} d^{3} a^{3} + x^{3} c^{3} b^{2} a + 3 x^{3} d c^{2} b a^{2} + x^{3} d^{2} c a^{3} + \frac{3}{2} x^{2} c^{3} b a^{2} + \frac{3}{2} x^{2} d c^{2} a^{3} + x c^{3} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.167046, size = 190, normalized size = 2.07 \[ a^{3} c^{3} x + \frac{b^{3} d^{3} x^{7}}{7} + x^{6} \left (\frac{a b^{2} d^{3}}{2} + \frac{b^{3} c d^{2}}{2}\right ) + x^{5} \left (\frac{3 a^{2} b d^{3}}{5} + \frac{9 a b^{2} c d^{2}}{5} + \frac{3 b^{3} c^{2} d}{5}\right ) + x^{4} \left (\frac{a^{3} d^{3}}{4} + \frac{9 a^{2} b c d^{2}}{4} + \frac{9 a b^{2} c^{2} d}{4} + \frac{b^{3} c^{3}}{4}\right ) + x^{3} \left (a^{3} c d^{2} + 3 a^{2} b c^{2} d + a b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{3} c^{2} d}{2} + \frac{3 a^{2} b c^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220735, size = 254, normalized size = 2.76 \[ \frac{1}{7} \, b^{3} d^{3} x^{7} + \frac{1}{2} \, b^{3} c d^{2} x^{6} + \frac{1}{2} \, a b^{2} d^{3} x^{6} + \frac{3}{5} \, b^{3} c^{2} d x^{5} + \frac{9}{5} \, a b^{2} c d^{2} x^{5} + \frac{3}{5} \, a^{2} b d^{3} x^{5} + \frac{1}{4} \, b^{3} c^{3} x^{4} + \frac{9}{4} \, a b^{2} c^{2} d x^{4} + \frac{9}{4} \, a^{2} b c d^{2} x^{4} + \frac{1}{4} \, a^{3} d^{3} x^{4} + a b^{2} c^{3} x^{3} + 3 \, a^{2} b c^{2} d x^{3} + a^{3} c d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{3} x^{2} + \frac{3}{2} \, a^{3} c^{2} d x^{2} + a^{3} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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