3.1259 \(\int (a+b x)^4 (c+d x)^3 \, dx\)

Optimal. Leaf size=92 \[ \frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4} \]

[Out]

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

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Rubi [A]  time = 0.255455, 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 \[ \frac{3 d^2 (a+b x)^7 (b c-a d)}{7 b^4}+\frac{d (a+b x)^6 (b c-a d)^2}{2 b^4}+\frac{(a+b x)^5 (b c-a d)^3}{5 b^4}+\frac{d^3 (a+b x)^8}{8 b^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^4*(c + d*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 36.1538, size = 80, normalized size = 0.87 \[ \frac{d^{3} \left (a + b x\right )^{8}}{8 b^{4}} - \frac{3 d^{2} \left (a + b x\right )^{7} \left (a d - b c\right )}{7 b^{4}} + \frac{d \left (a + b x\right )^{6} \left (a d - b c\right )^{2}}{2 b^{4}} - \frac{\left (a + b x\right )^{5} \left (a d - b c\right )^{3}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**4*(d*x+c)**3,x)

[Out]

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

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Mathematica [B]  time = 0.0465297, size = 217, normalized size = 2.36 \[ a^4 c^3 x+\frac{1}{2} a^3 c^2 x^2 (3 a d+4 b c)+\frac{1}{2} b^2 d x^6 \left (2 a^2 d^2+4 a b c d+b^2 c^2\right )+a^2 c x^3 \left (a^2 d^2+4 a b c d+2 b^2 c^2\right )+\frac{1}{5} b x^5 \left (4 a^3 d^3+18 a^2 b c d^2+12 a b^2 c^2 d+b^3 c^3\right )+\frac{1}{4} a x^4 \left (a^3 d^3+12 a^2 b c d^2+18 a b^2 c^2 d+4 b^3 c^3\right )+\frac{1}{7} b^3 d^2 x^7 (4 a d+3 b c)+\frac{1}{8} b^4 d^3 x^8 \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^4*(c + d*x)^3,x]

[Out]

a^4*c^3*x + (a^3*c^2*(4*b*c + 3*a*d)*x^2)/2 + a^2*c*(2*b^2*c^2 + 4*a*b*c*d + a^2
*d^2)*x^3 + (a*(4*b^3*c^3 + 18*a*b^2*c^2*d + 12*a^2*b*c*d^2 + a^3*d^3)*x^4)/4 +
(b*(b^3*c^3 + 12*a*b^2*c^2*d + 18*a^2*b*c*d^2 + 4*a^3*d^3)*x^5)/5 + (b^2*d*(b^2*
c^2 + 4*a*b*c*d + 2*a^2*d^2)*x^6)/2 + (b^3*d^2*(3*b*c + 4*a*d)*x^7)/7 + (b^4*d^3
*x^8)/8

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Maple [B]  time = 0.001, size = 229, normalized size = 2.5 \[{\frac{{b}^{4}{d}^{3}{x}^{8}}{8}}+{\frac{ \left ( 4\,a{b}^{3}{d}^{3}+3\,{b}^{4}c{d}^{2} \right ){x}^{7}}{7}}+{\frac{ \left ( 6\,{a}^{2}{b}^{2}{d}^{3}+12\,a{b}^{3}c{d}^{2}+3\,{b}^{4}{c}^{2}d \right ){x}^{6}}{6}}+{\frac{ \left ( 4\,{a}^{3}b{d}^{3}+18\,{a}^{2}{b}^{2}c{d}^{2}+12\,a{b}^{3}{c}^{2}d+{b}^{4}{c}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ({a}^{4}{d}^{3}+12\,{a}^{3}bc{d}^{2}+18\,{a}^{2}{b}^{2}{c}^{2}d+4\,a{b}^{3}{c}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,{a}^{4}c{d}^{2}+12\,{a}^{3}b{c}^{2}d+6\,{a}^{2}{b}^{2}{c}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,{a}^{4}{c}^{2}d+4\,{a}^{3}b{c}^{3} \right ){x}^{2}}{2}}+{a}^{4}{c}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^4*(d*x+c)^3,x)

[Out]

1/8*b^4*d^3*x^8+1/7*(4*a*b^3*d^3+3*b^4*c*d^2)*x^7+1/6*(6*a^2*b^2*d^3+12*a*b^3*c*
d^2+3*b^4*c^2*d)*x^6+1/5*(4*a^3*b*d^3+18*a^2*b^2*c*d^2+12*a*b^3*c^2*d+b^4*c^3)*x
^5+1/4*(a^4*d^3+12*a^3*b*c*d^2+18*a^2*b^2*c^2*d+4*a*b^3*c^3)*x^4+1/3*(3*a^4*c*d^
2+12*a^3*b*c^2*d+6*a^2*b^2*c^3)*x^3+1/2*(3*a^4*c^2*d+4*a^3*b*c^3)*x^2+a^4*c^3*x

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Maxima [A]  time = 1.35427, size = 304, normalized size = 3.3 \[ \frac{1}{8} \, b^{4} d^{3} x^{8} + a^{4} c^{3} x + \frac{1}{7} \,{\left (3 \, b^{4} c d^{2} + 4 \, a b^{3} d^{3}\right )} x^{7} + \frac{1}{2} \,{\left (b^{4} c^{2} d + 4 \, a b^{3} c d^{2} + 2 \, a^{2} b^{2} d^{3}\right )} x^{6} + \frac{1}{5} \,{\left (b^{4} c^{3} + 12 \, a b^{3} c^{2} d + 18 \, a^{2} b^{2} c d^{2} + 4 \, a^{3} b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, a b^{3} c^{3} + 18 \, a^{2} b^{2} c^{2} d + 12 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{4} +{\left (2 \, a^{2} b^{2} c^{3} + 4 \, a^{3} b c^{2} d + a^{4} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (4 \, a^{3} b c^{3} + 3 \, a^{4} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^3,x, algorithm="maxima")

[Out]

1/8*b^4*d^3*x^8 + a^4*c^3*x + 1/7*(3*b^4*c*d^2 + 4*a*b^3*d^3)*x^7 + 1/2*(b^4*c^2
*d + 4*a*b^3*c*d^2 + 2*a^2*b^2*d^3)*x^6 + 1/5*(b^4*c^3 + 12*a*b^3*c^2*d + 18*a^2
*b^2*c*d^2 + 4*a^3*b*d^3)*x^5 + 1/4*(4*a*b^3*c^3 + 18*a^2*b^2*c^2*d + 12*a^3*b*c
*d^2 + a^4*d^3)*x^4 + (2*a^2*b^2*c^3 + 4*a^3*b*c^2*d + a^4*c*d^2)*x^3 + 1/2*(4*a
^3*b*c^3 + 3*a^4*c^2*d)*x^2

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Fricas [A]  time = 0.190238, size = 1, normalized size = 0.01 \[ \frac{1}{8} x^{8} d^{3} b^{4} + \frac{3}{7} x^{7} d^{2} c b^{4} + \frac{4}{7} x^{7} d^{3} b^{3} a + \frac{1}{2} x^{6} d c^{2} b^{4} + 2 x^{6} d^{2} c b^{3} a + x^{6} d^{3} b^{2} a^{2} + \frac{1}{5} x^{5} c^{3} b^{4} + \frac{12}{5} x^{5} d c^{2} b^{3} a + \frac{18}{5} x^{5} d^{2} c b^{2} a^{2} + \frac{4}{5} x^{5} d^{3} b a^{3} + x^{4} c^{3} b^{3} a + \frac{9}{2} x^{4} d c^{2} b^{2} a^{2} + 3 x^{4} d^{2} c b a^{3} + \frac{1}{4} x^{4} d^{3} a^{4} + 2 x^{3} c^{3} b^{2} a^{2} + 4 x^{3} d c^{2} b a^{3} + x^{3} d^{2} c a^{4} + 2 x^{2} c^{3} b a^{3} + \frac{3}{2} x^{2} d c^{2} a^{4} + x c^{3} a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.199411, size = 243, normalized size = 2.64 \[ a^{4} c^{3} x + \frac{b^{4} d^{3} x^{8}}{8} + x^{7} \left (\frac{4 a b^{3} d^{3}}{7} + \frac{3 b^{4} c d^{2}}{7}\right ) + x^{6} \left (a^{2} b^{2} d^{3} + 2 a b^{3} c d^{2} + \frac{b^{4} c^{2} d}{2}\right ) + x^{5} \left (\frac{4 a^{3} b d^{3}}{5} + \frac{18 a^{2} b^{2} c d^{2}}{5} + \frac{12 a b^{3} c^{2} d}{5} + \frac{b^{4} c^{3}}{5}\right ) + x^{4} \left (\frac{a^{4} d^{3}}{4} + 3 a^{3} b c d^{2} + \frac{9 a^{2} b^{2} c^{2} d}{2} + a b^{3} c^{3}\right ) + x^{3} \left (a^{4} c d^{2} + 4 a^{3} b c^{2} d + 2 a^{2} b^{2} c^{3}\right ) + x^{2} \left (\frac{3 a^{4} c^{2} d}{2} + 2 a^{3} b c^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**4*(d*x+c)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.21844, size = 331, normalized size = 3.6 \[ \frac{1}{8} \, b^{4} d^{3} x^{8} + \frac{3}{7} \, b^{4} c d^{2} x^{7} + \frac{4}{7} \, a b^{3} d^{3} x^{7} + \frac{1}{2} \, b^{4} c^{2} d x^{6} + 2 \, a b^{3} c d^{2} x^{6} + a^{2} b^{2} d^{3} x^{6} + \frac{1}{5} \, b^{4} c^{3} x^{5} + \frac{12}{5} \, a b^{3} c^{2} d x^{5} + \frac{18}{5} \, a^{2} b^{2} c d^{2} x^{5} + \frac{4}{5} \, a^{3} b d^{3} x^{5} + a b^{3} c^{3} x^{4} + \frac{9}{2} \, a^{2} b^{2} c^{2} d x^{4} + 3 \, a^{3} b c d^{2} x^{4} + \frac{1}{4} \, a^{4} d^{3} x^{4} + 2 \, a^{2} b^{2} c^{3} x^{3} + 4 \, a^{3} b c^{2} d x^{3} + a^{4} c d^{2} x^{3} + 2 \, a^{3} b c^{3} x^{2} + \frac{3}{2} \, a^{4} c^{2} d x^{2} + a^{4} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^3,x, algorithm="giac")

[Out]

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