3.1247 \(\int (a+b x)^3 (c+d x)^2 \, dx\)

Optimal. Leaf size=65 \[ \frac{2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac{(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac{d^2 (a+b x)^6}{6 b^3} \]

[Out]

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

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Rubi [A]  time = 0.139477, antiderivative size = 65, 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 \[ \frac{2 d (a+b x)^5 (b c-a d)}{5 b^3}+\frac{(a+b x)^4 (b c-a d)^2}{4 b^3}+\frac{d^2 (a+b x)^6}{6 b^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^3*(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 19.7189, size = 56, normalized size = 0.86 \[ \frac{d^{2} \left (a + b x\right )^{6}}{6 b^{3}} - \frac{2 d \left (a + b x\right )^{5} \left (a d - b c\right )}{5 b^{3}} + \frac{\left (a + b x\right )^{4} \left (a d - b c\right )^{2}}{4 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0274062, size = 122, normalized size = 1.88 \[ a^3 c^2 x+\frac{1}{4} b x^4 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+\frac{1}{3} a x^3 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{2} a^2 c x^2 (2 a d+3 b c)+\frac{1}{5} b^2 d x^5 (3 a d+2 b c)+\frac{1}{6} b^3 d^2 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.002, size = 125, normalized size = 1.9 \[{\frac{{b}^{3}{d}^{2}{x}^{6}}{6}}+{\frac{ \left ( 3\,a{b}^{2}{d}^{2}+2\,{b}^{3}cd \right ){x}^{5}}{5}}+{\frac{ \left ( 3\,{a}^{2}b{d}^{2}+6\,a{b}^{2}cd+{b}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({a}^{3}{d}^{2}+6\,{a}^{2}bcd+3\,a{b}^{2}{c}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,{a}^{3}cd+3\,{a}^{2}b{c}^{2} \right ){x}^{2}}{2}}+{a}^{3}{c}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.35051, size = 167, normalized size = 2.57 \[ \frac{1}{6} \, b^{3} d^{2} x^{6} + a^{3} c^{2} x + \frac{1}{5} \,{\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} x^{5} + \frac{1}{4} \,{\left (b^{3} c^{2} + 6 \, a b^{2} c d + 3 \, a^{2} b d^{2}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, a b^{2} c^{2} + 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (3 \, a^{2} b c^{2} + 2 \, a^{3} c d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.179296, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} d^{2} b^{3} + \frac{2}{5} x^{5} d c b^{3} + \frac{3}{5} x^{5} d^{2} b^{2} a + \frac{1}{4} x^{4} c^{2} b^{3} + \frac{3}{2} x^{4} d c b^{2} a + \frac{3}{4} x^{4} d^{2} b a^{2} + x^{3} c^{2} b^{2} a + 2 x^{3} d c b a^{2} + \frac{1}{3} x^{3} d^{2} a^{3} + \frac{3}{2} x^{2} c^{2} b a^{2} + x^{2} d c a^{3} + x c^{2} a^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.157803, size = 133, normalized size = 2.05 \[ a^{3} c^{2} x + \frac{b^{3} d^{2} x^{6}}{6} + x^{5} \left (\frac{3 a b^{2} d^{2}}{5} + \frac{2 b^{3} c d}{5}\right ) + x^{4} \left (\frac{3 a^{2} b d^{2}}{4} + \frac{3 a b^{2} c d}{2} + \frac{b^{3} c^{2}}{4}\right ) + x^{3} \left (\frac{a^{3} d^{2}}{3} + 2 a^{2} b c d + a b^{2} c^{2}\right ) + x^{2} \left (a^{3} c d + \frac{3 a^{2} b c^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.215007, size = 176, normalized size = 2.71 \[ \frac{1}{6} \, b^{3} d^{2} x^{6} + \frac{2}{5} \, b^{3} c d x^{5} + \frac{3}{5} \, a b^{2} d^{2} x^{5} + \frac{1}{4} \, b^{3} c^{2} x^{4} + \frac{3}{2} \, a b^{2} c d x^{4} + \frac{3}{4} \, a^{2} b d^{2} x^{4} + a b^{2} c^{2} x^{3} + 2 \, a^{2} b c d x^{3} + \frac{1}{3} \, a^{3} d^{2} x^{3} + \frac{3}{2} \, a^{2} b c^{2} x^{2} + a^{3} c d x^{2} + a^{3} c^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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