3.1261 \(\int (a+b x)^2 (c+d x)^3 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.140652, 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 \[ -\frac{2 b (c+d x)^5 (b c-a d)}{5 d^3}+\frac{(c+d x)^4 (b c-a d)^2}{4 d^3}+\frac{b^2 (c+d x)^6}{6 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2*(c + d*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 20.1618, size = 56, normalized size = 0.86 \[ \frac{b^{2} \left (c + d x\right )^{6}}{6 d^{3}} + \frac{2 b \left (c + d x\right )^{5} \left (a d - b c\right )}{5 d^{3}} + \frac{\left (c + d x\right )^{4} \left (a d - b c\right )^{2}}{4 d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0256111, size = 122, normalized size = 1.88 \[ \frac{1}{4} d x^4 \left (a^2 d^2+6 a b c d+3 b^2 c^2\right )+\frac{1}{3} c x^3 \left (3 a^2 d^2+6 a b c d+b^2 c^2\right )+a^2 c^3 x+\frac{1}{2} a c^2 x^2 (3 a d+2 b c)+\frac{1}{5} b d^2 x^5 (2 a d+3 b c)+\frac{1}{6} b^2 d^3 x^6 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(d*x+c)^3,x)

[Out]

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

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Maxima [A]  time = 1.34897, size = 167, normalized size = 2.57 \[ \frac{1}{6} \, b^{2} d^{3} x^{6} + a^{2} c^{3} x + \frac{1}{5} \,{\left (3 \, b^{2} c d^{2} + 2 \, a b d^{3}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, b^{2} c^{2} d + 6 \, a b c d^{2} + a^{2} d^{3}\right )} x^{4} + \frac{1}{3} \,{\left (b^{2} c^{3} + 6 \, a b c^{2} d + 3 \, a^{2} c d^{2}\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c^{3} + 3 \, a^{2} c^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.186121, size = 1, normalized size = 0.02 \[ \frac{1}{6} x^{6} d^{3} b^{2} + \frac{3}{5} x^{5} d^{2} c b^{2} + \frac{2}{5} x^{5} d^{3} b a + \frac{3}{4} x^{4} d c^{2} b^{2} + \frac{3}{2} x^{4} d^{2} c b a + \frac{1}{4} x^{4} d^{3} a^{2} + \frac{1}{3} x^{3} c^{3} b^{2} + 2 x^{3} d c^{2} b a + x^{3} d^{2} c a^{2} + x^{2} c^{3} b a + \frac{3}{2} x^{2} d c^{2} a^{2} + x c^{3} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.152008, size = 133, normalized size = 2.05 \[ a^{2} c^{3} x + \frac{b^{2} d^{3} x^{6}}{6} + x^{5} \left (\frac{2 a b d^{3}}{5} + \frac{3 b^{2} c d^{2}}{5}\right ) + x^{4} \left (\frac{a^{2} d^{3}}{4} + \frac{3 a b c d^{2}}{2} + \frac{3 b^{2} c^{2} d}{4}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + \frac{b^{2} c^{3}}{3}\right ) + x^{2} \left (\frac{3 a^{2} c^{2} d}{2} + a b c^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.223482, size = 176, normalized size = 2.71 \[ \frac{1}{6} \, b^{2} d^{3} x^{6} + \frac{3}{5} \, b^{2} c d^{2} x^{5} + \frac{2}{5} \, a b d^{3} x^{5} + \frac{3}{4} \, b^{2} c^{2} d x^{4} + \frac{3}{2} \, a b c d^{2} x^{4} + \frac{1}{4} \, a^{2} d^{3} x^{4} + \frac{1}{3} \, b^{2} c^{3} x^{3} + 2 \, a b c^{2} d x^{3} + a^{2} c d^{2} x^{3} + a b c^{3} x^{2} + \frac{3}{2} \, a^{2} c^{2} d x^{2} + a^{2} c^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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