3.1099 \(\int (b d+2 c d x)^3 \left (a+b x+c x^2\right ) \, dx\)

Optimal. Leaf size=45 \[ \frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2} \]

[Out]

-((b^2 - 4*a*c)*d^3*(b + 2*c*x)^4)/(32*c^2) + (d^3*(b + 2*c*x)^6)/(48*c^2)

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Rubi [A]  time = 0.128156, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ \frac{d^3 (b+2 c x)^6}{48 c^2}-\frac{d^3 \left (b^2-4 a c\right ) (b+2 c x)^4}{32 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b^2 - 4*a*c)*d^3*(b + 2*c*x)^4)/(32*c^2) + (d^3*(b + 2*c*x)^6)/(48*c^2)

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Rubi in Sympy [A]  time = 21.911, size = 41, normalized size = 0.91 \[ \frac{d^{3} \left (b + 2 c x\right )^{6}}{48 c^{2}} - \frac{d^{3} \left (b + 2 c x\right )^{4} \left (- a c + \frac{b^{2}}{4}\right )}{8 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**3*(b + 2*c*x)**6/(48*c**2) - d**3*(b + 2*c*x)**4*(-a*c + b**2/4)/(8*c**2)

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Mathematica [A]  time = 0.0287172, size = 66, normalized size = 1.47 \[ \frac{1}{6} d^3 x (b+c x) \left (6 a \left (b^2+2 b c x+2 c^2 x^2\right )+x \left (3 b^3+11 b^2 c x+16 b c^2 x^2+8 c^3 x^3\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(d^3*x*(b + c*x)*(6*a*(b^2 + 2*b*c*x + 2*c^2*x^2) + x*(3*b^3 + 11*b^2*c*x + 16*b
*c^2*x^2 + 8*c^3*x^3)))/6

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Maple [B]  time = 0.001, size = 108, normalized size = 2.4 \[{\frac{4\,{c}^{4}{d}^{3}{x}^{6}}{3}}+4\,b{c}^{3}{d}^{3}{x}^{5}+{\frac{ \left ( 8\,{c}^{3}{d}^{3}a+18\,{b}^{2}{d}^{3}{c}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 12\,b{d}^{3}{c}^{2}a+7\,{b}^{3}{d}^{3}c \right ){x}^{3}}{3}}+{\frac{ \left ( 6\,{b}^{2}{d}^{3}ca+{b}^{4}{d}^{3} \right ){x}^{2}}{2}}+{b}^{3}{d}^{3}ax \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.683633, size = 131, normalized size = 2.91 \[ \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + a b^{3} d^{3} x + \frac{1}{2} \,{\left (9 \, b^{2} c^{2} + 4 \, a c^{3}\right )} d^{3} x^{4} + \frac{1}{3} \,{\left (7 \, b^{3} c + 12 \, a b c^{2}\right )} d^{3} x^{3} + \frac{1}{2} \,{\left (b^{4} + 6 \, a b^{2} c\right )} d^{3} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.189681, size = 1, normalized size = 0.02 \[ \frac{4}{3} x^{6} d^{3} c^{4} + 4 x^{5} d^{3} c^{3} b + \frac{9}{2} x^{4} d^{3} c^{2} b^{2} + 2 x^{4} d^{3} c^{3} a + \frac{7}{3} x^{3} d^{3} c b^{3} + 4 x^{3} d^{3} c^{2} b a + \frac{1}{2} x^{2} d^{3} b^{4} + 3 x^{2} d^{3} c b^{2} a + x d^{3} b^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.152311, size = 114, normalized size = 2.53 \[ a b^{3} d^{3} x + 4 b c^{3} d^{3} x^{5} + \frac{4 c^{4} d^{3} x^{6}}{3} + x^{4} \left (2 a c^{3} d^{3} + \frac{9 b^{2} c^{2} d^{3}}{2}\right ) + x^{3} \left (4 a b c^{2} d^{3} + \frac{7 b^{3} c d^{3}}{3}\right ) + x^{2} \left (3 a b^{2} c d^{3} + \frac{b^{4} d^{3}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.211873, size = 146, normalized size = 3.24 \[ \frac{4}{3} \, c^{4} d^{3} x^{6} + 4 \, b c^{3} d^{3} x^{5} + \frac{9}{2} \, b^{2} c^{2} d^{3} x^{4} + 2 \, a c^{3} d^{3} x^{4} + \frac{7}{3} \, b^{3} c d^{3} x^{3} + 4 \, a b c^{2} d^{3} x^{3} + \frac{1}{2} \, b^{4} d^{3} x^{2} + 3 \, a b^{2} c d^{3} x^{2} + a b^{3} d^{3} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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