3.1750 \(\int (a+b x) \left (a c+(b c+a d) x+b d x^2\right ) \, dx\)

Optimal. Leaf size=38 \[ \frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0851836, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{(a+b x)^3 (b c-a d)}{3 b^2}+\frac{d (a+b x)^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.8786, size = 31, normalized size = 0.82 \[ \frac{d \left (a + b x\right )^{4}}{4 b^{2}} - \frac{\left (a + b x\right )^{3} \left (a d - b c\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0158388, size = 46, normalized size = 1.21 \[ \frac{1}{12} x \left (6 a^2 (2 c+d x)+4 a b x (3 c+2 d x)+b^2 x^2 (4 c+3 d x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 55, normalized size = 1.5 \[{\frac{{b}^{2}d{x}^{4}}{4}}+{\frac{ \left ( bda+b \left ( ad+bc \right ) \right ){x}^{3}}{3}}+{\frac{ \left ( a \left ( ad+bc \right ) +abc \right ){x}^{2}}{2}}+{a}^{2}cx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.71696, size = 65, normalized size = 1.71 \[ \frac{1}{4} \, b^{2} d x^{4} + a^{2} c x + \frac{1}{3} \,{\left (b^{2} c + 2 \, a b d\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.178109, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} d b^{2} + \frac{1}{3} x^{3} c b^{2} + \frac{2}{3} x^{3} d b a + x^{2} c b a + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.106881, size = 49, normalized size = 1.29 \[ a^{2} c x + \frac{b^{2} d x^{4}}{4} + x^{3} \left (\frac{2 a b d}{3} + \frac{b^{2} c}{3}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209138, size = 66, normalized size = 1.74 \[ \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{3} \, b^{2} c x^{3} + \frac{2}{3} \, a b d x^{3} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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