3.1443 \(\int (d+e x) \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0779613, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{(a+b x)^3 (b d-a e)}{3 b^2}+\frac{e (a+b x)^4}{4 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.238, size = 31, normalized size = 0.82 \[ \frac{e \left (a + b x\right )^{4}}{4 b^{2}} - \frac{\left (a + b x\right )^{3} \left (a e - b d\right )}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0172308, size = 46, normalized size = 1.21 \[ \frac{1}{12} x \left (6 a^2 (2 d+e x)+4 a b x (3 d+2 e x)+b^2 x^2 (4 d+3 e x)\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 49, normalized size = 1.3 \[{\frac{{b}^{2}e{x}^{4}}{4}}+{\frac{ \left ( 2\,bea+{b}^{2}d \right ){x}^{3}}{3}}+{\frac{ \left ( e{a}^{2}+2\,bda \right ){x}^{2}}{2}}+{a}^{2}dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.680064, size = 65, normalized size = 1.71 \[ \frac{1}{4} \, b^{2} e x^{4} + a^{2} d x + \frac{1}{3} \,{\left (b^{2} d + 2 \, a b e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, a b d + a^{2} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.17767, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} e b^{2} + \frac{1}{3} x^{3} d b^{2} + \frac{2}{3} x^{3} e b a + x^{2} d b a + \frac{1}{2} x^{2} e a^{2} + x d a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.10759, size = 49, normalized size = 1.29 \[ a^{2} d x + \frac{b^{2} e x^{4}}{4} + x^{3} \left (\frac{2 a b e}{3} + \frac{b^{2} d}{3}\right ) + x^{2} \left (\frac{a^{2} e}{2} + a b d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21, size = 70, normalized size = 1.84 \[ \frac{1}{4} \, b^{2} x^{4} e + \frac{1}{3} \, b^{2} d x^{3} + \frac{2}{3} \, a b x^{3} e + a b d x^{2} + \frac{1}{2} \, a^{2} x^{2} e + a^{2} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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