3.54 \(\int x (a+b x)^2 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

[Out]

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

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Rubi [A]  time = 0.0255135, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[x*(a + b*x)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a^{2} \int x\, dx + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(b*x+a)**2,x)

[Out]

a**2*Integral(x, x) + 2*a*b*x**3/3 + b**2*x**4/4

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Mathematica [A]  time = 0.00187126, size = 30, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[x*(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0., size = 25, normalized size = 0.8 \[{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.33277, size = 32, normalized size = 1.07 \[ \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.177388, size = 1, normalized size = 0.03 \[ \frac{1}{4} x^{4} b^{2} + \frac{2}{3} x^{3} b a + \frac{1}{2} x^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.080362, size = 26, normalized size = 0.87 \[ \frac{a^{2} x^{2}}{2} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(b*x+a)**2,x)

[Out]

a**2*x**2/2 + 2*a*b*x**3/3 + b**2*x**4/4

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GIAC/XCAS [A]  time = 0.208858, size = 32, normalized size = 1.07 \[ \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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