3.2111 \(\int \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=46 \[ a^2 x+\frac{1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

[Out]

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

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Rubi [A]  time = 0.0553718, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ a^2 x+\frac{1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a b \int x\, dx + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5} + x^{3} \left (\frac{2 a c}{3} + \frac{b^{2}}{3}\right ) + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*b*Integral(x, x) + b*c*x**4/2 + c**2*x**5/5 + x**3*(2*a*c/3 + b**2/3) + Inte
gral(a**2, x)

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Mathematica [A]  time = 0.0123411, size = 46, normalized size = 1. \[ a^2 x+\frac{1}{3} x^3 \left (2 a c+b^2\right )+a b x^2+\frac{1}{2} b c x^4+\frac{c^2 x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 41, normalized size = 0.9 \[{a}^{2}x+ab{x}^{2}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{3}}{3}}+{\frac{bc{x}^{4}}{2}}+{\frac{{c}^{2}{x}^{5}}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.814393, size = 61, normalized size = 1.33 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} + a^{2} x + \frac{1}{3} \,{\left (2 \, c x^{3} + 3 \, b x^{2}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.181306, size = 1, normalized size = 0.02 \[ \frac{1}{5} x^{5} c^{2} + \frac{1}{2} x^{4} c b + \frac{1}{3} x^{3} b^{2} + \frac{2}{3} x^{3} c a + x^{2} b a + x a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.101438, size = 42, normalized size = 0.91 \[ a^{2} x + a b x^{2} + \frac{b c x^{4}}{2} + \frac{c^{2} x^{5}}{5} + x^{3} \left (\frac{2 a c}{3} + \frac{b^{2}}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.202969, size = 57, normalized size = 1.24 \[ \frac{1}{5} \, c^{2} x^{5} + \frac{1}{2} \, b c x^{4} + \frac{1}{3} \, b^{2} x^{3} + \frac{2}{3} \, a c x^{3} + a b x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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