3.1498 \(\int (b+2 c x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=16 \[ \frac{1}{2} \left (a+b x+c x^2\right )^2 \]

[Out]

(a + b*x + c*x^2)^2/2

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Rubi [A]  time = 0.0036006, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {629} \[ \frac{1}{2} \left (a+b x+c x^2\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(b + 2*c*x)*(a + b*x + c*x^2),x]

[Out]

(a + b*x + c*x^2)^2/2

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int (b+2 c x) \left (a+b x+c x^2\right ) \, dx &=\frac{1}{2} \left (a+b x+c x^2\right )^2\\ \end{align*}

Mathematica [A]  time = 0.0046234, size = 21, normalized size = 1.31 \[ \frac{1}{2} x (b+c x) (2 a+x (b+c x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b + c*x)*(2*a + x*(b + c*x)))/2

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Maple [B]  time = 0.002, size = 33, normalized size = 2.1 \begin{align*}{\frac{{c}^{2}{x}^{4}}{2}}+bc{x}^{3}+{\frac{ \left ( 2\,ac+{b}^{2} \right ){x}^{2}}{2}}+abx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x+a),x)

[Out]

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

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Maxima [A]  time = 0.980479, size = 19, normalized size = 1.19 \begin{align*} \frac{1}{2} \,{\left (c x^{2} + b x + a\right )}^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x + a)^2

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Fricas [B]  time = 1.56808, size = 74, normalized size = 4.62 \begin{align*} \frac{1}{2} x^{4} c^{2} + x^{3} c b + \frac{1}{2} x^{2} b^{2} + x^{2} c a + x b a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.064064, size = 31, normalized size = 1.94 \begin{align*} a b x + b c x^{3} + \frac{c^{2} x^{4}}{2} + x^{2} \left (a c + \frac{b^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a),x)

[Out]

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

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Giac [B]  time = 1.14083, size = 45, normalized size = 2.81 \begin{align*} \frac{1}{2} \, c^{2} x^{4} + b c x^{3} + \frac{1}{2} \, b^{2} x^{2} + a c x^{2} + a b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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