3.1112 \(\int (b d+2 c d x) (a+b x+c x^2) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{2} d \left (a+b x+c x^2\right )^2 \]

[Out]

(d*(a + b*x + c*x^2)^2)/2

________________________________________________________________________________________

Rubi [A]  time = 0.004063, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {629} \[ \frac{1}{2} d \left (a+b x+c x^2\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0047767, size = 22, normalized size = 1.29 \[ \frac{1}{2} d x (b+c x) (2 a+x (b+c x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d*x*(b + c*x)*(2*a + x*(b + c*x)))/2

________________________________________________________________________________________

Maple [B]  time = 0.038, size = 39, normalized size = 2.3 \begin{align*}{\frac{{c}^{2}d{x}^{4}}{2}}+bcd{x}^{3}+{\frac{ \left ( 2\,acd+{b}^{2}d \right ){x}^{2}}{2}}+bdax \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.17061, size = 20, normalized size = 1.18 \begin{align*} \frac{1}{2} \,{\left (c x^{2} + b x + a\right )}^{2} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(c*x^2 + b*x + a)^2*d

________________________________________________________________________________________

Fricas [B]  time = 1.942, size = 88, normalized size = 5.18 \begin{align*} \frac{1}{2} x^{4} d c^{2} + x^{3} d c b + \frac{1}{2} x^{2} d b^{2} + x^{2} d c a + x d b a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.097042, size = 39, normalized size = 2.29 \begin{align*} a b d x + b c d x^{3} + \frac{c^{2} d x^{4}}{2} + x^{2} \left (a c d + \frac{b^{2} d}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.24304, size = 51, normalized size = 3. \begin{align*} \frac{1}{2} \, c^{2} d x^{4} + b c d x^{3} + \frac{1}{2} \, b^{2} d x^{2} + a c d x^{2} + a b d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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