3.1539 \(\int (d+e x) (9+12 x+4 x^2) \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{12} (2 x+3)^3 (2 d-3 e)+\frac{1}{16} e (2 x+3)^4 \]

[Out]

((2*d - 3*e)*(3 + 2*x)^3)/12 + (e*(3 + 2*x)^4)/16

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Rubi [A]  time = 0.0206185, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {27, 43} \[ \frac{1}{12} (2 x+3)^3 (2 d-3 e)+\frac{1}{16} e (2 x+3)^4 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2),x]

[Out]

((2*d - 3*e)*(3 + 2*x)^3)/12 + (e*(3 + 2*x)^4)/16

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx &=\int (3+2 x)^2 (d+e x) \, dx\\ &=\int \left (\frac{1}{2} (2 d-3 e) (3+2 x)^2+\frac{1}{2} e (3+2 x)^3\right ) \, dx\\ &=\frac{1}{12} (2 d-3 e) (3+2 x)^3+\frac{1}{16} e (3+2 x)^4\\ \end{align*}

Mathematica [A]  time = 0.005751, size = 36, normalized size = 1.16 \[ \frac{4}{3} x^3 (d+3 e)+\frac{3}{2} x^2 (4 d+3 e)+9 d x+e x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(9 + 12*x + 4*x^2),x]

[Out]

9*d*x + (3*(4*d + 3*e)*x^2)/2 + (4*(d + 3*e)*x^3)/3 + e*x^4

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Maple [A]  time = 0.038, size = 35, normalized size = 1.1 \begin{align*} e{x}^{4}+{\frac{ \left ( 4\,d+12\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 12\,d+9\,e \right ){x}^{2}}{2}}+9\,dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(4*x^2+12*x+9),x)

[Out]

e*x^4+1/3*(4*d+12*e)*x^3+1/2*(12*d+9*e)*x^2+9*d*x

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Maxima [A]  time = 1.11375, size = 43, normalized size = 1.39 \begin{align*} e x^{4} + \frac{4}{3} \,{\left (d + 3 \, e\right )} x^{3} + \frac{3}{2} \,{\left (4 \, d + 3 \, e\right )} x^{2} + 9 \, d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9),x, algorithm="maxima")

[Out]

e*x^4 + 4/3*(d + 3*e)*x^3 + 3/2*(4*d + 3*e)*x^2 + 9*d*x

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Fricas [A]  time = 1.36334, size = 80, normalized size = 2.58 \begin{align*} x^{4} e + 4 x^{3} e + \frac{4}{3} x^{3} d + \frac{9}{2} x^{2} e + 6 x^{2} d + 9 x d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9),x, algorithm="fricas")

[Out]

x^4*e + 4*x^3*e + 4/3*x^3*d + 9/2*x^2*e + 6*x^2*d + 9*x*d

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Sympy [A]  time = 0.077802, size = 32, normalized size = 1.03 \begin{align*} 9 d x + e x^{4} + x^{3} \left (\frac{4 d}{3} + 4 e\right ) + x^{2} \left (6 d + \frac{9 e}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x**2+12*x+9),x)

[Out]

9*d*x + e*x**4 + x**3*(4*d/3 + 4*e) + x**2*(6*d + 9*e/2)

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Giac [A]  time = 1.11945, size = 50, normalized size = 1.61 \begin{align*} x^{4} e + \frac{4}{3} \, d x^{3} + 4 \, x^{3} e + 6 \, d x^{2} + \frac{9}{2} \, x^{2} e + 9 \, d x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(4*x^2+12*x+9),x, algorithm="giac")

[Out]

x^4*e + 4/3*d*x^3 + 4*x^3*e + 6*d*x^2 + 9/2*x^2*e + 9*d*x