\(\int (d+e x) (9+12 x+4 x^2) \, dx\) [1539]

Optimal result

Integrand size = 16, antiderivative size = 31 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=\frac {1}{12} (2 d-3 e) (3+2 x)^3+\frac {1}{16} e (3+2 x)^4 \]

[Out]

1/12*(2*d-3*e)*(3+2*x)^3+1/16*e*(3+2*x)^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {27, 45} \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=\frac {1}{12} (2 x+3)^3 (2 d-3 e)+\frac {1}{16} e (2 x+3)^4 \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=9 d x+\frac {3}{2} (4 d+3 e) x^2+\frac {4}{3} (d+3 e) x^3+e x^4 \]

[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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06

method	result	size
norman	\(e \,x^{4}+\left (\frac {4 d}{3}+4 e \right ) x^{3}+\left (6 d +\frac {9 e}{2}\right ) x^{2}+9 d x\)	\(33\)
gosper	\(\frac {x \left (6 e \,x^{3}+8 d \,x^{2}+24 e \,x^{2}+36 d x +27 e x +54 d \right )}{6}\)	\(34\)
default	\(e \,x^{4}+\frac {\left (4 d +12 e \right ) x^{3}}{3}+\frac {\left (12 d +9 e \right ) x^{2}}{2}+9 d x\)	\(35\)
risch	\(e \,x^{4}+\frac {4}{3} d \,x^{3}+4 e \,x^{3}+6 d \,x^{2}+\frac {9}{2} e \,x^{2}+9 d x\)	\(35\)
parallelrisch	\(e \,x^{4}+\frac {4}{3} d \,x^{3}+4 e \,x^{3}+6 d \,x^{2}+\frac {9}{2} e \,x^{2}+9 d x\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=e x^{4} + \frac {4}{3} \, {\left (d + 3 \, e\right )} x^{3} + \frac {3}{2} \, {\left (4 \, d + 3 \, e\right )} x^{2} + 9 \, d x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=9 d x + e x^{4} + x^{3} \cdot \left (\frac {4 d}{3} + 4 e\right ) + x^{2} \cdot \left (6 d + \frac {9 e}{2}\right ) \]

[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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=e x^{4} + \frac {4}{3} \, {\left (d + 3 \, e\right )} x^{3} + \frac {3}{2} \, {\left (4 \, d + 3 \, e\right )} x^{2} + 9 \, d x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=e x^{4} + \frac {4}{3} \, d x^{3} + 4 \, e x^{3} + 6 \, d x^{2} + \frac {9}{2} \, e x^{2} + 9 \, d x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int (d+e x) \left (9+12 x+4 x^2\right ) \, dx=e\,x^4+\left (\frac {4\,d}{3}+4\,e\right )\,x^3+\left (6\,d+\frac {9\,e}{2}\right )\,x^2+9\,d\,x \]

[In]

int((d + e*x)*(12*x + 4*x^2 + 9),x)

[Out]

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