\(\int (d+e x) (9+12 x+4 x^2)^2 \, dx\) [1538]

Optimal result

Integrand size = 18, antiderivative size = 31 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {1}{20} (2 d-3 e) (3+2 x)^5+\frac {1}{24} e (3+2 x)^6 \]

[Out]

1/20*(2*d-3*e)*(3+2*x)^5+1/24*e*(3+2*x)^6

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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {27, 45} \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {1}{20} (2 x+3)^5 (2 d-3 e)+\frac {1}{24} e (2 x+3)^6 \]

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^2,x]

[Out]

((2*d - 3*e)*(3 + 2*x)^5)/20 + (e*(3 + 2*x)^6)/24

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)^4 (d+e x) \, dx \\ & = \int \left (\frac {1}{2} (2 d-3 e) (3+2 x)^4+\frac {1}{2} e (3+2 x)^5\right ) \, dx \\ & = \frac {1}{20} (2 d-3 e) (3+2 x)^5+\frac {1}{24} e (3+2 x)^6 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=81 d x+\frac {27}{2} (8 d+3 e) x^2+72 (d+e) x^3+6 (4 d+9 e) x^4+\frac {16}{5} (d+6 e) x^5+\frac {8 e x^6}{3} \]

[In]

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

[Out]

81*d*x + (27*(8*d + 3*e)*x^2)/2 + 72*(d + e)*x^3 + 6*(4*d + 9*e)*x^4 + (16*(d + 6*e)*x^5)/5 + (8*e*x^6)/3

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).

Time = 2.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81

method	result	size
norman	\(\frac {8 e \,x^{6}}{3}+\left (\frac {16 d}{5}+\frac {96 e}{5}\right ) x^{5}+\left (24 d +54 e \right ) x^{4}+\left (72 d +72 e \right ) x^{3}+\left (108 d +\frac {81 e}{2}\right ) x^{2}+81 d x\)	\(56\)
gosper	\(\frac {x \left (80 e \,x^{5}+96 d \,x^{4}+576 e \,x^{4}+720 d \,x^{3}+1620 e \,x^{3}+2160 d \,x^{2}+2160 e \,x^{2}+3240 d x +1215 e x +2430 d \right )}{30}\)	\(58\)
default	\(\frac {8 e \,x^{6}}{3}+\frac {\left (16 d +96 e \right ) x^{5}}{5}+\frac {\left (96 d +216 e \right ) x^{4}}{4}+\frac {\left (216 d +216 e \right ) x^{3}}{3}+\frac {\left (216 d +81 e \right ) x^{2}}{2}+81 d x\)	\(60\)
risch	\(\frac {8}{3} e \,x^{6}+\frac {16}{5} d \,x^{5}+\frac {96}{5} e \,x^{5}+24 d \,x^{4}+54 e \,x^{4}+72 d \,x^{3}+72 e \,x^{3}+108 d \,x^{2}+\frac {81}{2} e \,x^{2}+81 d x\)	\(60\)
parallelrisch	\(\frac {8}{3} e \,x^{6}+\frac {16}{5} d \,x^{5}+\frac {96}{5} e \,x^{5}+24 d \,x^{4}+54 e \,x^{4}+72 d \,x^{3}+72 e \,x^{3}+108 d \,x^{2}+\frac {81}{2} e \,x^{2}+81 d x\)	\(60\)

[In]

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

[Out]

8/3*e*x^6+(16/5*d+96/5*e)*x^5+(24*d+54*e)*x^4+(72*d+72*e)*x^3+(108*d+81/2*e)*x^2+81*d*x

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {8}{3} \, e x^{6} + \frac {16}{5} \, {\left (d + 6 \, e\right )} x^{5} + 6 \, {\left (4 \, d + 9 \, e\right )} x^{4} + 72 \, {\left (d + e\right )} x^{3} + \frac {27}{2} \, {\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \]

[In]

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

[Out]

8/3*e*x^6 + 16/5*(d + 6*e)*x^5 + 6*(4*d + 9*e)*x^4 + 72*(d + e)*x^3 + 27/2*(8*d + 3*e)*x^2 + 81*d*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.87 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=81 d x + \frac {8 e x^{6}}{3} + x^{5} \cdot \left (\frac {16 d}{5} + \frac {96 e}{5}\right ) + x^{4} \cdot \left (24 d + 54 e\right ) + x^{3} \cdot \left (72 d + 72 e\right ) + x^{2} \cdot \left (108 d + \frac {81 e}{2}\right ) \]

[In]

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

[Out]

81*d*x + 8*e*x**6/3 + x**5*(16*d/5 + 96*e/5) + x**4*(24*d + 54*e) + x**3*(72*d + 72*e) + x**2*(108*d + 81*e/2)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {8}{3} \, e x^{6} + \frac {16}{5} \, {\left (d + 6 \, e\right )} x^{5} + 6 \, {\left (4 \, d + 9 \, e\right )} x^{4} + 72 \, {\left (d + e\right )} x^{3} + \frac {27}{2} \, {\left (8 \, d + 3 \, e\right )} x^{2} + 81 \, d x \]

[In]

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

[Out]

8/3*e*x^6 + 16/5*(d + 6*e)*x^5 + 6*(4*d + 9*e)*x^4 + 72*(d + e)*x^3 + 27/2*(8*d + 3*e)*x^2 + 81*d*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (27) = 54\).

Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {8}{3} \, e x^{6} + \frac {16}{5} \, d x^{5} + \frac {96}{5} \, e x^{5} + 24 \, d x^{4} + 54 \, e x^{4} + 72 \, d x^{3} + 72 \, e x^{3} + 108 \, d x^{2} + \frac {81}{2} \, e x^{2} + 81 \, d x \]

[In]

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

[Out]

8/3*e*x^6 + 16/5*d*x^5 + 96/5*e*x^5 + 24*d*x^4 + 54*e*x^4 + 72*d*x^3 + 72*e*x^3 + 108*d*x^2 + 81/2*e*x^2 + 81*
d*x

Mupad [B] (verification not implemented)

Time = 9.81 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int (d+e x) \left (9+12 x+4 x^2\right )^2 \, dx=\frac {8\,e\,x^6}{3}+\left (\frac {16\,d}{5}+\frac {96\,e}{5}\right )\,x^5+\left (24\,d+54\,e\right )\,x^4+\left (72\,d+72\,e\right )\,x^3+\left (108\,d+\frac {81\,e}{2}\right )\,x^2+81\,d\,x \]

[In]

int((d + e*x)*(12*x + 4*x^2 + 9)^2,x)

[Out]

x^4*(24*d + 54*e) + x^5*((16*d)/5 + (96*e)/5) + x^3*(72*d + 72*e) + x^2*(108*d + (81*e)/2) + 81*d*x + (8*e*x^6
)/3