### 3.1537 $$\int (d+e x) (9+12 x+4 x^2)^3 \, dx$$

Optimal. Leaf size=31 $\frac{1}{28} (2 x+3)^7 (2 d-3 e)+\frac{1}{32} e (2 x+3)^8$

[Out]

((2*d - 3*e)*(3 + 2*x)^7)/28 + (e*(3 + 2*x)^8)/32

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Rubi [A]  time = 0.0115803, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {27, 43} $\frac{1}{28} (2 x+3)^7 (2 d-3 e)+\frac{1}{32} e (2 x+3)^8$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^3,x]

[Out]

((2*d - 3*e)*(3 + 2*x)^7)/28 + (e*(3 + 2*x)^8)/32

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 )^3 \, dx &=\int (3+2 x)^6 (d+e x) \, dx\\ &=\int \left (\frac{1}{2} (2 d-3 e) (3+2 x)^6+\frac{1}{2} e (3+2 x)^7\right ) \, dx\\ &=\frac{1}{28} (2 d-3 e) (3+2 x)^7+\frac{1}{32} e (3+2 x)^8\\ \end{align*}

Mathematica [B]  time = 0.0150449, size = 81, normalized size = 2.61 $\frac{64}{7} x^7 (d+9 e)+24 x^6 (4 d+15 e)+432 x^5 (d+2 e)+135 x^4 (8 d+9 e)+324 x^3 (5 d+3 e)+\frac{729}{2} x^2 (4 d+e)+729 d x+8 e x^8$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)*(9 + 12*x + 4*x^2)^3,x]

[Out]

729*d*x + (729*(4*d + e)*x^2)/2 + 324*(5*d + 3*e)*x^3 + 135*(8*d + 9*e)*x^4 + 432*(d + 2*e)*x^5 + 24*(4*d + 15
*e)*x^6 + (64*(d + 9*e)*x^7)/7 + 8*e*x^8

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Maple [B]  time = 0.04, size = 84, normalized size = 2.7 \begin{align*} 8\,e{x}^{8}+{\frac{ \left ( 64\,d+576\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 576\,d+2160\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 2160\,d+4320\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 4320\,d+4860\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 4860\,d+2916\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 2916\,d+729\,e \right ){x}^{2}}{2}}+729\,dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(4*x^2+12*x+9)^3,x)

[Out]

8*e*x^8+1/7*(64*d+576*e)*x^7+1/6*(576*d+2160*e)*x^6+1/5*(2160*d+4320*e)*x^5+1/4*(4320*d+4860*e)*x^4+1/3*(4860*
d+2916*e)*x^3+1/2*(2916*d+729*e)*x^2+729*d*x

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Maxima [B]  time = 1.09033, size = 104, normalized size = 3.35 \begin{align*} 8 \, e x^{8} + \frac{64}{7} \,{\left (d + 9 \, e\right )} x^{7} + 24 \,{\left (4 \, d + 15 \, e\right )} x^{6} + 432 \,{\left (d + 2 \, e\right )} x^{5} + 135 \,{\left (8 \, d + 9 \, e\right )} x^{4} + 324 \,{\left (5 \, d + 3 \, e\right )} x^{3} + \frac{729}{2} \,{\left (4 \, d + e\right )} x^{2} + 729 \, 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)^3,x, algorithm="maxima")

[Out]

8*e*x^8 + 64/7*(d + 9*e)*x^7 + 24*(4*d + 15*e)*x^6 + 432*(d + 2*e)*x^5 + 135*(8*d + 9*e)*x^4 + 324*(5*d + 3*e)
*x^3 + 729/2*(4*d + e)*x^2 + 729*d*x

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Fricas [B]  time = 1.34382, size = 231, normalized size = 7.45 \begin{align*} 8 x^{8} e + \frac{576}{7} x^{7} e + \frac{64}{7} x^{7} d + 360 x^{6} e + 96 x^{6} d + 864 x^{5} e + 432 x^{5} d + 1215 x^{4} e + 1080 x^{4} d + 972 x^{3} e + 1620 x^{3} d + \frac{729}{2} x^{2} e + 1458 x^{2} d + 729 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)^3,x, algorithm="fricas")

[Out]

8*x^8*e + 576/7*x^7*e + 64/7*x^7*d + 360*x^6*e + 96*x^6*d + 864*x^5*e + 432*x^5*d + 1215*x^4*e + 1080*x^4*d +
972*x^3*e + 1620*x^3*d + 729/2*x^2*e + 1458*x^2*d + 729*x*d

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Sympy [B]  time = 0.089094, size = 76, normalized size = 2.45 \begin{align*} 729 d x + 8 e x^{8} + x^{7} \left (\frac{64 d}{7} + \frac{576 e}{7}\right ) + x^{6} \left (96 d + 360 e\right ) + x^{5} \left (432 d + 864 e\right ) + x^{4} \left (1080 d + 1215 e\right ) + x^{3} \left (1620 d + 972 e\right ) + x^{2} \left (1458 d + \frac{729 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)**3,x)

[Out]

729*d*x + 8*e*x**8 + x**7*(64*d/7 + 576*e/7) + x**6*(96*d + 360*e) + x**5*(432*d + 864*e) + x**4*(1080*d + 121
5*e) + x**3*(1620*d + 972*e) + x**2*(1458*d + 729*e/2)

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Giac [B]  time = 1.13701, size = 122, normalized size = 3.94 \begin{align*} 8 \, x^{8} e + \frac{64}{7} \, d x^{7} + \frac{576}{7} \, x^{7} e + 96 \, d x^{6} + 360 \, x^{6} e + 432 \, d x^{5} + 864 \, x^{5} e + 1080 \, d x^{4} + 1215 \, x^{4} e + 1620 \, d x^{3} + 972 \, x^{3} e + 1458 \, d x^{2} + \frac{729}{2} \, x^{2} e + 729 \, 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)^3,x, algorithm="giac")

[Out]

8*x^8*e + 64/7*d*x^7 + 576/7*x^7*e + 96*d*x^6 + 360*x^6*e + 432*d*x^5 + 864*x^5*e + 1080*d*x^4 + 1215*x^4*e +
1620*d*x^3 + 972*x^3*e + 1458*d*x^2 + 729/2*x^2*e + 729*d*x