### 3.1758 $$\int (d+e x) (9+12 x+4 x^2)^p \, dx$$

Optimal. Leaf size=60 $\frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)}$

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^p)/(4*(1 + 2*p)) + (e*(9 + 12*x + 4*x^2)^(1 + p))/(8*(1 + p))

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Rubi [A]  time = 0.0149501, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.111, Rules used = {640, 609} $\frac{(2 x+3) (2 d-3 e) \left (4 x^2+12 x+9\right )^p}{4 (2 p+1)}+\frac{e \left (4 x^2+12 x+9\right )^{p+1}}{8 (p+1)}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)*(9 + 12*x + 4*x^2)^p,x]

[Out]

((2*d - 3*e)*(3 + 2*x)*(9 + 12*x + 4*x^2)^p)/(4*(1 + 2*p)) + (e*(9 + 12*x + 4*x^2)^(1 + p))/(8*(1 + p))

Rule 640

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(a + b*x + c*x^2)^(p +
1))/(2*c*(p + 1)), x] + Dist[(2*c*d - b*e)/(2*c), Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}
, x] && NeQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 609

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p + 1
)), x] /; FreeQ[{a, b, c, p}, x] && EqQ[b^2 - 4*a*c, 0] && NeQ[p, -2^(-1)]

Rubi steps

\begin{align*} \int (d+e x) \left (9+12 x+4 x^2\right )^p \, dx &=\frac{e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}+\frac{1}{2} (2 d-3 e) \int \left (9+12 x+4 x^2\right )^p \, dx\\ &=\frac{(2 d-3 e) (3+2 x) \left (9+12 x+4 x^2\right )^p}{4 (1+2 p)}+\frac{e \left (9+12 x+4 x^2\right )^{1+p}}{8 (1+p)}\\ \end{align*}

Mathematica [A]  time = 0.0246458, size = 48, normalized size = 0.8 $\frac{(2 x+3) \left ((2 x+3)^2\right )^p (4 d (p+1)+e ((4 p+2) x-3))}{8 (p+1) (2 p+1)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((3 + 2*x)*((3 + 2*x)^2)^p*(4*d*(1 + p) + e*(-3 + (2 + 4*p)*x)))/(8*(1 + p)*(1 + 2*p))

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Maple [A]  time = 0.042, size = 52, normalized size = 0.9 \begin{align*}{\frac{ \left ( 4\,{x}^{2}+12\,x+9 \right ) ^{p} \left ( 4\,epx+4\,dp+2\,ex+4\,d-3\,e \right ) \left ( 3+2\,x \right ) }{16\,{p}^{2}+24\,p+8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*x^2+12*x+9)^p*(4*e*p*x+4*d*p+2*e*x+4*d-3*e)*(3+2*x)/(2*p^2+3*p+1)

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Maxima [A]  time = 1.07015, size = 88, normalized size = 1.47 \begin{align*} \frac{{\left (4 \,{\left (2 \, p + 1\right )} x^{2} + 12 \, p x - 9\right )} e{\left (2 \, x + 3\right )}^{2 \, p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} + \frac{d{\left (2 \, x + 3\right )}^{2 \, p}{\left (2 \, x + 3\right )}}{2 \,{\left (2 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(2*p + 1)*x^2 + 12*p*x - 9)*e*(2*x + 3)^(2*p)/(2*p^2 + 3*p + 1) + 1/2*d*(2*x + 3)^(2*p)*(2*x + 3)/(2*p
+ 1)

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Fricas [A]  time = 1.8246, size = 154, normalized size = 2.57 \begin{align*} \frac{{\left (4 \,{\left (2 \, e p + e\right )} x^{2} + 12 \, d p + 4 \,{\left ({\left (2 \, d + 3 \, e\right )} p + 2 \, d\right )} x + 12 \, d - 9 \, e\right )}{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p}}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*(2*e*p + e)*x^2 + 12*d*p + 4*((2*d + 3*e)*p + 2*d)*x + 12*d - 9*e)*(4*x^2 + 12*x + 9)^p/(2*p^2 + 3*p +
1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [B]  time = 1.16227, size = 205, normalized size = 3.42 \begin{align*} \frac{8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x^{2} e + 8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p x + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} p x e + 4 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} x^{2} e + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d p + 8 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d x + 12 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} d - 9 \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{p} e}{8 \,{\left (2 \, p^{2} + 3 \, p + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(8*(4*x^2 + 12*x + 9)^p*p*x^2*e + 8*(4*x^2 + 12*x + 9)^p*d*p*x + 12*(4*x^2 + 12*x + 9)^p*p*x*e + 4*(4*x^2
+ 12*x + 9)^p*x^2*e + 12*(4*x^2 + 12*x + 9)^p*d*p + 8*(4*x^2 + 12*x + 9)^p*d*x + 12*(4*x^2 + 12*x + 9)^p*d - 9
*(4*x^2 + 12*x + 9)^p*e)/(2*p^2 + 3*p + 1)