### 3.209 $$\int x (9+12 x+4 x^2)^{3/2} \, dx$$

Optimal. Leaf size=42 $\frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}$

[Out]

(-3*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (9 + 12*x + 4*x^2)^(5/2)/20

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Rubi [A]  time = 0.008506, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.125, Rules used = {640, 609} $\frac{1}{20} \left (4 x^2+12 x+9\right )^{5/2}-\frac{3}{16} (2 x+3) \left (4 x^2+12 x+9\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*(9 + 12*x + 4*x^2)^(3/2),x]

[Out]

(-3*(3 + 2*x)*(9 + 12*x + 4*x^2)^(3/2))/16 + (9 + 12*x + 4*x^2)^(5/2)/20

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 x \left (9+12 x+4 x^2\right )^{3/2} \, dx &=\frac{1}{20} \left (9+12 x+4 x^2\right )^{5/2}-\frac{3}{2} \int \left (9+12 x+4 x^2\right )^{3/2} \, dx\\ &=-\frac{3}{16} (3+2 x) \left (9+12 x+4 x^2\right )^{3/2}+\frac{1}{20} \left (9+12 x+4 x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0116702, size = 37, normalized size = 0.88 $\frac{x^2 \sqrt{(2 x+3)^2} \left (16 x^3+90 x^2+180 x+135\right )}{20 x+30}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(135 + 180*x + 90*x^2 + 16*x^3))/(30 + 20*x)

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Maple [A]  time = 0.067, size = 37, normalized size = 0.9 \begin{align*}{\frac{{x}^{2} \left ( 16\,{x}^{3}+90\,{x}^{2}+180\,x+135 \right ) }{10\, \left ( 3+2\,x \right ) ^{3}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^2*(16*x^3+90*x^2+180*x+135)*((3+2*x)^2)^(3/2)/(3+2*x)^3

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Maxima [A]  time = 1.71987, size = 59, normalized size = 1.4 \begin{align*} \frac{1}{20} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} x - \frac{9}{16} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x^2+12*x+9)^(3/2),x, algorithm="maxima")

[Out]

1/20*(4*x^2 + 12*x + 9)^(5/2) - 3/8*(4*x^2 + 12*x + 9)^(3/2)*x - 9/16*(4*x^2 + 12*x + 9)^(3/2)

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Fricas [A]  time = 1.65142, size = 50, normalized size = 1.19 \begin{align*} \frac{8}{5} \, x^{5} + 9 \, x^{4} + 18 \, x^{3} + \frac{27}{2} \, x^{2} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x^2+12*x+9)^(3/2),x, algorithm="fricas")

[Out]

8/5*x^5 + 9*x^4 + 18*x^3 + 27/2*x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (\left (2 x + 3\right )^{2}\right )^{\frac{3}{2}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x**2+12*x+9)**(3/2),x)

[Out]

Integral(x*((2*x + 3)**2)**(3/2), x)

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Giac [A]  time = 1.26485, size = 72, normalized size = 1.71 \begin{align*} \frac{8}{5} \, x^{5} \mathrm{sgn}\left (2 \, x + 3\right ) + 9 \, x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 18 \, x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{27}{2} \, x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) - \frac{243}{80} \, \mathrm{sgn}\left (2 \, x + 3\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x^2+12*x+9)^(3/2),x, algorithm="giac")

[Out]

8/5*x^5*sgn(2*x + 3) + 9*x^4*sgn(2*x + 3) + 18*x^3*sgn(2*x + 3) + 27/2*x^2*sgn(2*x + 3) - 243/80*sgn(2*x + 3)