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

Optimal. Leaf size=23 \[ \frac{1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \]

[Out]

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

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Rubi [A]  time = 0.0028427, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {609} \[ \frac{1}{12} (3 x+2) \left (9 x^2+12 x+4\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(4 + 12*x + 9*x^2)^(3/2),x]

[Out]

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

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

Mathematica [A]  time = 0.0087792, size = 20, normalized size = 0.87 \[ \frac{1}{12} (3 x+2) \left ((3 x+2)^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

Integrate[(4 + 12*x + 9*x^2)^(3/2),x]

[Out]

((2 + 3*x)*((2 + 3*x)^2)^(3/2))/12

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Maple [A]  time = 0.071, size = 35, normalized size = 1.5 \begin{align*}{\frac{x \left ( 27\,{x}^{3}+72\,{x}^{2}+72\,x+32 \right ) }{4\, \left ( 2+3\,x \right ) ^{3}} \left ( \left ( 2+3\,x \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x*(27*x^3+72*x^2+72*x+32)*((2+3*x)^2)^(3/2)/(2+3*x)^3

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Maxima [A]  time = 1.70056, size = 41, normalized size = 1.78 \begin{align*} \frac{1}{4} \,{\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac{3}{2}} x + \frac{1}{6} \,{\left (9 \, x^{2} + 12 \, x + 4\right )}^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(9*x^2 + 12*x + 4)^(3/2)*x + 1/6*(9*x^2 + 12*x + 4)^(3/2)

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Fricas [A]  time = 2.08727, size = 46, normalized size = 2. \begin{align*} \frac{27}{4} \, x^{4} + 18 \, x^{3} + 18 \, x^{2} + 8 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/4*x^4 + 18*x^3 + 18*x^2 + 8*x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (9 x^{2} + 12 x + 4\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20392, size = 69, normalized size = 3. \begin{align*} \frac{27}{4} \, x^{4} \mathrm{sgn}\left (3 \, x + 2\right ) + 18 \, x^{3} \mathrm{sgn}\left (3 \, x + 2\right ) + 18 \, x^{2} \mathrm{sgn}\left (3 \, x + 2\right ) + 8 \, x \mathrm{sgn}\left (3 \, x + 2\right ) + \frac{4}{3} \, \mathrm{sgn}\left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/4*x^4*sgn(3*x + 2) + 18*x^3*sgn(3*x + 2) + 18*x^2*sgn(3*x + 2) + 8*x*sgn(3*x + 2) + 4/3*sgn(3*x + 2)