3.208 \(\int x (9+12 x+4 x^2)^{5/2} \, dx\)

Optimal. Leaf size=42 \[ \frac{1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac{1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]

[Out]

-((3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/8 + (9 + 12*x + 4*x^2)^(7/2)/28

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Rubi [A]  time = 0.0088291, 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} \[ \frac{1}{28} \left (4 x^2+12 x+9\right )^{7/2}-\frac{1}{8} (2 x+3) \left (4 x^2+12 x+9\right )^{5/2} \]

Antiderivative was successfully verified.

[In]

Int[x*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

-((3 + 2*x)*(9 + 12*x + 4*x^2)^(5/2))/8 + (9 + 12*x + 4*x^2)^(7/2)/28

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 )^{5/2} \, dx &=\frac{1}{28} \left (9+12 x+4 x^2\right )^{7/2}-\frac{3}{2} \int \left (9+12 x+4 x^2\right )^{5/2} \, dx\\ &=-\frac{1}{8} (3+2 x) \left (9+12 x+4 x^2\right )^{5/2}+\frac{1}{28} \left (9+12 x+4 x^2\right )^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0174827, size = 47, normalized size = 1.12 \[ \frac{x^2 \sqrt{(2 x+3)^2} \left (64 x^5+560 x^4+2016 x^3+3780 x^2+3780 x+1701\right )}{28 x+42} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(9 + 12*x + 4*x^2)^(5/2),x]

[Out]

(x^2*Sqrt[(3 + 2*x)^2]*(1701 + 3780*x + 3780*x^2 + 2016*x^3 + 560*x^4 + 64*x^5))/(42 + 28*x)

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Maple [A]  time = 0.064, size = 47, normalized size = 1.1 \begin{align*}{\frac{{x}^{2} \left ( 64\,{x}^{5}+560\,{x}^{4}+2016\,{x}^{3}+3780\,{x}^{2}+3780\,x+1701 \right ) }{14\, \left ( 3+2\,x \right ) ^{5}} \left ( \left ( 3+2\,x \right ) ^{2} \right ) ^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(4*x^2+12*x+9)^(5/2),x)

[Out]

1/14*x^2*(64*x^5+560*x^4+2016*x^3+3780*x^2+3780*x+1701)*((3+2*x)^2)^(5/2)/(3+2*x)^5

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Maxima [A]  time = 1.70609, size = 59, normalized size = 1.4 \begin{align*} \frac{1}{28} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{7}{2}} - \frac{1}{4} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} x - \frac{3}{8} \,{\left (4 \, x^{2} + 12 \, x + 9\right )}^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x^2+12*x+9)^(5/2),x, algorithm="maxima")

[Out]

1/28*(4*x^2 + 12*x + 9)^(7/2) - 1/4*(4*x^2 + 12*x + 9)^(5/2)*x - 3/8*(4*x^2 + 12*x + 9)^(5/2)

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Fricas [A]  time = 1.61433, size = 82, normalized size = 1.95 \begin{align*} \frac{32}{7} \, x^{7} + 40 \, x^{6} + 144 \, x^{5} + 270 \, x^{4} + 270 \, x^{3} + \frac{243}{2} \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x^2+12*x+9)^(5/2),x, algorithm="fricas")

[Out]

32/7*x^7 + 40*x^6 + 144*x^5 + 270*x^4 + 270*x^3 + 243/2*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{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(4*x**2+12*x+9)**(5/2),x)

[Out]

Integral(x*((2*x + 3)**2)**(5/2), x)

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Giac [B]  time = 1.30584, size = 101, normalized size = 2.4 \begin{align*} \frac{32}{7} \, x^{7} \mathrm{sgn}\left (2 \, x + 3\right ) + 40 \, x^{6} \mathrm{sgn}\left (2 \, x + 3\right ) + 144 \, x^{5} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{4} \mathrm{sgn}\left (2 \, x + 3\right ) + 270 \, x^{3} \mathrm{sgn}\left (2 \, x + 3\right ) + \frac{243}{2} \, x^{2} \mathrm{sgn}\left (2 \, x + 3\right ) - \frac{729}{56} \, \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)^(5/2),x, algorithm="giac")

[Out]

32/7*x^7*sgn(2*x + 3) + 40*x^6*sgn(2*x + 3) + 144*x^5*sgn(2*x + 3) + 270*x^4*sgn(2*x + 3) + 270*x^3*sgn(2*x +
3) + 243/2*x^2*sgn(2*x + 3) - 729/56*sgn(2*x + 3)