### 3.140 $$\int x^3 \sqrt{4-9 x^2} \, dx$$

Optimal. Leaf size=31 $\frac{1}{405} \left (4-9 x^2\right )^{5/2}-\frac{4}{243} \left (4-9 x^2\right )^{3/2}$

[Out]

(-4*(4 - 9*x^2)^(3/2))/243 + (4 - 9*x^2)^(5/2)/405

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Rubi [A]  time = 0.0139225, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.133, Rules used = {266, 43} $\frac{1}{405} \left (4-9 x^2\right )^{5/2}-\frac{4}{243} \left (4-9 x^2\right )^{3/2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Sqrt[4 - 9*x^2],x]

[Out]

(-4*(4 - 9*x^2)^(3/2))/243 + (4 - 9*x^2)^(5/2)/405

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 x^3 \sqrt{4-9 x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \sqrt{4-9 x} x \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{4}{9} \sqrt{4-9 x}-\frac{1}{9} (4-9 x)^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac{4}{243} \left (4-9 x^2\right )^{3/2}+\frac{1}{405} \left (4-9 x^2\right )^{5/2}\\ \end{align*}

Mathematica [A]  time = 0.0105125, size = 22, normalized size = 0.71 $-\frac{\left (4-9 x^2\right )^{3/2} \left (27 x^2+8\right )}{1215}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Sqrt[4 - 9*x^2],x]

[Out]

-((4 - 9*x^2)^(3/2)*(8 + 27*x^2))/1215

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Maple [A]  time = 0.004, size = 29, normalized size = 0.9 \begin{align*}{\frac{ \left ( -2+3\,x \right ) \left ( 2+3\,x \right ) \left ( 27\,{x}^{2}+8 \right ) }{1215}\sqrt{-9\,{x}^{2}+4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(-9*x^2+4)^(1/2),x)

[Out]

1/1215*(-2+3*x)*(2+3*x)*(27*x^2+8)*(-9*x^2+4)^(1/2)

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Maxima [A]  time = 1.39839, size = 35, normalized size = 1.13 \begin{align*} -\frac{1}{45} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} x^{2} - \frac{8}{1215} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-9*x^2+4)^(1/2),x, algorithm="maxima")

[Out]

-1/45*(-9*x^2 + 4)^(3/2)*x^2 - 8/1215*(-9*x^2 + 4)^(3/2)

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Fricas [A]  time = 2.56603, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{1215} \,{\left (243 \, x^{4} - 36 \, x^{2} - 32\right )} \sqrt{-9 \, x^{2} + 4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-9*x^2+4)^(1/2),x, algorithm="fricas")

[Out]

1/1215*(243*x^4 - 36*x^2 - 32)*sqrt(-9*x^2 + 4)

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Sympy [A]  time = 0.592645, size = 44, normalized size = 1.42 \begin{align*} \frac{x^{4} \sqrt{4 - 9 x^{2}}}{5} - \frac{4 x^{2} \sqrt{4 - 9 x^{2}}}{135} - \frac{32 \sqrt{4 - 9 x^{2}}}{1215} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(-9*x**2+4)**(1/2),x)

[Out]

x**4*sqrt(4 - 9*x**2)/5 - 4*x**2*sqrt(4 - 9*x**2)/135 - 32*sqrt(4 - 9*x**2)/1215

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Giac [A]  time = 1.06412, size = 43, normalized size = 1.39 \begin{align*} \frac{1}{405} \,{\left (9 \, x^{2} - 4\right )}^{2} \sqrt{-9 \, x^{2} + 4} - \frac{4}{243} \,{\left (-9 \, x^{2} + 4\right )}^{\frac{3}{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(-9*x^2+4)^(1/2),x, algorithm="giac")

[Out]

1/405*(9*x^2 - 4)^2*sqrt(-9*x^2 + 4) - 4/243*(-9*x^2 + 4)^(3/2)