3.943 \(\int \frac{x-2 x^3}{\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=53 \[ -\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2} \]

[Out]

(-4*Sqrt[2 + 3*x])/81 - (10*(2 + 3*x)^(3/2))/81 + (8*(2 + 3*x)^(5/2))/135 - (4*(2 + 3*x)^(7/2))/567

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Rubi [A]  time = 0.0208054, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1593, 772} \[ -\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2} \]

Antiderivative was successfully verified.

[In]

Int[(x - 2*x^3)/Sqrt[2 + 3*x],x]

[Out]

(-4*Sqrt[2 + 3*x])/81 - (10*(2 + 3*x)^(3/2))/81 + (8*(2 + 3*x)^(5/2))/135 - (4*(2 + 3*x)^(7/2))/567

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{x-2 x^3}{\sqrt{2+3 x}} \, dx &=\int \frac{x \left (1-2 x^2\right )}{\sqrt{2+3 x}} \, dx\\ &=\int \left (-\frac{2}{27 \sqrt{2+3 x}}-\frac{5}{9} \sqrt{2+3 x}+\frac{4}{9} (2+3 x)^{3/2}-\frac{2}{27} (2+3 x)^{5/2}\right ) \, dx\\ &=-\frac{4}{81} \sqrt{2+3 x}-\frac{10}{81} (2+3 x)^{3/2}+\frac{8}{135} (2+3 x)^{5/2}-\frac{4}{567} (2+3 x)^{7/2}\\ \end{align*}

Mathematica [A]  time = 0.0250221, size = 28, normalized size = 0.53 \[ -\frac{2 \sqrt{3 x+2} \left (270 x^3-216 x^2-123 x+164\right )}{2835} \]

Antiderivative was successfully verified.

[In]

Integrate[(x - 2*x^3)/Sqrt[2 + 3*x],x]

[Out]

(-2*Sqrt[2 + 3*x]*(164 - 123*x - 216*x^2 + 270*x^3))/2835

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Maple [A]  time = 0.005, size = 25, normalized size = 0.5 \begin{align*} -{\frac{540\,{x}^{3}-432\,{x}^{2}-246\,x+328}{2835}\sqrt{2+3\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2835*(270*x^3-216*x^2-123*x+164)*(2+3*x)^(1/2)

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Maxima [A]  time = 1.0009, size = 50, normalized size = 0.94 \begin{align*} -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/567*(3*x + 2)^(7/2) + 8/135*(3*x + 2)^(5/2) - 10/81*(3*x + 2)^(3/2) - 4/81*sqrt(3*x + 2)

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Fricas [A]  time = 1.44073, size = 77, normalized size = 1.45 \begin{align*} -\frac{2}{2835} \,{\left (270 \, x^{3} - 216 \, x^{2} - 123 \, x + 164\right )} \sqrt{3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/2835*(270*x^3 - 216*x^2 - 123*x + 164)*sqrt(3*x + 2)

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Sympy [A]  time = 24.6686, size = 46, normalized size = 0.87 \begin{align*} - \frac{4 \left (3 x + 2\right )^{\frac{7}{2}}}{567} + \frac{8 \left (3 x + 2\right )^{\frac{5}{2}}}{135} - \frac{10 \left (3 x + 2\right )^{\frac{3}{2}}}{81} - \frac{4 \sqrt{3 x + 2}}{81} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*(3*x + 2)**(7/2)/567 + 8*(3*x + 2)**(5/2)/135 - 10*(3*x + 2)**(3/2)/81 - 4*sqrt(3*x + 2)/81

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Giac [A]  time = 1.1018, size = 50, normalized size = 0.94 \begin{align*} -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/567*(3*x + 2)^(7/2) + 8/135*(3*x + 2)^(5/2) - 10/81*(3*x + 2)^(3/2) - 4/81*sqrt(3*x + 2)