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

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 = 0.0502457, 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 \[ -\frac{4}{567} (3 x+2)^{7/2}+\frac{8}{135} (3 x+2)^{5/2}-\frac{10}{81} (3 x+2)^{3/2}-\frac{4}{81} \sqrt{3 x+2} \]

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

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Rubi in Sympy [A]  time = 3.45853, size = 46, normalized size = 0.87 \[ - \frac{4 \left (3 x + 2\right )^{\frac{7}{2}}}{567} + \frac{8 \left (3 x + 2\right )^{\frac{5}{2}}}{135} - \frac{10 \left (3 x + 2\right )^{\frac{3}{2}}}{81} - \frac{4 \sqrt{3 x + 2}}{81} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_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*sq
rt(3*x + 2)/81

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Mathematica [A]  time = 0.0135814, 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 = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{540\,{x}^{3}-432\,{x}^{2}-246\,x+328}{2835}\sqrt{2+3\,x}} \]

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 = 0.691183, size = 50, normalized size = 0.94 \[ -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^3 - x)/sqrt(3*x + 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*sq
rt(3*x + 2)

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Fricas [A]  time = 0.272769, size = 32, normalized size = 0.6 \[ -\frac{2}{2835} \,{\left (270 \, x^{3} - 216 \, x^{2} - 123 \, x + 164\right )} \sqrt{3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^3 - x)/sqrt(3*x + 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 [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.263622, size = 50, normalized size = 0.94 \[ -\frac{4}{567} \,{\left (3 \, x + 2\right )}^{\frac{7}{2}} + \frac{8}{135} \,{\left (3 \, x + 2\right )}^{\frac{5}{2}} - \frac{10}{81} \,{\left (3 \, x + 2\right )}^{\frac{3}{2}} - \frac{4}{81} \, \sqrt{3 \, x + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^3 - x)/sqrt(3*x + 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*sq
rt(3*x + 2)