3.680 \(\int \frac{x+\left (1-9 x^2\right )^{3/2}}{\sqrt{1-9 x^2}} \, dx\)

Optimal. Leaf size=22 \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]

[Out]

x - 3*x^3 - Sqrt[1 - 9*x^2]/9

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Rubi [A]  time = 0.132968, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]

Antiderivative was successfully verified.

[In]  Int[(x + (1 - 9*x^2)^(3/2))/Sqrt[1 - 9*x^2],x]

[Out]

x - 3*x^3 - Sqrt[1 - 9*x^2]/9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + \left (- 9 x^{2} + 1\right )^{\frac{3}{2}}}{\sqrt{- 9 x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+(-9*x**2+1)**(3/2))/(-9*x**2+1)**(1/2),x)

[Out]

Integral((x + (-9*x**2 + 1)**(3/2))/sqrt(-9*x**2 + 1), x)

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Mathematica [A]  time = 0.00293968, size = 22, normalized size = 1. \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]

Antiderivative was successfully verified.

[In]  Integrate[(x + (1 - 9*x^2)^(3/2))/Sqrt[1 - 9*x^2],x]

[Out]

x - 3*x^3 - Sqrt[1 - 9*x^2]/9

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Maple [A]  time = 0.002, size = 19, normalized size = 0.9 \[ x-3\,{x}^{3}-{\frac{1}{9}\sqrt{-9\,{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x+(-9*x^2+1)^(3/2))/(-9*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.699771, size = 24, normalized size = 1.09 \[ -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-9*x^2 + 1)^(3/2) + x)/sqrt(-9*x^2 + 1),x, algorithm="maxima")

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)

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Fricas [A]  time = 0.262978, size = 62, normalized size = 2.82 \[ \frac{3 \, x^{3} + x^{2} -{\left (3 \, x^{3} - x\right )} \sqrt{-9 \, x^{2} + 1} - x}{\sqrt{-9 \, x^{2} + 1} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-9*x^2 + 1)^(3/2) + x)/sqrt(-9*x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.42675, size = 17, normalized size = 0.77 \[ - 3 x^{3} + x - \frac{\sqrt{- 9 x^{2} + 1}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x+(-9*x**2+1)**(3/2))/(-9*x**2+1)**(1/2),x)

[Out]

-3*x**3 + x - sqrt(-9*x**2 + 1)/9

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GIAC/XCAS [A]  time = 0.279793, size = 24, normalized size = 1.09 \[ -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-9*x^2 + 1)^(3/2) + x)/sqrt(-9*x^2 + 1),x, algorithm="giac")

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)