3.835 \(\int (1-9 x^2+\frac{x}{\sqrt{1-9 x^2}}) \, dx\)

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 = 0.0050603, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {261} \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]

Antiderivative was successfully verified.

[In]

Int[1 - 9*x^2 + x/Sqrt[1 - 9*x^2],x]

[Out]

x - 3*x^3 - Sqrt[1 - 9*x^2]/9

Rule 261

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

Rubi steps

\begin{align*} \int \left (1-9 x^2+\frac{x}{\sqrt{1-9 x^2}}\right ) \, dx &=x-3 x^3+\int \frac{x}{\sqrt{1-9 x^2}} \, dx\\ &=x-3 x^3-\frac{1}{9} \sqrt{1-9 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0073615, size = 22, normalized size = 1. \[ -3 x^3-\frac{1}{9} \sqrt{1-9 x^2}+x \]

Antiderivative was successfully verified.

[In]

Integrate[1 - 9*x^2 + x/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 \begin{align*} x-3\,{x}^{3}-{\frac{1}{9}\sqrt{-9\,{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1-9*x^2+x/(-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 = 1.14436, size = 24, normalized size = 1.09 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)

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Fricas [A]  time = 1.75207, size = 47, normalized size = 2.14 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)

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Sympy [A]  time = 0.13385, size = 17, normalized size = 0.77 \begin{align*} - 3 x^{3} + x - \frac{\sqrt{1 - 9 x^{2}}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x**3 + x - sqrt(1 - 9*x**2)/9

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Giac [A]  time = 1.10133, size = 24, normalized size = 1.09 \begin{align*} -3 \, x^{3} + x - \frac{1}{9} \, \sqrt{-9 \, x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)