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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 = 0.01, size = 22, normalized size = 1.00 \[ -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \]

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

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]

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)

________________________________________________________________________________________

giac [A]  time = 0.34, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 19, normalized size = 0.86 \[ -3 x^{3}+x -\frac {\sqrt {-9 x^{2}+1}}{9} \]

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)

________________________________________________________________________________________

maxima [A]  time = 0.58, size = 18, normalized size = 0.82 \[ -3 \, x^{3} + x - \frac {1}{9} \, \sqrt {-9 \, x^{2} + 1} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 18, normalized size = 0.82 \[ x-3\,x^3-\frac {\sqrt {\frac {1}{9}-x^2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - 3*x^3 - (1/9 - x^2)^(1/2)/3

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 17, normalized size = 0.77 \[ - 3 x^{3} + x - \frac {\sqrt {1 - 9 x^{2}}}{9} \]

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

________________________________________________________________________________________