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

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

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6742, 261} \[ -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \]

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

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ -3 x^3-\frac {1}{9} \sqrt {1-9 x^2}+x \]

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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fricas [A]  time = 0.40, 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((x+(-9*x^2+1)^(3/2))/(-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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giac [A]  time = 0.57, 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((x+(-9*x^2+1)^(3/2))/(-9*x^2+1)^(1/2),x, algorithm="giac")

[Out]

-3*x^3 + x - 1/9*sqrt(-9*x^2 + 1)

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

[Out]

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

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maxima [A]  time = 0.82, 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((x+(-9*x^2+1)^(3/2))/(-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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mupad [B]  time = 0.03, 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)^(3/2))/(1 - 9*x^2)^(1/2),x)

[Out]

x - 3*x^3 - (1/9 - x^2)^(1/2)/3

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sympy [A]  time = 1.17, 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((x+(-9*x**2+1)**(3/2))/(-9*x**2+1)**(1/2),x)

[Out]

-3*x**3 + x - sqrt(1 - 9*x**2)/9

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