3.671 \(\int \frac{1}{3-3 x^2-5 \sqrt{1-x^2}-4 x \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=31 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

[Out]

3/(5*(4 + 5*x)) + Sqrt[1 - x^2]/(4 + 5*x)

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Rubi [A]  time = 0.300407, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

Antiderivative was successfully verified.

[In]  Int[(3 - 3*x^2 - 5*Sqrt[1 - x^2] - 4*x*Sqrt[1 - x^2])^(-1),x]

[Out]

3/(5*(4 + 5*x)) + Sqrt[1 - x^2]/(4 + 5*x)

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Rubi in Sympy [A]  time = 4.96378, size = 46, normalized size = 1.48 \[ \frac{4 + \frac{8 \left (- \sqrt{- x^{2} + 1} + 1\right )}{x}}{8 \left (1 - \frac{4 \left (\sqrt{- x^{2} + 1} - 1\right )}{x} + \frac{4 \left (\sqrt{- x^{2} + 1} - 1\right )^{2}}{x^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(3-3*x**2-5*(-x**2+1)**(1/2)-4*x*(-x**2+1)**(1/2)),x)

[Out]

(4 + 8*(-sqrt(-x**2 + 1) + 1)/x)/(8*(1 - 4*(sqrt(-x**2 + 1) - 1)/x + 4*(sqrt(-x*
*2 + 1) - 1)**2/x**2))

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Mathematica [A]  time = 0.0285025, size = 23, normalized size = 0.74 \[ \frac{5 \sqrt{1-x^2}+3}{25 x+20} \]

Antiderivative was successfully verified.

[In]  Integrate[(3 - 3*x^2 - 5*Sqrt[1 - x^2] - 4*x*Sqrt[1 - x^2])^(-1),x]

[Out]

(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)

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Maple [B]  time = 0.004, size = 81, normalized size = 2.6 \[{\frac{3}{20+25\,x}}-{\frac{1}{2}\sqrt{- \left ( 1+x \right ) ^{2}+2+2\,x}}+{\frac{5}{9} \left ( - \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}} \right ) ^{{\frac{3}{2}}} \left ( x+{\frac{4}{5}} \right ) ^{-1}}+{\frac{5\,x}{9}\sqrt{- \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}}}}+{\frac{1}{18}\sqrt{- \left ( -1+x \right ) ^{2}-2\,x+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(3-3*x^2-5*(-x^2+1)^(1/2)-4*x*(-x^2+1)^(1/2)),x)

[Out]

3/5/(4+5*x)-1/2*(-(1+x)^2+2+2*x)^(1/2)+5/9/(x+4/5)*(-(x+4/5)^2+8/5*x+41/25)^(3/2
)+5/9*x*(-(x+4/5)^2+8/5*x+41/25)^(1/2)+1/18*(-(-1+x)^2-2*x+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{1}{3 \, x^{2} + 4 \, \sqrt{-x^{2} + 1} x + 5 \, \sqrt{-x^{2} + 1} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 + 4*sqrt(-x^2 + 1)*x + 5*sqrt(-x^2 + 1) - 3),x, algorithm="maxima")

[Out]

-integrate(1/(3*x^2 + 4*sqrt(-x^2 + 1)*x + 5*sqrt(-x^2 + 1) - 3), x)

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Fricas [A]  time = 0.266187, size = 68, normalized size = 2.19 \[ -\frac{20 \, x^{2} - \sqrt{-x^{2} + 1}{\left (25 \, x + 12\right )} + 25 \, x + 12}{20 \,{\left (\sqrt{-x^{2} + 1}{\left (5 \, x + 4\right )} - 5 \, x - 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 + 4*sqrt(-x^2 + 1)*x + 5*sqrt(-x^2 + 1) - 3),x, algorithm="fricas")

[Out]

-1/20*(20*x^2 - sqrt(-x^2 + 1)*(25*x + 12) + 25*x + 12)/(sqrt(-x^2 + 1)*(5*x + 4
) - 5*x - 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{1}{3 x^{2} + 4 x \sqrt{- x^{2} + 1} + 5 \sqrt{- x^{2} + 1} - 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(3-3*x**2-5*(-x**2+1)**(1/2)-4*x*(-x**2+1)**(1/2)),x)

[Out]

-Integral(1/(3*x**2 + 4*x*sqrt(-x**2 + 1) + 5*sqrt(-x**2 + 1) - 3), x)

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GIAC/XCAS [A]  time = 0.268847, size = 92, normalized size = 2.97 \[ \frac{\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - 4}{4 \,{\left (\frac{5 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 2\right )}} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/(3*x^2 + 4*sqrt(-x^2 + 1)*x + 5*sqrt(-x^2 + 1) - 3),x, algorithm="giac")

[Out]

1/4*(5*(sqrt(-x^2 + 1) - 1)/x - 4)/(5*(sqrt(-x^2 + 1) - 1)/x - 2*(sqrt(-x^2 + 1)
- 1)^2/x^2 - 2) + 3/5/(5*x + 4)