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

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 = 0.520675, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.0308358, size = 23, normalized size = 0.74 \[ \frac{5 \sqrt{1-x^2}+3}{25 x+20} \]

Antiderivative was successfully verified.

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

[Out]

(3 + 5*Sqrt[1 - x^2])/(20 + 25*x)

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Maple [A]  time = 0.003, size = 32, normalized size = 1. \[{\frac{1}{5}\sqrt{- \left ( x+{\frac{4}{5}} \right ) ^{2}+{\frac{8\,x}{5}}+{\frac{41}{25}}} \left ( x+{\frac{4}{5}} \right ) ^{-1}}+{\frac{3}{20+25\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.795728, size = 34, normalized size = 1.1 \[ \frac{5 \, \sqrt{x + 1} \sqrt{-x + 1} + 3}{5 \,{\left (5 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.2666, 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(-(4*x + 3*sqrt(-x^2 + 1) + 5)/(sqrt(-x^2 + 1)*(5*x + 4)^2),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. \[ - \int \frac{4 x}{25 x^{2} \sqrt{- x^{2} + 1} + 40 x \sqrt{- x^{2} + 1} + 16 \sqrt{- x^{2} + 1}}\, dx - \int \frac{3 \sqrt{- x^{2} + 1}}{25 x^{2} \sqrt{- x^{2} + 1} + 40 x \sqrt{- x^{2} + 1} + 16 \sqrt{- x^{2} + 1}}\, dx - \int \frac{5}{25 x^{2} \sqrt{- x^{2} + 1} + 40 x \sqrt{- x^{2} + 1} + 16 \sqrt{- x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(4*x/(25*x**2*sqrt(-x**2 + 1) + 40*x*sqrt(-x**2 + 1) + 16*sqrt(-x**2 +
1)), x) - Integral(3*sqrt(-x**2 + 1)/(25*x**2*sqrt(-x**2 + 1) + 40*x*sqrt(-x**2
+ 1) + 16*sqrt(-x**2 + 1)), x) - Integral(5/(25*x**2*sqrt(-x**2 + 1) + 40*x*sqrt
(-x**2 + 1) + 16*sqrt(-x**2 + 1)), x)

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GIAC/XCAS [A]  time = 0.28688, size = 74, normalized size = 2.39 \[ -\frac{1}{5} \, i{\rm sign}\left (\frac{1}{5 \, x + 4}\right ) + \frac{\sqrt{\frac{8}{5 \, x + 4} + \frac{9}{{\left (5 \, x + 4\right )}^{2}} - 1}}{5 \,{\rm sign}\left (\frac{1}{5 \, x + 4}\right )} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*i*sign(1/(5*x + 4)) + 1/5*sqrt(8/(5*x + 4) + 9/(5*x + 4)^2 - 1)/sign(1/(5*x
+ 4)) + 3/5/(5*x + 4)