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

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 = 0.0580792, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029 \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

Antiderivative was successfully verified.

[In]  Int[-3/(4 + 5*x)^2 - (5 + 4*x)/((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 [A]  time = 2.61917, size = 19, normalized size = 0.61 \[ \frac{\sqrt{- x^{2} + 1}}{5 x + 4} + \frac{3}{5 \left (5 x + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0532263, size = 23, normalized size = 0.74 \[ \frac{5 \sqrt{1-x^2}+3}{25 x+20} \]

Antiderivative was successfully verified.

[In]  Integrate[-3/(4 + 5*x)^2 - (5 + 4*x)/((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 = 0.013, 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(-3/(4+5*x)^2+(-5-4*x)/(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 = 1.04935, size = 36, normalized size = 1.16 \[ \frac{\sqrt{-x^{2} + 1}}{5 \, x + 4} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.266149, 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 + 5)/(sqrt(-x^2 + 1)*(5*x + 4)^2) - 3/(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(-3/(4+5*x)**2+(-5-4*x)/(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 [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{4 \, x + 5}{\sqrt{-x^{2} + 1}{\left (5 \, x + 4\right )}^{2}} - \frac{3}{{\left (5 \, x + 4\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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