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

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 = 0.0152564, 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, Rules used = {803} \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

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)

Rule 803

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Simp[((e*f - d*g
)*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 + a*e^2)), x] /; FreeQ[{a, c, d, e, f, g, m, p}, x]
&& NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplify[m + 2*p + 3], 0] && EqQ[c*d*f + a*e*g, 0]

Rubi steps

\begin{align*} \int \left (-\frac{3}{(4+5 x)^2}-\frac{5+4 x}{(4+5 x)^2 \sqrt{1-x^2}}\right ) \, dx &=\frac{3}{5 (4+5 x)}-\int \frac{5+4 x}{(4+5 x)^2 \sqrt{1-x^2}} \, dx\\ &=\frac{3}{5 (4+5 x)}+\frac{\sqrt{1-x^2}}{4+5 x}\\ \end{align*}

Mathematica [A]  time = 0.189383, 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 = 0.009, size = 32, normalized size = 1. \begin{align*}{\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}} \end{align*}

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 = 1.57145, size = 36, normalized size = 1.16 \begin{align*} \frac{\sqrt{-x^{2} + 1}}{5 \, x + 4} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.67726, size = 65, normalized size = 2.1 \begin{align*} \frac{25 \, x + 20 \, \sqrt{-x^{2} + 1} + 32}{20 \,{\left (5 \, x + 4\right )}} \end{align*}

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, algorithm="fricas")

[Out]

1/20*(25*x + 20*sqrt(-x^2 + 1) + 32)/(5*x + 4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{4 x}{25 x^{2} \sqrt{1 - x^{2}} + 40 x \sqrt{1 - x^{2}} + 16 \sqrt{1 - x^{2}}}\, dx - \int \frac{3 \sqrt{1 - x^{2}}}{25 x^{2} \sqrt{1 - x^{2}} + 40 x \sqrt{1 - x^{2}} + 16 \sqrt{1 - x^{2}}}\, dx - \int \frac{5}{25 x^{2} \sqrt{1 - x^{2}} + 40 x \sqrt{1 - x^{2}} + 16 \sqrt{1 - x^{2}}}\, dx \end{align*}

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, algorithm="giac")

[Out]

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