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

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 = 0.287747, 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, Rules used = {6742, 731, 725, 206, 807} \[ \frac{\sqrt{1-x^2}}{5 x+4}+\frac{3}{5 (5 x+4)} \]

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)

Rule 6742

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

Rule 731

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

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 807

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] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e^2
), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], 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]

Rubi steps

\begin{align*} \int \frac{-5-4 x-3 \sqrt{1-x^2}}{(4+5 x)^2 \sqrt{1-x^2}} \, dx &=\int \left (-\frac{3}{(4+5 x)^2}-\frac{5}{(4+5 x)^2 \sqrt{1-x^2}}-\frac{4 x}{(4+5 x)^2 \sqrt{1-x^2}}\right ) \, dx\\ &=\frac{3}{5 (4+5 x)}-4 \int \frac{x}{(4+5 x)^2 \sqrt{1-x^2}} \, dx-5 \int \frac{1}{(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 = 0.145681, 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 = 0.001, 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((-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 = 1.27767, size = 34, normalized size = 1.1 \begin{align*} \frac{5 \, \sqrt{x + 1} \sqrt{-x + 1} + 3}{5 \,{\left (5 \, x + 4\right )}} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.74197, 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((-5-4*x-3*(-x^2+1)^(1/2))/(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((-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(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 [B]  time = 1.13998, size = 74, normalized size = 2.39 \begin{align*} -\frac{1}{5} \, i \mathrm{sgn}\left (\frac{1}{5 \, x + 4}\right ) + \frac{\sqrt{\frac{8}{5 \, x + 4} + \frac{9}{{\left (5 \, x + 4\right )}^{2}} - 1}}{5 \, \mathrm{sgn}\left (\frac{1}{5 \, x + 4}\right )} + \frac{3}{5 \,{\left (5 \, x + 4\right )}} \end{align*}

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, algorithm="giac")

[Out]

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