3.609 \(\int \frac{1-\sqrt{2+3 x}}{1+\sqrt{2+3 x}} \, dx\)

Optimal. Leaf size=33 \[ -x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left (\sqrt{3 x+2}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0216975, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {431, 376, 77} \[ -x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left (\sqrt{3 x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - Sqrt[2 + 3*x])/(1 + Sqrt[2 + 3*x]),x]

[Out]

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

Rule 431

Int[((a_.) + (b_.)*(u_)^(n_))^(p_.)*((c_.) + (d_.)*(u_)^(n_))^(q_.), x_Symbol] :> Dist[1/Coefficient[u, x, 1],
Subst[Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x, u], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && LinearQ[u, x] && N
eQ[u, x]

Rule 376

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, Dis
t[g, Subst[Int[x^(g - 1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b, c, d, p, q}
, x] && NeQ[b*c - a*d, 0] && FractionQ[n]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0144503, size = 33, normalized size = 1. \[ -x+\frac{4}{3} \sqrt{3 x+2}-\frac{4}{3} \log \left (\sqrt{3 x+2}+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Sqrt[2 + 3*x])/(1 + Sqrt[2 + 3*x]),x]

[Out]

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

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*}{\frac{4}{3}\sqrt{2+3\,x}}-x-{\frac{2}{3}}-{\frac{4}{3}\ln \left ( 1+\sqrt{2+3\,x} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-(2+3*x)^(1/2))/(1+(2+3*x)^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.0267, size = 35, normalized size = 1.06 \begin{align*} -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \, \log \left (\sqrt{3 \, x + 2} + 1\right ) - \frac{2}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(2+3*x)^(1/2))/(1+(2+3*x)^(1/2)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58899, size = 72, normalized size = 2.18 \begin{align*} -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \, \log \left (\sqrt{3 \, x + 2} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(2+3*x)^(1/2))/(1+(2+3*x)^(1/2)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.144897, size = 27, normalized size = 0.82 \begin{align*} - x + \frac{4 \sqrt{3 x + 2}}{3} - \frac{4 \log{\left (\sqrt{3 x + 2} + 1 \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(2+3*x)**(1/2))/(1+(2+3*x)**(1/2)),x)

[Out]

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

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Giac [A]  time = 1.10943, size = 35, normalized size = 1.06 \begin{align*} -x + \frac{4}{3} \, \sqrt{3 \, x + 2} - \frac{4}{3} \, \log \left (\sqrt{3 \, x + 2} + 1\right ) - \frac{2}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-(2+3*x)^(1/2))/(1+(2+3*x)^(1/2)),x, algorithm="giac")

[Out]

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