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

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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 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]))))

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 33, normalized size = 1.00 \[ -x+\frac {4}{3} \sqrt {3 x+2}-\frac {4}{3} \log \left (\sqrt {3 x+2}+1\right ) \]

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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fricas [A]  time = 0.40, size = 25, normalized size = 0.76 \[ -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) \]

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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giac [A]  time = 0.33, size = 26, normalized size = 0.79 \[ -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \]

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

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maple [A]  time = 0.01, size = 27, normalized size = 0.82 \[ -x -\frac {4 \ln \left (1+\sqrt {3 x +2}\right )}{3}+\frac {4 \sqrt {3 x +2}}{3}-\frac {2}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.88, size = 26, normalized size = 0.79 \[ -x + \frac {4}{3} \, \sqrt {3 \, x + 2} - \frac {4}{3} \, \log \left (\sqrt {3 \, x + 2} + 1\right ) - \frac {2}{3} \]

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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mupad [B]  time = 3.10, size = 25, normalized size = 0.76 \[ \frac {4\,\sqrt {3\,x+2}}{3}-\frac {4\,\ln \left (\sqrt {3\,x+2}+1\right )}{3}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.17, size = 27, normalized size = 0.82 \[ - x + \frac {4 \sqrt {3 x + 2}}{3} - \frac {4 \log {\left (\sqrt {3 x + 2} + 1 \right )}}{3} \]

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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