3.2812 \(\int \frac {1}{\sqrt {-(2+3 x)^2}} \, dx\)

Optimal. Leaf size=28 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {247, 15, 29} \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-(2 + 3*x)^2],x]

[Out]

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

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 28, normalized size = 1.00 \[ \frac {(3 x+2) \log (3 x+2)}{3 \sqrt {-(3 x+2)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-(2 + 3*x)^2],x]

[Out]

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

________________________________________________________________________________________

fricas [C]  time = 0.81, size = 6, normalized size = 0.21 \[ -\frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*log(x + 2/3)

________________________________________________________________________________________

giac [C]  time = 0.24, size = 23, normalized size = 0.82 \[ \frac {i \, \log \left ({\left (-3 i \, x - 2 i\right )} \mathrm {sgn}\left (-3 \, x - 2\right )\right )}{3 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*log((-3*I*x - 2*I)*sgn(-3*x - 2))/sgn(-3*x - 2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 25, normalized size = 0.89 \[ \frac {\left (3 x +2\right ) \ln \left (3 x +2\right )}{3 \sqrt {-\left (3 x +2\right )^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 1.29, size = 6, normalized size = 0.21 \[ \frac {1}{3} i \, \log \left (x + \frac {2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*I*log(x + 2/3)

________________________________________________________________________________________

mupad [B]  time = 1.36, size = 15, normalized size = 0.54 \[ -\frac {\ln \left (-3\,x-2\right )\,\mathrm {sign}\left (3\,x+2\right )\,1{}\mathrm {i}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-(3*x + 2)^2)^(1/2),x)

[Out]

-(log(- 3*x - 2)*sign(3*x + 2)*1i)/3

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (3 x + 2\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(-(3*x + 2)**2), x)

________________________________________________________________________________________