3.2810 \(\int \frac{1}{\sqrt{(3+5 x)^2}} \, dx\)

Optimal. Leaf size=26 \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]

[Out]

((3 + 5*x)*Log[3 + 5*x])/(5*Sqrt[(3 + 5*x)^2])

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Rubi [A]  time = 0.0048987, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 15, 29} \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(3 + 5*x)^2],x]

[Out]

((3 + 5*x)*Log[3 + 5*x])/(5*Sqrt[(3 + 5*x)^2])

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.0066036, size = 26, normalized size = 1. \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(3 + 5*x)^2],x]

[Out]

((3 + 5*x)*Log[3 + 5*x])/(5*Sqrt[(3 + 5*x)^2])

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Maple [A]  time = 0.006, size = 23, normalized size = 0.9 \begin{align*}{\frac{ \left ( 3+5\,x \right ) \ln \left ( 3+5\,x \right ) }{5}{\frac{1}{\sqrt{ \left ( 3+5\,x \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((3+5*x)^2)^(1/2),x)

[Out]

1/5*(3+5*x)*ln(3+5*x)/((3+5*x)^2)^(1/2)

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Maxima [A]  time = 1.44625, size = 8, normalized size = 0.31 \begin{align*} \frac{1}{5} \, \log \left (x + \frac{3}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((3+5*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/5*log(x + 3/5)

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Fricas [A]  time = 1.19683, size = 24, normalized size = 0.92 \begin{align*} \frac{1}{5} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((3+5*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/5*log(5*x + 3)

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Sympy [A]  time = 0.073199, size = 7, normalized size = 0.27 \begin{align*} \frac{\log{\left (5 x + 3 \right )}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((3+5*x)**2)**(1/2),x)

[Out]

log(5*x + 3)/5

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Giac [A]  time = 1.10842, size = 20, normalized size = 0.77 \begin{align*} \frac{1}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \mathrm{sgn}\left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((3+5*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/5*log(abs(5*x + 3))*sgn(5*x + 3)