\(\int \frac {1}{\sqrt {(6+10 x)^2}} \, dx\) [2811]

Optimal result

Integrand size = 11, antiderivative size = 26 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {(3+5 x) \log (3+5 x)}{10 \sqrt {(3+5 x)^2}} \]

[Out]

1/10*(3+5*x)*ln(3+5*x)/((3+5*x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {253, 15, 29} \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {(5 x+3) \log (10 x+6)}{10 \sqrt {(5 x+3)^2}} \]

[In]

Int[1/Sqrt[(6 + 10*x)^2],x]

[Out]

((3 + 5*x)*Log[6 + 10*x])/(10*Sqrt[(3 + 5*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 253

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*} \text {integral}& = \frac {1}{10} \text {Subst}\left (\int \frac {1}{\sqrt {x^2}} \, dx,x,6+10 x\right ) \\ & = \frac {(6+10 x) \text {Subst}\left (\int \frac {1}{x} \, dx,x,6+10 x\right )}{10 \sqrt {(6+10 x)^2}} \\ & = \frac {(3+5 x) \log (3+5 x)}{10 \sqrt {(3+5 x)^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {(6+10 x) \log (6+10 x)}{10 \sqrt {(6+10 x)^2}} \]

[In]

Integrate[1/Sqrt[(6 + 10*x)^2],x]

[Out]

((6 + 10*x)*Log[6 + 10*x])/(10*Sqrt[(6 + 10*x)^2])

Maple [A] (verified)

Time = 6.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {\sqrt {\left (3+5 x \right )^{2}}\, \ln \left (3+5 x \right )}{30+50 x}\)	\(25\)
default	\(\frac {\left (3+5 x \right ) \sqrt {4}\, \ln \left (3+5 x \right )}{20 \sqrt {\left (3+5 x \right )^{2}}}\)	\(26\)
meijerg	\(\frac {3 \ln \left (1+\frac {5 x}{3}\right )}{5 \sqrt {\left (6+10 x \right )^{2}}}+\frac {x \ln \left (1+\frac {5 x}{3}\right )}{\sqrt {\left (6+10 x \right )^{2}}}\)	\(36\)

[In]

int(1/((6+10*x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/10*((3+5*x)^2)^(1/2)/(3+5*x)*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.31 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {1}{10} \, \log \left (5 \, x + 3\right ) \]

[In]

integrate(1/((6+10*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/10*log(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {\left (x + \frac {3}{5}\right ) \log {\left (x + \frac {3}{5} \right )}}{10 \sqrt {\left (x + \frac {3}{5}\right )^{2}}} \]

[In]

integrate(1/((6+10*x)**2)**(1/2),x)

[Out]

(x + 3/5)*log(x + 3/5)/(10*sqrt((x + 3/5)**2))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {1}{10} \, \log \left (x + \frac {3}{5}\right ) \]

[In]

integrate(1/((6+10*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/10*log(x + 3/5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {1}{10} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \mathrm {sgn}\left (5 \, x + 3\right ) \]

[In]

integrate(1/((6+10*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/10*log(abs(5*x + 3))*sgn(5*x + 3)

Mupad [B] (verification not implemented)

Time = 6.81 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.54 \[ \int \frac {1}{\sqrt {(6+10 x)^2}} \, dx=\frac {\ln \left (10\,x+6\right )\,\mathrm {sign}\left (10\,x+6\right )}{10} \]

[In]

int(1/((10*x + 6)^2)^(1/2),x)

[Out]

(log(10*x + 6)*sign(10*x + 6))/10