3.2804 \(\int \frac{1}{\sqrt{(6+10 x)^2}} \, dx\)

Optimal. Leaf size=26 \[ \frac{(5 x+3) \log (5 x+3)}{10 \sqrt{(5 x+3)^2}} \]

[Out]

((3 + 5*x)*Log[3 + 5*x])/(10*Sqrt[(3 + 5*x)^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0165697, 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 \[ \frac{(5 x+3) \log (10 x+6)}{10 \sqrt{(5 x+3)^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[(6 + 10*x)^2],x]

[Out]

((3 + 5*x)*Log[6 + 10*x])/(10*Sqrt[(3 + 5*x)^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.77435, size = 26, normalized size = 1. \[ \frac{\left (100 x + 60\right ) \log{\left (5 x + 3 \right )}}{100 \sqrt{100 x^{2} + 120 x + 36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((6+10*x)**2)**(1/2),x)

[Out]

(100*x + 60)*log(5*x + 3)/(100*sqrt(100*x**2 + 120*x + 36))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0121235, size = 26, normalized size = 1. \[ \frac{(10 x+6) \log (10 x+6)}{10 \sqrt{(10 x+6)^2}} \]

Antiderivative was successfully verified.

[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]  time = 0.005, size = 26, normalized size = 1. \[{\frac{ \left ( 3+5\,x \right ) \sqrt{4}\ln \left ( 3+5\,x \right ) }{20}{\frac{1}{\sqrt{ \left ( 3+5\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((6+10*x)^2)^(1/2),x)

[Out]

1/20/((3+5*x)^2)^(1/2)*(3+5*x)*4^(1/2)*ln(3+5*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.47944, size = 8, normalized size = 0.31 \[ \frac{1}{10} \, \log \left (x + \frac{3}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*log(x + 3/5)

_______________________________________________________________________________________

Fricas [A]  time = 0.211059, size = 11, normalized size = 0.42 \[ \frac{1}{10} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*log(5*x + 3)

_______________________________________________________________________________________

Sympy [A]  time = 0.101199, size = 7, normalized size = 0.27 \[ \frac{\log{\left (10 x + 6 \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(10*x + 6)/10

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215447, size = 20, normalized size = 0.77 \[ \frac{1}{10} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ){\rm sign}\left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*ln(abs(5*x + 3))*sign(5*x + 3)