3.2803 \(\int \frac{1}{\sqrt{(3+5 x)^2}} \, dx\)

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 = 0.0162103, 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 \[ \frac{(5 x+3) \log (5 x+3)}{5 \sqrt{(5 x+3)^2}} \]

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])

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Rubi in Sympy [A]  time = 1.75355, size = 26, normalized size = 1. \[ \frac{\left (25 x + 15\right ) \log{\left (5 x + 3 \right )}}{25 \sqrt{25 x^{2} + 30 x + 9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((3+5*x)**2)**(1/2),x)

[Out]

(25*x + 15)*log(5*x + 3)/(25*sqrt(25*x**2 + 30*x + 9))

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Mathematica [A]  time = 0.0130783, 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 = 0.008, size = 23, normalized size = 0.9 \[{\frac{ \left ( 3+5\,x \right ) \ln \left ( 3+5\,x \right ) }{5}{\frac{1}{\sqrt{ \left ( 3+5\,x \right ) ^{2}}}}} \]

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 = 1.4816, size = 8, normalized size = 0.31 \[ \frac{1}{5} \, \log \left (x + \frac{3}{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*log(x + 3/5)

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Fricas [A]  time = 0.214323, size = 11, normalized size = 0.42 \[ \frac{1}{5} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*log(5*x + 3)

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Sympy [A]  time = 0.106889, size = 7, normalized size = 0.27 \[ \frac{\log{\left (5 x + 3 \right )}}{5} \]

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/XCAS [A]  time = 0.216018, size = 20, normalized size = 0.77 \[ \frac{1}{5} \,{\rm ln}\left ({\left | 5 \, x + 3 \right |}\right ){\rm sign}\left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*ln(abs(5*x + 3))*sign(5*x + 3)