3.185 \(\int \frac{1}{\sqrt{a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

[Out]

((a + b*x)*Log[a + b*x])/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi [A]  time = 0.0254914, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{a^2+2 a b x+b^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*Log[a + b*x])/(b*Sqrt[a^2 + 2*a*b*x + b^2*x^2])

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Rubi in Sympy [A]  time = 3.06716, size = 32, normalized size = 0.91 \[ \frac{\left (a + b x\right ) \log{\left (a + b x \right )}}{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((b*x+a)**2)**(1/2),x)

[Out]

(a + b*x)*log(a + b*x)/(b*sqrt(a**2 + 2*a*b*x + b**2*x**2))

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Mathematica [A]  time = 0.0134335, size = 26, normalized size = 0.74 \[ \frac{(a+b x) \log (a+b x)}{b \sqrt{(a+b x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

((a + b*x)*Log[a + b*x])/(b*Sqrt[(a + b*x)^2])

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Maple [A]  time = 0.005, size = 25, normalized size = 0.7 \[{\frac{ \left ( bx+a \right ) \ln \left ( bx+a \right ) }{b}{\frac{1}{\sqrt{ \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((b*x+a)^2)^(1/2),x)

[Out]

(b*x+a)*ln(b*x+a)/b/((b*x+a)^2)^(1/2)

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Maxima [A]  time = 0.718246, size = 19, normalized size = 0.54 \[ \sqrt{\frac{1}{b^{2}}} \log \left (x + \frac{a}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(b^(-2))*log(x + a/b)

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Fricas [A]  time = 0.217344, size = 14, normalized size = 0.4 \[ \frac{\log \left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(b*x + a)/b

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Sympy [A]  time = 0.141737, size = 7, normalized size = 0.2 \[ \frac{\log{\left (a + b x \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x+a)**2)**(1/2),x)

[Out]

log(a + b*x)/b

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GIAC/XCAS [A]  time = 0.205192, size = 23, normalized size = 0.66 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (b x + a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(abs(b*x + a))*sign(b*x + a)/b