3.1249 \(\int \frac{1}{(a+b x) \sqrt{1+a^2+2 a b x+b^2 x^2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b} \]

[Out]

-(ArcTanh[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]/b)

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Rubi [A]  time = 0.0660912, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{\tanh ^{-1}\left (\sqrt{a^2+2 a b x+b^2 x^2+1}\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTanh[Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]]/b)

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Rubi in Sympy [A]  time = 18.9949, size = 26, normalized size = 0.96 \[ - \frac{\operatorname{atanh}{\left (\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1} \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-atanh(sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1))/b

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Mathematica [A]  time = 0.020756, size = 32, normalized size = 1.19 \[ \frac{\log (a+b x)}{b}-\frac{\log \left (\sqrt{(a+b x)^2+1}+1\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*x]/b - Log[1 + Sqrt[1 + (a + b*x)^2]]/b

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Maple [A]  time = 0.011, size = 24, normalized size = 0.9 \[ -{\frac{1}{b}{\it Artanh} \left ({\frac{1}{\sqrt{ \left ( x+{\frac{a}{b}} \right ) ^{2}{b}^{2}+1}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b*arctanh(1/((x+a/b)^2*b^2+1)^(1/2))

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Maxima [A]  time = 0.757424, size = 19, normalized size = 0.7 \[ -\frac{\operatorname{arsinh}\left (\frac{1}{{\left | b x + a \right |}}\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsinh(1/abs(b*x + a))/b

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Fricas [A]  time = 0.216567, size = 89, normalized size = 3.3 \[ -\frac{\log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right ) - \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1) + 1) - log(-b*x - a + sqrt(b^
2*x^2 + 2*a*b*x + a^2 + 1) - 1))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)), x)

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GIAC/XCAS [A]  time = 0.241197, size = 120, normalized size = 4.44 \[ \frac{{\rm ln}\left (\frac{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} - 2 \,{\left | b \right |} \right |}}{{\left | -2 \,{\left (x{\left | b \right |} - \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} b - 2 \, a{\left | b \right |} + 2 \,{\left | b \right |} \right |}}\right )}{{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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