3.2415 \(\int \frac{1}{x \sqrt{-a^2-2 a b x-b^2 x^2}} \, dx\)

Optimal. Leaf size=74 \[ \frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0863762, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\log (x) (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}}-\frac{(a+b x) \log (a+b x)}{a \sqrt{-a^2-2 a b x-b^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.245, size = 66, normalized size = 0.89 \[ - \frac{\sqrt{- a^{2} - 2 a b x - b^{2} x^{2}} \log{\left (x \right )}}{a \left (a + b x\right )} + \frac{\sqrt{- a^{2} - 2 a b x - b^{2} x^{2}} \log{\left (a + b x \right )}}{a \left (a + b x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0218459, size = 33, normalized size = 0.45 \[ \frac{(a+b x) (\log (x)-\log (a+b x))}{a \sqrt{-(a+b x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 33, normalized size = 0.5 \[ -{\frac{ \left ( bx+a \right ) \left ( -\ln \left ( x \right ) +\ln \left ( bx+a \right ) \right ) }{a}{\frac{1}{\sqrt{- \left ( bx+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-(b*x+a)^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218006, size = 92, normalized size = 1.24 \[ \sqrt{-\frac{1}{a^{2}}} \log \left (-\frac{i \,{\left (a^{2} \sqrt{-\frac{1}{a^{2}}} + 2 i \, b x + i \, a\right )}}{2 \, b}\right ) - \sqrt{-\frac{1}{a^{2}}} \log \left (\frac{i \,{\left (a^{2} \sqrt{-\frac{1}{a^{2}}} - 2 i \, b x - i \, a\right )}}{2 \, b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef