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

Optimal. Leaf size=65 \[ \frac{b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0664, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b \log (x) \sqrt{a^2+2 a b x+b^2 x^2}}{a+b x}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.1007, size = 58, normalized size = 0.89 \[ - \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{x \left (a + b x\right )} + \frac{b \sqrt{a^{2} + 2 a b x + b^{2} x^{2}} \log{\left (x \right )}}{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0155224, size = 31, normalized size = 0.48 \[ \frac{\sqrt{(a+b x)^2} (b x \log (x)-a)}{x (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.015, size = 22, normalized size = 0.3 \[{\frac{{\it csgn} \left ( bx+a \right ) \left ( \ln \left ( bx \right ) xb-a \right ) }{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x^2,x)

[Out]

csgn(b*x+a)*(ln(b*x)*x*b-a)/x

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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(sqrt((b*x + a)^2)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230738, size = 18, normalized size = 0.28 \[ \frac{b x \log \left (x\right ) - a}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*log(x) - a)/x

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Sympy [A]  time = 1.06347, size = 7, normalized size = 0.11 \[ - \frac{a}{x} + b \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/x + b*log(x)

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GIAC/XCAS [A]  time = 0.207618, size = 32, normalized size = 0.49 \[ b{\rm ln}\left ({\left | x \right |}\right ){\rm sign}\left (b x + a\right ) - \frac{a{\rm sign}\left (b x + a\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*ln(abs(x))*sign(b*x + a) - a*sign(b*x + a)/x