3.144 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0497282, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{(a+b x) \sqrt{a^2+2 a b x+b^2 x^2}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.74039, size = 36, normalized size = 1.03 \[ - \frac{\left (2 a + 2 b x\right ) \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*a + 2*b*x)*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(4*a*x**2)

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Mathematica [A]  time = 0.0107399, size = 31, normalized size = 0.89 \[ -\frac{\sqrt{(a+b x)^2} (a+2 b x)}{2 x^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^3,x]

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.8 \[ -{\frac{2\,bx+a}{2\,{x}^{2} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x^3,x)

[Out]

-1/2*(2*b*x+a)*((b*x+a)^2)^(1/2)/x^2/(b*x+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(sqrt((b*x + a)^2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227406, size = 15, normalized size = 0.43 \[ -\frac{2 \, b x + a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*x + a)/x^2

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Sympy [A]  time = 1.08921, size = 12, normalized size = 0.34 \[ - \frac{a + 2 b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/x**3,x)

[Out]

-(a + 2*b*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.208175, size = 53, normalized size = 1.51 \[ -\frac{b^{2}{\rm sign}\left (b x + a\right )}{2 \, a} - \frac{2 \, b x{\rm sign}\left (b x + a\right ) + a{\rm sign}\left (b x + a\right )}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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