3.145 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^4} \, dx\)

Optimal. Leaf size=71 \[ -\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(2*x^2*(a + b*x))

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Rubi [A]  time = 0.0670105, antiderivative size = 71, 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 \[ -\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{2 x^2 (a+b x)}-\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^4,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*x^3*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(2*x^2*(a + b*x))

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Rubi in Sympy [A]  time = 9.05182, size = 56, normalized size = 0.79 \[ \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{6 x^{3} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{2 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/x**4,x)

[Out]

a*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(6*x**3*(a + b*x)) - sqrt(a**2 + 2*a*b*x + b*
*2*x**2)/(2*x**3)

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Mathematica [A]  time = 0.0144389, size = 33, normalized size = 0.46 \[ -\frac{\sqrt{(a+b x)^2} (2 a+3 b x)}{6 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^4,x]

[Out]

-(Sqrt[(a + b*x)^2]*(2*a + 3*b*x))/(6*x^3*(a + b*x))

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Maple [A]  time = 0.004, size = 30, normalized size = 0.4 \[ -{\frac{3\,bx+2\,a}{6\,{x}^{3} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x^4,x)

[Out]

-1/6*(3*b*x+2*a)*((b*x+a)^2)^(1/2)/x^3/(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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219781, size = 18, normalized size = 0.25 \[ -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^4,x, algorithm="fricas")

[Out]

-1/6*(3*b*x + 2*a)/x^3

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Sympy [A]  time = 1.12786, size = 14, normalized size = 0.2 \[ - \frac{2 a + 3 b x}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/x**4,x)

[Out]

-(2*a + 3*b*x)/(6*x**3)

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GIAC/XCAS [A]  time = 0.211137, size = 54, normalized size = 0.76 \[ \frac{b^{3}{\rm sign}\left (b x + a\right )}{6 \, a^{2}} - \frac{3 \, b x{\rm sign}\left (b x + a\right ) + 2 \, a{\rm sign}\left (b x + a\right )}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^4,x, algorithm="giac")

[Out]

1/6*b^3*sign(b*x + a)/a^2 - 1/6*(3*b*x*sign(b*x + a) + 2*a*sign(b*x + a))/x^3