3.147 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^6} \, dx\)

Optimal. Leaf size=71 \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(4*x^4*(a + b*x))

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Rubi [A]  time = 0.0689451, 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 \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{5 x^5 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^6,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*x^5*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(4*x^4*(a + b*x))

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Rubi in Sympy [A]  time = 9.00913, size = 56, normalized size = 0.79 \[ \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{20 x^{5} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{4 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/x**6,x)

[Out]

a*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(20*x**5*(a + b*x)) - sqrt(a**2 + 2*a*b*x + b
**2*x**2)/(4*x**5)

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Mathematica [A]  time = 0.0119318, size = 33, normalized size = 0.46 \[ -\frac{\sqrt{(a+b x)^2} (4 a+5 b x)}{20 x^5 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^6,x]

[Out]

-(Sqrt[(a + b*x)^2]*(4*a + 5*b*x))/(20*x^5*(a + b*x))

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Maple [A]  time = 0.005, size = 30, normalized size = 0.4 \[ -{\frac{5\,bx+4\,a}{20\,{x}^{5} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x^6,x)

[Out]

-1/20*(5*b*x+4*a)*((b*x+a)^2)^(1/2)/x^5/(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^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218058, size = 18, normalized size = 0.25 \[ -\frac{5 \, b x + 4 \, a}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^6,x, algorithm="fricas")

[Out]

-1/20*(5*b*x + 4*a)/x^5

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Sympy [A]  time = 1.20217, size = 14, normalized size = 0.2 \[ - \frac{4 a + 5 b x}{20 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/x**6,x)

[Out]

-(4*a + 5*b*x)/(20*x**5)

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GIAC/XCAS [A]  time = 0.211061, size = 54, normalized size = 0.76 \[ \frac{b^{5}{\rm sign}\left (b x + a\right )}{20 \, a^{4}} - \frac{5 \, b x{\rm sign}\left (b x + a\right ) + 4 \, a{\rm sign}\left (b x + a\right )}{20 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^6,x, algorithm="giac")

[Out]

1/20*b^5*sign(b*x + a)/a^4 - 1/20*(5*b*x*sign(b*x + a) + 4*a*sign(b*x + a))/x^5