3.146 \(\int \frac{\sqrt{a^2+2 a b x+b^2 x^2}}{x^5} \, dx\)

Optimal. Leaf size=71 \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(3*x^3*(a + b*x))

_______________________________________________________________________________________

Rubi [A]  time = 0.0679753, 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 \[ -\frac{a \sqrt{a^2+2 a b x+b^2 x^2}}{4 x^4 (a+b x)}-\frac{b \sqrt{a^2+2 a b x+b^2 x^2}}{3 x^3 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^5,x]

[Out]

-(a*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*x^4*(a + b*x)) - (b*Sqrt[a^2 + 2*a*b*x + b
^2*x^2])/(3*x^3*(a + b*x))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 8.99, size = 56, normalized size = 0.79 \[ \frac{a \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{12 x^{4} \left (a + b x\right )} - \frac{\sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{3 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(((b*x+a)**2)**(1/2)/x**5,x)

[Out]

a*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(12*x**4*(a + b*x)) - sqrt(a**2 + 2*a*b*x + b
**2*x**2)/(3*x**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0134754, size = 33, normalized size = 0.46 \[ -\frac{\sqrt{(a+b x)^2} (3 a+4 b x)}{12 x^4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a^2 + 2*a*b*x + b^2*x^2]/x^5,x]

[Out]

-(Sqrt[(a + b*x)^2]*(3*a + 4*b*x))/(12*x^4*(a + b*x))

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 30, normalized size = 0.4 \[ -{\frac{4\,bx+3\,a}{12\,{x}^{4} \left ( bx+a \right ) }\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((b*x+a)^2)^(1/2)/x^5,x)

[Out]

-1/12*(4*b*x+3*a)*((b*x+a)^2)^(1/2)/x^4/(b*x+a)

_______________________________________________________________________________________

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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223462, size = 18, normalized size = 0.25 \[ -\frac{4 \, b x + 3 \, a}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^5,x, algorithm="fricas")

[Out]

-1/12*(4*b*x + 3*a)/x^4

_______________________________________________________________________________________

Sympy [A]  time = 1.15682, size = 14, normalized size = 0.2 \[ - \frac{3 a + 4 b x}{12 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((b*x+a)**2)**(1/2)/x**5,x)

[Out]

-(3*a + 4*b*x)/(12*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210254, size = 54, normalized size = 0.76 \[ -\frac{b^{4}{\rm sign}\left (b x + a\right )}{12 \, a^{3}} - \frac{4 \, b x{\rm sign}\left (b x + a\right ) + 3 \, a{\rm sign}\left (b x + a\right )}{12 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((b*x + a)^2)/x^5,x, algorithm="giac")

[Out]

-1/12*b^4*sign(b*x + a)/a^3 - 1/12*(4*b*x*sign(b*x + a) + 3*a*sign(b*x + a))/x^4