3.431 \(\int \sqrt{\frac{1+x}{x^5}} \, dx\)

Optimal. Leaf size=18 \[ -\frac{2}{3} \left (\frac{1}{x^5}+\frac{1}{x^4}\right )^{3/2} x^6 \]

[Out]

(-2*(x^(-5) + x^(-4))^(3/2)*x^6)/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0155205, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{2}{3} \left (\frac{1}{x^5}+\frac{1}{x^4}\right )^{3/2} x^6 \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(1 + x)/x^5],x]

[Out]

(-2*(x^(-5) + x^(-4))^(3/2)*x^6)/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.18497, size = 20, normalized size = 1.11 \[ - \frac{2 x^{6} \left (\frac{1}{x^{4}} + \frac{1}{x^{5}}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**6*(x**(-4) + x**(-5))**(3/2)/3

_______________________________________________________________________________________

Mathematica [A]  time = 0.0119901, size = 18, normalized size = 1. \[ -\frac{2}{3} x^6 \left (\frac{x+1}{x^5}\right )^{3/2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(1 + x)/x^5],x]

[Out]

(-2*x^6*((1 + x)/x^5)^(3/2))/3

_______________________________________________________________________________________

Maple [A]  time = 0.006, size = 16, normalized size = 0.9 \[ -{\frac{2\,x \left ( 1+x \right ) }{3}\sqrt{{\frac{1+x}{{x}^{5}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(((1+x)/x^5)^(1/2),x)

[Out]

-2/3*x*(1+x)*((1+x)/x^5)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.41484, size = 14, normalized size = 0.78 \[ -\frac{2 \,{\left (x + 1\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x + 1)/x^5),x, algorithm="maxima")

[Out]

-2/3*(x + 1)^(3/2)/x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.227398, size = 22, normalized size = 1.22 \[ -\frac{2}{3} \,{\left (x^{2} + x\right )} \sqrt{\frac{x + 1}{x^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x + 1)/x^5),x, algorithm="fricas")

[Out]

-2/3*(x^2 + x)*sqrt((x + 1)/x^5)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{x + 1}{x^{5}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x + 1)/x**5), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.229375, size = 68, normalized size = 3.78 \[ \frac{2 \,{\left (3 \,{\left (x - \sqrt{x^{2} + x}\right )}^{2}{\rm sign}\left (x\right ) + 3 \,{\left (x - \sqrt{x^{2} + x}\right )}{\rm sign}\left (x\right ) +{\rm sign}\left (x\right )\right )}}{3 \,{\left (x - \sqrt{x^{2} + x}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((x + 1)/x^5),x, algorithm="giac")

[Out]

2/3*(3*(x - sqrt(x^2 + x))^2*sign(x) + 3*(x - sqrt(x^2 + x))*sign(x) + sign(x))/
(x - sqrt(x^2 + x))^3