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

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

[Out]

(-2*(x^(-5) + x^(-4))^(3/2)*x^6)/3

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Rubi [A]  time = 0.006895, 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, Rules used = {1979, 2000} \[ -\frac{2}{3} \left (\frac{1}{x^4}+\frac{1}{x^5}\right )^{3/2} x^6 \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x)/x^5],x]

[Out]

(-2*(x^(-5) + x^(-4))^(3/2)*x^6)/3

Rule 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2000

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + b*x^n)^(p + 1)/(b*(n - j)*(p + 1)*x
^(n - 1)), x] /; FreeQ[{a, b, j, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && EqQ[j*p - n + j + 1, 0]

Rubi steps

\begin{align*} \int \sqrt{\frac{1+x}{x^5}} \, dx &=\int \sqrt{\frac{1}{x^5}+\frac{1}{x^4}} \, dx\\ &=-\frac{2}{3} \left (\frac{1}{x^5}+\frac{1}{x^4}\right )^{3/2} x^6\\ \end{align*}

Mathematica [A]  time = 0.0052456, size = 19, normalized size = 1.06 \[ -\frac{2}{3} x (x+1) \sqrt{\frac{x+1}{x^5}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[(1 + x)/x^5],x]

[Out]

(-2*x*(1 + x)*Sqrt[(1 + x)/x^5])/3

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Maple [A]  time = 0.024, size = 16, normalized size = 0.9 \begin{align*} -{\frac{2\,x \left ( 1+x \right ) }{3}\sqrt{{\frac{1+x}{{x}^{5}}}}} \end{align*}

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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\frac{x + 1}{x^{5}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/x^5)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt((x + 1)/x^5), x)

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Fricas [A]  time = 0.92097, size = 46, normalized size = 2.56 \begin{align*} -\frac{2}{3} \,{\left (x^{2} + x\right )} \sqrt{\frac{x + 1}{x^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/x^5)^(1/2),x, algorithm="fricas")

[Out]

-2/3*(x^2 + x)*sqrt((x + 1)/x^5)

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

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)

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Giac [B]  time = 1.2883, size = 68, normalized size = 3.78 \begin{align*} \frac{2 \,{\left (3 \,{\left (x - \sqrt{x^{2} + x}\right )}^{2} \mathrm{sgn}\left (x\right ) + 3 \,{\left (x - \sqrt{x^{2} + x}\right )} \mathrm{sgn}\left (x\right ) + \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left (x - \sqrt{x^{2} + x}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((1+x)/x^5)^(1/2),x, algorithm="giac")

[Out]

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