3.5.31 \(\int \sqrt {\frac {1+x}{x^5}} \, dx\) [431]

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

[Out]

-2/3*(1/x^5+1/x^4)^(3/2)*x^6

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 = {2004, 2025} \begin {gather*} -\frac {2}{3} \left (\frac {1}{x^5}+\frac {1}{x^4}\right )^{3/2} x^6 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[(1 + x)/x^5],x]

[Out]

(-2*(x^(-5) + x^(-4))^(3/2)*x^6)/3

Rule 2004

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

Rule 2025

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*}

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Mathematica [A]
time = 0.02, size = 19, normalized size = 1.06 \begin {gather*} -\frac {2}{3} x (1+x) \sqrt {\frac {1+x}{x^5}} \end {gather*}

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 = 0.06, size = 26, normalized size = 1.44

method	result	size
gosper	\(-\frac {2 x \left (x +1\right ) \sqrt {\frac {x +1}{x^{5}}}}{3}\)	\(16\)
trager	\(-\frac {2 x \left (x +1\right ) \sqrt {-\frac {-1-x}{x^{5}}}}{3}\)	\(19\)
default	\(-\frac {2 \sqrt {\frac {x +1}{x^{5}}}\, \left (x^{2}+x \right )^{\frac {3}{2}}}{3 \sqrt {x \left (x +1\right )}}\)	\(26\)
risch	\(-\frac {2 \sqrt {\frac {x +1}{x^{5}}}\, x \left (x^{2}+2 x +1\right )}{3 \left (x +1\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x+1)/x^5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 1.35, size = 16, normalized size = 0.89 \begin {gather*} -\frac {2}{3} \, {\left (x^{2} + x\right )} \sqrt {\frac {x + 1}{x^{5}}} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {x + 1}{x^{5}}}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (14) = 28\).
time = 1.31, size = 50, normalized size = 2.78 \begin {gather*} \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 {gather*}

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

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Mupad [B]
time = 5.26, size = 15, normalized size = 0.83 \begin {gather*} -\frac {2\,x\,\sqrt {\frac {x+1}{x^5}}\,\left (x+1\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x + 1)/x^5)^(1/2),x)

[Out]

-(2*x*((x + 1)/x^5)^(1/2)*(x + 1))/3

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