Optimal. Leaf size=18 \[ -\frac{2}{3} \left (\frac{1}{x^4}+\frac{1}{x^5}\right )^{3/2} x^6 \]
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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.
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Rule 1979
Rule 2000
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.
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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.
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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.
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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.
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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.
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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.
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