Optimal. Leaf size=18 \[ -\frac {2}{3} \left (\frac {1}{x^5}+\frac {1}{x^4}\right )^{3/2} x^6 \]
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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.
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Rule 2004
Rule 2025
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.
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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.
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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.
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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.
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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.
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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.
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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.
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