Optimal. Leaf size=24 \[ -\sqrt {1+x^3}+\sqrt {1-2 x+x^5} \]
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Rubi [A]
time = 0.22, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {12, 6820, 267,
2124} \begin {gather*} \sqrt {x^5-2 x+1}-\sqrt {x^3+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 267
Rule 2124
Rule 6820
Rubi steps
\begin {align*} \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{2 \sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx &=\frac {1}{2} \int \frac {-2 \sqrt {1+x^3}+5 x^4 \sqrt {1+x^3}-3 x^2 \sqrt {1-2 x+x^5}}{\sqrt {1+x^3} \sqrt {1-2 x+x^5}} \, dx\\ &=\frac {1}{2} \int \left (-\frac {3 x^2}{\sqrt {1+x^3}}-\frac {2}{\sqrt {1-2 x+x^5}}+\frac {5 x^4}{\sqrt {1-2 x+x^5}}\right ) \, dx\\ &=-\left (\frac {3}{2} \int \frac {x^2}{\sqrt {1+x^3}} \, dx\right )+\frac {5}{2} \int \frac {x^4}{\sqrt {1-2 x+x^5}} \, dx-\int \frac {1}{\sqrt {1-2 x+x^5}} \, dx\\ &=-\sqrt {1+x^3}+\sqrt {1-2 x+x^5}\\ \end {align*}
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Mathematica [A]
time = 23.84, size = 24, normalized size = 1.00 \begin {gather*} -\sqrt {1+x^3}+\sqrt {1-2 x+x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 21, normalized size = 0.88
method | result | size |
default | \(-\sqrt {x^{3}+1}+\sqrt {x^{5}-2 x +1}\) | \(21\) |
elliptic | \(-\sqrt {x^{3}+1}+\frac {\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x -1\right )}{\sqrt {x^{5}-2 x +1}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.52, size = 30, normalized size = 1.25 \begin {gather*} \sqrt {x^{4} + x^{3} + x^{2} + x - 1} \sqrt {x - 1} - \sqrt {x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.65, size = 20, normalized size = 0.83 \begin {gather*} \sqrt {x^{5} - 2 \, x + 1} - \sqrt {x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.28, size = 19, normalized size = 0.79 \begin {gather*} - \sqrt {x^{3} + 1} + \sqrt {x^{5} - 2 x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 20, normalized size = 0.83 \begin {gather*} \sqrt {x^5-2\,x+1}-\sqrt {x^3+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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