Optimal. Leaf size=20 \[ \frac {4 \left (x^5-x^3\right )^{7/4}}{7 x^7} \]
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Rubi [B] time = 0.15, antiderivative size = 41, normalized size of antiderivative = 2.05, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2052, 2025, 2011, 365, 364, 2024} \begin {gather*} \frac {4 \left (x^5-x^3\right )^{3/4}}{7 x^2}-\frac {4 \left (x^5-x^3\right )^{3/4}}{7 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2011
Rule 2024
Rule 2025
Rule 2052
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^2 \sqrt [4]{-x^3+x^5}} \, dx &=\int \left (-\frac {1}{x^2 \sqrt [4]{-x^3+x^5}}+\frac {x^2}{\sqrt [4]{-x^3+x^5}}\right ) \, dx\\ &=-\int \frac {1}{x^2 \sqrt [4]{-x^3+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{-x^3+x^5}} \, dx\\ &=-\frac {4 \left (-x^3+x^5\right )^{3/4}}{7 x^4}+\frac {4 \left (-x^3+x^5\right )^{3/4}}{7 x^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3 \left (x^2-1\right )\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 20, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^5-x^3\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 21, normalized size = 1.05 \begin {gather*} \frac {4 \, {\left (x^{5} - x^{3}\right )}^{\frac {3}{4}} {\left (x^{2} - 1\right )}}{7 \, x^{4}} \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*} \int \frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 1.40 \begin {gather*} \frac {4 \left (x^{2}-1\right ) \left (-1+x \right ) \left (1+x \right )}{7 \left (x^{5}-x^{3}\right )^{\frac {1}{4}} x} \end {gather*}
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*} \int \frac {x^{4} - 1}{{\left (x^{5} - x^{3}\right )}^{\frac {1}{4}} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 35, normalized size = 1.75 \begin {gather*} \frac {4\,x^2\,{\left (x^5-x^3\right )}^{3/4}-4\,{\left (x^5-x^3\right )}^{3/4}}{7\,x^4} \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 \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt [4]{x^{3} \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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