Optimal. Leaf size=18 \[ -\frac {4 \left (x^5+x\right )^{3/4}}{x^4+1} \]
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Rubi [A] time = 0.07, antiderivative size = 12, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2056, 449} \begin {gather*} -\frac {4 x}{\sqrt [4]{x^5+x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{x+x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {-3+x^4}{\sqrt [4]{x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=-\frac {4 x}{\sqrt [4]{x+x^5}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 60, normalized size = 3.33 \begin {gather*} \frac {4 \sqrt [4]{x^4+1} \left (x^5 \, _2F_1\left (\frac {19}{16},\frac {5}{4};\frac {35}{16};-x^4\right )-19 x \, _2F_1\left (\frac {3}{16},\frac {5}{4};\frac {19}{16};-x^4\right )\right )}{19 \sqrt [4]{x^5+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 18, normalized size = 1.00 \begin {gather*} -\frac {4 \left (x^5+x\right )^{3/4}}{x^4+1} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 16, normalized size = 0.89 \begin {gather*} -\frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}}}{x^{4} + 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*} \int \frac {x^{4} - 3}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 11, normalized size = 0.61 \begin {gather*} -\frac {4 x}{\left (x^{5}+x \right )^{\frac {1}{4}}} \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} - 3}{{\left (x^{5} + x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 16, normalized size = 0.89 \begin {gather*} -\frac {4\,{\left (x^5+x\right )}^{3/4}}{x^4+1} \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 {x^{4} - 3}{\sqrt [4]{x \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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