Optimal. Leaf size=18 \[ \frac {4 \left (x^5+x^3\right )^{9/4}}{9 x^9} \]
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Rubi [B] time = 0.18, antiderivative size = 53, normalized size of antiderivative = 2.94, number of steps used = 9, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2052, 2004, 2032, 364, 2020, 2025} \begin {gather*} \frac {4}{9} \sqrt [4]{x^5+x^3} x+\frac {8 \sqrt [4]{x^5+x^3}}{9 x}+\frac {4 \sqrt [4]{x^5+x^3}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 2004
Rule 2020
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{x^3+x^5}}{x^4} \, dx &=\int \left (\sqrt [4]{x^3+x^5}-\frac {\sqrt [4]{x^3+x^5}}{x^4}\right ) \, dx\\ &=\int \sqrt [4]{x^3+x^5} \, dx-\int \frac {\sqrt [4]{x^3+x^5}}{x^4} \, dx\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {4}{9} x \sqrt [4]{x^3+x^5}-\frac {2}{9} \int \frac {x}{\left (x^3+x^5\right )^{3/4}} \, dx+\frac {2}{9} \int \frac {x^3}{\left (x^3+x^5\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}-\frac {2}{9} \int \frac {x^3}{\left (x^3+x^5\right )^{3/4}} \, dx+\frac {\left (2 x^{9/4} \left (1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{9 \left (x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}+\frac {8 x^4 \left (1+x^2\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{63 \left (x^3+x^5\right )^{3/4}}-\frac {\left (2 x^{9/4} \left (1+x^2\right )^{3/4}\right ) \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{9 \left (x^3+x^5\right )^{3/4}}\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {8 \sqrt [4]{x^3+x^5}}{9 x}+\frac {4}{9} x \sqrt [4]{x^3+x^5}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 65, normalized size = 3.61 \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3} \left (7 \, _2F_1\left (-\frac {9}{8},-\frac {1}{4};-\frac {1}{8};-x^2\right )+9 x^4 \, _2F_1\left (-\frac {1}{4},\frac {7}{8};\frac {15}{8};-x^2\right )\right )}{63 x^3 \sqrt [4]{x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^5+x^3\right )^{9/4}}{9 x^9} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 24, normalized size = 1.33 \begin {gather*} \frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{2} + 1\right )}}{9 \, x^{3}} \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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 1.22 \begin {gather*} \frac {4 \left (x^{2}+1\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}} \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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 41, normalized size = 2.28 \begin {gather*} \frac {4\,x\,{\left (x^5+x^3\right )}^{1/4}}{9}+\frac {8\,{\left (x^5+x^3\right )}^{1/4}}{9\,x}+\frac {4\,{\left (x^5+x^3\right )}^{1/4}}{9\,x^3} \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 {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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