Optimal. Leaf size=38 \[ \frac {3 \left (x^4+1\right )^{2/3} \left (5 x^8+8 x^7+10 x^4+8 x^3+5\right )}{40 x^8} \]
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Rubi [A] time = 0.09, antiderivative size = 33, normalized size of antiderivative = 0.87, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1833, 1584, 449, 1474, 847, 74} \begin {gather*} \frac {3 \left (x^4+1\right )^{8/3}}{8 x^8}+\frac {3 \left (x^4+1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 74
Rule 449
Rule 847
Rule 1474
Rule 1584
Rule 1833
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (1+x^3+x^4\right )}{x^9} \, dx &=\int \left (\frac {\left (1+x^4\right )^{2/3} \left (-3 x^2+x^6\right )}{x^8}+\frac {\left (1+x^4\right )^{2/3} \left (-3-2 x^4+x^8\right )}{x^9}\right ) \, dx\\ &=\int \frac {\left (1+x^4\right )^{2/3} \left (-3 x^2+x^6\right )}{x^8} \, dx+\int \frac {\left (1+x^4\right )^{2/3} \left (-3-2 x^4+x^8\right )}{x^9} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {(1+x)^{2/3} \left (-3-2 x+x^2\right )}{x^3} \, dx,x,x^4\right )+\int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx\\ &=\frac {3 \left (1+x^4\right )^{5/3}}{5 x^5}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {(-3+x) (1+x)^{5/3}}{x^3} \, dx,x,x^4\right )\\ &=\frac {3 \left (1+x^4\right )^{5/3}}{5 x^5}+\frac {3 \left (1+x^4\right )^{8/3}}{8 x^8}\\ \end {align*}
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Mathematica [C] time = 0.23, size = 158, normalized size = 4.16 \begin {gather*} \frac {24 \, _2F_1\left (-\frac {5}{4},-\frac {2}{3};-\frac {1}{4};-x^4\right )+x^4 \left (x \left (-12 \left (x^4+1\right )^{5/3} \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};x^4+1\right )+18 \left (x^4+1\right )^{5/3} \, _2F_1\left (\frac {5}{3},3;\frac {8}{3};x^4+1\right )+5 \left (3 \left (\left (x^4+1\right )^{2/3}+\log \left (1-\sqrt [3]{x^4+1}\right )\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^4+1}+1}{\sqrt {3}}\right )-4 \log (x)\right )\right )-40 \, _2F_1\left (-\frac {2}{3},-\frac {1}{4};\frac {3}{4};-x^4\right )\right )}{40 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 28, normalized size = 0.74 \begin {gather*} \frac {3 \left (x^4+1\right )^{5/3} \left (5 x^4+8 x^3+5\right )}{40 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (5 \, x^{8} + 8 \, x^{7} + 10 \, x^{4} + 8 \, x^{3} + 5\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{40 \, x^{8}} \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^{4} + x^{3} + 1\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \begin {gather*} \frac {3 \left (x^{4}+1\right )^{\frac {5}{3}} \left (5 x^{4}+8 x^{3}+5\right )}{40 x^{8}} \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*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {2 \, {\left (x^{4} + 1\right )}^{\frac {5}{3}} + {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{8 \, {\left (2 \, x^{4} - {\left (x^{4} + 1\right )}^{2} + 1\right )}} + \int \frac {{\left (x^{5} + x^{4} - 2 \, x - 3\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{x^{6}}\,{d x} - \frac {1}{24} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {2}{3}} + {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{12} \, \log \left ({\left (x^{4} + 1\right )}^{\frac {1}{3}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 58, normalized size = 1.53 \begin {gather*} \frac {3\,{\left (x^4+1\right )}^{2/3}}{8}+\frac {3\,{\left (x^4+1\right )}^{2/3}}{5\,x}+\frac {3\,{\left (x^4+1\right )}^{2/3}}{4\,x^4}+\frac {3\,{\left (x^4+1\right )}^{2/3}}{5\,x^5}+\frac {3\,{\left (x^4+1\right )}^{2/3}}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.46, size = 180, normalized size = 4.74 \begin {gather*} - \frac {x^{\frac {8}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} + \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{2 x^{\frac {4}{3}} \Gamma \left (\frac {4}{3}\right )} + \frac {3 \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{4}}} \right )}}{4 x^{\frac {16}{3}} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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