Optimal. Leaf size=74 \[ \frac {5}{6} \sqrt {x^4+1} x-\frac {x^5}{2 \sqrt {x^4+1}}-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 321, 220} \[ -\frac {x^5}{2 \sqrt {x^4+1}}+\frac {5}{6} \sqrt {x^4+1} x-\frac {5 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^8}{\left (1+x^4\right )^{3/2}} \, dx &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{2} \int \frac {x^4}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{6} x \sqrt {1+x^4}-\frac {5}{6} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=-\frac {x^5}{2 \sqrt {1+x^4}}+\frac {5}{6} x \sqrt {1+x^4}-\frac {5 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{12 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 47, normalized size = 0.64 \[ \frac {x \left (-5 \sqrt {x^4+1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-x^4\right )+2 x^4+5\right )}{6 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1} x^{8}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]
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 \[ \int \frac {x^{8}}{{\left (x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 82, normalized size = 1.11 \[ \frac {x}{2 \sqrt {x^{4}+1}}+\frac {\sqrt {x^{4}+1}\, x}{3}-\frac {5 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{6 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
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 \[ \int \frac {x^{8}}{{\left (x^{4} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^8}{{\left (x^4+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.40, size = 29, normalized size = 0.39 \[ \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {13}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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