Optimal. Leaf size=72 \[ \frac {1}{7} x \left (x^4+1\right )^{3/2}+\frac {2}{7} x \sqrt {x^4+1}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{7 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {195, 220} \[ \frac {1}{7} x \left (x^4+1\right )^{3/2}+\frac {2}{7} x \sqrt {x^4+1}+\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{7 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 195
Rule 220
Rubi steps
\begin {align*} \int \left (1+x^4\right )^{3/2} \, dx &=\frac {1}{7} x \left (1+x^4\right )^{3/2}+\frac {6}{7} \int \sqrt {1+x^4} \, dx\\ &=\frac {2}{7} x \sqrt {1+x^4}+\frac {1}{7} x \left (1+x^4\right )^{3/2}+\frac {4}{7} \int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {2}{7} x \sqrt {1+x^4}+\frac {1}{7} x \left (1+x^4\right )^{3/2}+\frac {2 \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 )}{7 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 17, normalized size = 0.24 \[ x \, _2F_1\left (-\frac {3}{2},\frac {1}{4};\frac {5}{4};-x^4\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{4} + 1\right )}^{\frac {3}{2}}, x\right ) \]
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 \[ \int {\left (x^{4} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.05, size = 84, normalized size = 1.17 \[ \frac {\sqrt {x^{4}+1}\, x^{5}}{7}+\frac {3 \sqrt {x^{4}+1}\, x}{7}+\frac {4 \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )}{7 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
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 \[ \int {\left (x^{4} + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.02, size = 12, normalized size = 0.17 \[ x\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{4};\ \frac {5}{4};\ -x^4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.01, size = 29, normalized size = 0.40 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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