Optimal. Leaf size=27 \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\frac{\sqrt{x^4+1}}{2} \]
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Rubi [A] time = 0.0098072, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{6} \left (x^4+1\right )^{3/2}-\frac{\sqrt{x^4+1}}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt{1+x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{1}{\sqrt{1+x}}+\sqrt{1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{2} \sqrt{1+x^4}+\frac{1}{6} \left (1+x^4\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0050074, size = 18, normalized size = 0.67 \[ \frac{1}{6} \left (x^4-2\right ) \sqrt{x^4+1} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}-2}{6}\sqrt{{x}^{4}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05061, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44162, size = 39, normalized size = 1.44 \begin{align*} \frac{1}{6} \, \sqrt{x^{4} + 1}{\left (x^{4} - 2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.587138, size = 22, normalized size = 0.81 \begin{align*} \frac{x^{4} \sqrt{x^{4} + 1}}{6} - \frac{\sqrt{x^{4} + 1}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16796, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{6} \,{\left (x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{x^{4} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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