Optimal. Leaf size=27 \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0096739, 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} \[ \frac{\sqrt{x^4+1}}{2}+\frac{1}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\left (1+x^4\right )^{3/2}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{(1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{1}{(1+x)^{3/2}}+\frac{1}{\sqrt{1+x}}\right ) \, dx,x,x^4\right )\\ &=\frac{1}{2 \sqrt{1+x^4}}+\frac{\sqrt{1+x^4}}{2}\\ \end{align*}
Mathematica [A] time = 0.0052901, size = 18, normalized size = 0.67 \[ \frac{x^4+2}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 15, normalized size = 0.6 \begin{align*}{\frac{{x}^{4}+2}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982288, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} + \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 = 1.45773, size = 39, normalized size = 1.44 \begin{align*} \frac{x^{4} + 2}{2 \, \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.896962, size = 22, normalized size = 0.81 \begin{align*} \frac{x^{4}}{2 \sqrt{x^{4} + 1}} + \frac{1}{\sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16527, size = 26, normalized size = 0.96 \begin{align*} \frac{1}{2} \, \sqrt{x^{4} + 1} + \frac{1}{2 \, \sqrt{x^{4} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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