Optimal. Leaf size=136 \[ \frac{32 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{2}{11} \sqrt{x^3+1} x^4-\frac{16}{55} \sqrt{x^3+1} x \]
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Rubi [A] time = 0.0340231, antiderivative size = 136, 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 = {321, 218} \[ \frac{2}{11} \sqrt{x^3+1} x^4-\frac{16}{55} \sqrt{x^3+1} x+\frac{32 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 218
Rubi steps
\begin{align*} \int \frac{x^6}{\sqrt{1+x^3}} \, dx &=\frac{2}{11} x^4 \sqrt{1+x^3}-\frac{8}{11} \int \frac{x^3}{\sqrt{1+x^3}} \, dx\\ &=-\frac{16}{55} x \sqrt{1+x^3}+\frac{2}{11} x^4 \sqrt{1+x^3}+\frac{16}{55} \int \frac{1}{\sqrt{1+x^3}} \, dx\\ &=-\frac{16}{55} x \sqrt{1+x^3}+\frac{2}{11} x^4 \sqrt{1+x^3}+\frac{32 \sqrt{2+\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{55 \sqrt [4]{3} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0115907, size = 40, normalized size = 0.29 \[ \frac{2}{55} x \left (8 \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{4}{3};-x^3\right )+\sqrt{x^3+1} \left (5 x^3-8\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 139, normalized size = 1. \begin{align*}{\frac{2\,{x}^{4}}{11}\sqrt{{x}^{3}+1}}-{\frac{16\,x}{55}\sqrt{{x}^{3}+1}}+{\frac{48-16\,i\sqrt{3}}{55}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{6}}{\sqrt{x^{3} + 1}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.671079, size = 29, normalized size = 0.21 \begin{align*} \frac{x^{7} \Gamma \left (\frac{7}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{3} \\ \frac{10}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{10}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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