Optimal. Leaf size=262 \[ \frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right ),-7-4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}+\frac{2}{13} \sqrt{x^3+1} x^5-\frac{20}{91} \sqrt{x^3+1} x^2+\frac{80 \sqrt{x^3+1}}{91 \left (x+\sqrt{3}+1\right )}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
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Rubi [A] time = 0.0701099, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {321, 303, 218, 1877} \[ \frac{2}{13} \sqrt{x^3+1} x^5-\frac{20}{91} \sqrt{x^3+1} x^2+\frac{80 \sqrt{x^3+1}}{91 \left (x+\sqrt{3}+1\right )}+\frac{80 \sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
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Rule 321
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt{1+x^3}} \, dx &=\frac{2}{13} x^5 \sqrt{1+x^3}-\frac{10}{13} \int \frac{x^4}{\sqrt{1+x^3}} \, dx\\ &=-\frac{20}{91} x^2 \sqrt{1+x^3}+\frac{2}{13} x^5 \sqrt{1+x^3}+\frac{40}{91} \int \frac{x}{\sqrt{1+x^3}} \, dx\\ &=-\frac{20}{91} x^2 \sqrt{1+x^3}+\frac{2}{13} x^5 \sqrt{1+x^3}+\frac{40}{91} \int \frac{1-\sqrt{3}+x}{\sqrt{1+x^3}} \, dx+\frac{1}{91} \left (40 \sqrt{2 \left (2-\sqrt{3}\right )}\right ) \int \frac{1}{\sqrt{1+x^3}} \, dx\\ &=-\frac{20}{91} x^2 \sqrt{1+x^3}+\frac{2}{13} x^5 \sqrt{1+x^3}+\frac{80 \sqrt{1+x^3}}{91 \left (1+\sqrt{3}+x\right )}-\frac{40 \sqrt [4]{3} \sqrt{2-\sqrt{3}} (1+x) \sqrt{\frac{1-x+x^2}{\left (1+\sqrt{3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac{1-\sqrt{3}+x}{1+\sqrt{3}+x}\right )|-7-4 \sqrt{3}\right )}{91 \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}+\frac{80 \sqrt{2} (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 )}{91 \sqrt [4]{3} \sqrt{\frac{1+x}{\left (1+\sqrt{3}+x\right )^2}} \sqrt{1+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0130941, size = 42, normalized size = 0.16 \[ \frac{2}{91} x^2 \left (10 \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{5}{3};-x^3\right )+\sqrt{x^3+1} \left (7 x^3-10\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 198, normalized size = 0.8 \begin{align*}{\frac{2\,{x}^{5}}{13}\sqrt{{x}^{3}+1}}-{\frac{20\,{x}^{2}}{91}\sqrt{{x}^{3}+1}}+{\frac{120-40\,i\sqrt{3}}{91}\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 ) }} \left ( \left ( -{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \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 ) + \left ({\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 ) \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^{7}}{\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^{7}}{\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 = 0.985831, size = 29, normalized size = 0.11 \begin{align*} \frac{x^{8} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac{11}{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^{7}}{\sqrt{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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