Optimal. Leaf size=63 \[ -\frac{5 \sqrt{x^3+1}}{24 x^3}+\frac{5 \sqrt{x^3+1}}{36 x^6}-\frac{\sqrt{x^3+1}}{9 x^9}+\frac{5}{24} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
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Rubi [A] time = 0.0201826, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ -\frac{5 \sqrt{x^3+1}}{24 x^3}+\frac{5 \sqrt{x^3+1}}{36 x^6}-\frac{\sqrt{x^3+1}}{9 x^9}+\frac{5}{24} \tanh ^{-1}\left (\sqrt{x^3+1}\right ) \]
Antiderivative was successfully verified.
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Rule 266
Rule 51
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{1}{x^{10} \sqrt{1+x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^4 \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{9 x^9}-\frac{5}{18} \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{9 x^9}+\frac{5 \sqrt{1+x^3}}{36 x^6}+\frac{5}{24} \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{9 x^9}+\frac{5 \sqrt{1+x^3}}{36 x^6}-\frac{5 \sqrt{1+x^3}}{24 x^3}-\frac{5}{48} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1+x}} \, dx,x,x^3\right )\\ &=-\frac{\sqrt{1+x^3}}{9 x^9}+\frac{5 \sqrt{1+x^3}}{36 x^6}-\frac{5 \sqrt{1+x^3}}{24 x^3}-\frac{5}{24} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sqrt{1+x^3}\right )\\ &=-\frac{\sqrt{1+x^3}}{9 x^9}+\frac{5 \sqrt{1+x^3}}{36 x^6}-\frac{5 \sqrt{1+x^3}}{24 x^3}+\frac{5}{24} \tanh ^{-1}\left (\sqrt{1+x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0049084, size = 26, normalized size = 0.41 \[ \frac{2}{3} \sqrt{x^3+1} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};x^3+1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 48, normalized size = 0.8 \begin{align*}{\frac{5}{24}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{9\,{x}^{9}}\sqrt{{x}^{3}+1}}+{\frac{5}{36\,{x}^{6}}\sqrt{{x}^{3}+1}}-{\frac{5}{24\,{x}^{3}}\sqrt{{x}^{3}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.980731, size = 108, normalized size = 1.71 \begin{align*} -\frac{15 \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} + 1}}{72 \,{\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \,{\left (x^{3} + 1\right )}^{2} + 2\right )}} + \frac{5}{48} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{5}{48} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49432, size = 155, normalized size = 2.46 \begin{align*} \frac{15 \, x^{9} \log \left (\sqrt{x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \,{\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt{x^{3} + 1}}{144 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.27561, size = 85, normalized size = 1.35 \begin{align*} \frac{5 \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} - \frac{5}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{5}{72 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{1}{36 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{1}{9 x^{\frac{21}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13281, size = 80, normalized size = 1.27 \begin{align*} -\frac{15 \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{x^{3} + 1}}{72 \, x^{9}} + \frac{5}{48} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) - \frac{5}{48} \, \log \left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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