Optimal. Leaf size=151 \[ \frac{7 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right ),4 \sqrt{3}-7\right )}{20 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}-\frac{7 \sqrt{-x^3-1}}{20 x^2}+\frac{\sqrt{-x^3-1}}{5 x^5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0310589, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 219} \[ -\frac{7 \sqrt{-x^3-1}}{20 x^2}+\frac{\sqrt{-x^3-1}}{5 x^5}+\frac{7 \sqrt{2-\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 219
Rubi steps
\begin{align*} \int \frac{1}{x^6 \sqrt{-1-x^3}} \, dx &=\frac{\sqrt{-1-x^3}}{5 x^5}-\frac{7}{10} \int \frac{1}{x^3 \sqrt{-1-x^3}} \, dx\\ &=\frac{\sqrt{-1-x^3}}{5 x^5}-\frac{7 \sqrt{-1-x^3}}{20 x^2}+\frac{7}{40} \int \frac{1}{\sqrt{-1-x^3}} \, dx\\ &=\frac{\sqrt{-1-x^3}}{5 x^5}-\frac{7 \sqrt{-1-x^3}}{20 x^2}+\frac{7 \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 )}{20 \sqrt [4]{3} \sqrt{-\frac{1+x}{\left (1-\sqrt{3}+x\right )^2}} \sqrt{-1-x^3}}\\ \end{align*}
Mathematica [C] time = 0.0048459, size = 42, normalized size = 0.28 \[ -\frac{\sqrt{x^3+1} \, _2F_1\left (-\frac{5}{3},\frac{1}{2};-\frac{2}{3};-x^3\right )}{5 x^5 \sqrt{-x^3-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 136, normalized size = 0.9 \begin{align*}{\frac{1}{5\,{x}^{5}}\sqrt{-{x}^{3}-1}}-{\frac{7}{20\,{x}^{2}}\sqrt{-{x}^{3}-1}}-{{\frac{7\,i}{60}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{3} - 1} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-x^{3} - 1}}{x^{9} + x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.06515, size = 39, normalized size = 0.26 \begin{align*} - \frac{i \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{-x^{3} - 1} x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]