Optimal. Leaf size=41 \[ -\frac{2 \sqrt{x+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{1}{3}}}\right ),\frac{4}{3}\right )}{\sqrt{3} \sqrt{-x-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0107321, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {121, 118} \[ -\frac{2 \sqrt{x+1} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{3}+\frac{1}{3}}}\right )|\frac{4}{3}\right )}{\sqrt{3} \sqrt{-x-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 121
Rule 118
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-1-x} \sqrt{-3+x} \sqrt{-2+x}} \, dx &=\frac{\sqrt{1+x} \int \frac{1}{\sqrt{\frac{1}{3}+\frac{x}{3}} \sqrt{-3+x} \sqrt{-2+x}} \, dx}{\sqrt{3} \sqrt{-1-x}}\\ &=-\frac{2 \sqrt{1+x} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{1}{3}+\frac{x}{3}}}\right )|\frac{4}{3}\right )}{\sqrt{3} \sqrt{-1-x}}\\ \end{align*}
Mathematica [C] time = 0.0576922, size = 72, normalized size = 1.76 \[ \frac{2 i \sqrt{\frac{x-3}{x-2}} \sqrt{\frac{x-2}{x+1}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{3}}{\sqrt{-x-1}}\right ),\frac{4}{3}\right )}{\sqrt{3} \sqrt{\frac{x-3}{x+1}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.039, size = 68, normalized size = 1.7 \begin{align*} -{\frac{2\,\sqrt{3}}{3\,{x}^{3}-12\,{x}^{2}+3\,x+18}\sqrt{-1-x}\sqrt{-3+x}\sqrt{-2+x}\sqrt{1+x}\sqrt{2-x}\sqrt{3-x}{\it EllipticF} \left ({\frac{1}{2}\sqrt{1+x}},{\frac{2\,\sqrt{3}}{3}} \right ) } \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 - 2} \sqrt{x - 3} \sqrt{-x - 1}}\,{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 - 2} \sqrt{x - 3} \sqrt{-x - 1}}{x^{3} - 4 \, x^{2} + x + 6}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{- x - 1} \sqrt{x - 3} \sqrt{x - 2}}\, dx \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 - 2} \sqrt{x - 3} \sqrt{-x - 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]