Optimal. Leaf size=57 \[ -\frac{\sqrt{x+3} \text{EllipticF}\left (\sin ^{-1}\left (\frac{2}{\sqrt{x+3}}\right ),\frac{5}{4}\right )}{\sqrt{-x-3}}-\frac{i K\left (-\frac{1}{4}\right ) \sqrt{x+3}}{\sqrt{-x-3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0099321, antiderivative size = 36, normalized size of antiderivative = 0.63, 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} \[ -\frac{\sqrt{x+3} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{x}{4}+\frac{3}{4}}}\right )|\frac{5}{4}\right )}{\sqrt{-x-3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 121
Rule 118
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{-3-x} \sqrt{-2+x} \sqrt{-1+x}} \, dx &=\frac{\sqrt{3+x} \int \frac{1}{\sqrt{\frac{3}{4}+\frac{x}{4}} \sqrt{-2+x} \sqrt{-1+x}} \, dx}{2 \sqrt{-3-x}}\\ &=-\frac{\sqrt{3+x} F\left (\sin ^{-1}\left (\frac{1}{\sqrt{\frac{3}{4}+\frac{x}{4}}}\right )|\frac{5}{4}\right )}{\sqrt{-3-x}}\\ \end{align*}
Mathematica [A] time = 0.0934196, size = 63, normalized size = 1.11 \[ \frac{i \sqrt{\frac{x-2}{x-1}} \sqrt{\frac{x-1}{x+3}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2}{\sqrt{-x-3}}\right ),\frac{5}{4}\right )}{\sqrt{\frac{x-2}{x+3}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.035, size = 65, normalized size = 1.1 \begin{align*}{\frac{1}{-{x}^{3}+7\,x-6}\sqrt{-3-x}\sqrt{-2+x}\sqrt{-1+x}\sqrt{3+x}\sqrt{1-x}\sqrt{2-x}{\it EllipticF} \left ({\frac{1}{5}\sqrt{15+5\,x}},{\frac{\sqrt{5}}{2}} \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 - 1} \sqrt{x - 2} \sqrt{-x - 3}}\,{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 - 1} \sqrt{x - 2} \sqrt{-x - 3}}{x^{3} - 7 \, 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 - 3} \sqrt{x - 2} \sqrt{x - 1}}\, 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 - 1} \sqrt{x - 2} \sqrt{-x - 3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]