Optimal. Leaf size=29 \[ -\sqrt{-x^2-4 x+5}-\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0129247, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {640, 619, 216} \[ -\sqrt{-x^2-4 x+5}-\sin ^{-1}\left (\frac{1}{3} (-x-2)\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 640
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{3+x}{\sqrt{5-4 x-x^2}} \, dx &=-\sqrt{5-4 x-x^2}+\int \frac{1}{\sqrt{5-4 x-x^2}} \, dx\\ &=-\sqrt{5-4 x-x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{36}}} \, dx,x,-4-2 x\right )\\ &=-\sqrt{5-4 x-x^2}-\sin ^{-1}\left (\frac{1}{3} (-2-x)\right )\\ \end{align*}
Mathematica [A] time = 0.0081645, size = 25, normalized size = 0.86 \[ \sin ^{-1}\left (\frac{x+2}{3}\right )-\sqrt{-x^2-4 x+5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.044, size = 22, normalized size = 0.8 \begin{align*} \arcsin \left ({\frac{2}{3}}+{\frac{x}{3}} \right ) -\sqrt{-{x}^{2}-4\,x+5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.48641, size = 31, normalized size = 1.07 \begin{align*} -\sqrt{-x^{2} - 4 \, x + 5} - \arcsin \left (-\frac{1}{3} \, x - \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.34188, size = 105, normalized size = 3.62 \begin{align*} -\sqrt{-x^{2} - 4 \, x + 5} - \arctan \left (\frac{\sqrt{-x^{2} - 4 \, x + 5}{\left (x + 2\right )}}{x^{2} + 4 \, x - 5}\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{x + 3}{\sqrt{- \left (x - 1\right ) \left (x + 5\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.51581, size = 28, normalized size = 0.97 \begin{align*} -\sqrt{-x^{2} - 4 \, x + 5} + \arcsin \left (\frac{1}{3} \, x + \frac{2}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]