Optimal. Leaf size=87 \[ -\frac{2 (197-837 x)}{3887 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{169 (2 x+1)}+\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.092342, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1646, 806, 724, 206} \[ -\frac{2 (197-837 x)}{3887 \sqrt{3 x^2-x+2}}-\frac{4 \sqrt{3 x^2-x+2}}{169 (2 x+1)}+\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1646
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{1+3 x+4 x^2}{(1+2 x)^2 \left (2-x+3 x^2\right )^{3/2}} \, dx &=-\frac{2 (197-837 x)}{3887 \sqrt{2-x+3 x^2}}+\frac{2}{23} \int \frac{\frac{184}{169}-\frac{230 x}{169}}{(1+2 x)^2 \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (197-837 x)}{3887 \sqrt{2-x+3 x^2}}-\frac{4 \sqrt{2-x+3 x^2}}{169 (1+2 x)}-\frac{2}{169} \int \frac{1}{(1+2 x) \sqrt{2-x+3 x^2}} \, dx\\ &=-\frac{2 (197-837 x)}{3887 \sqrt{2-x+3 x^2}}-\frac{4 \sqrt{2-x+3 x^2}}{169 (1+2 x)}+\frac{4}{169} \operatorname{Subst}\left (\int \frac{1}{52-x^2} \, dx,x,\frac{9-8 x}{\sqrt{2-x+3 x^2}}\right )\\ &=-\frac{2 (197-837 x)}{3887 \sqrt{2-x+3 x^2}}-\frac{4 \sqrt{2-x+3 x^2}}{169 (1+2 x)}+\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{2-x+3 x^2}}\right )}{169 \sqrt{13}}\\ \end{align*}
Mathematica [A] time = 0.0459473, size = 74, normalized size = 0.85 \[ \frac{2 \left (1536 x^2+489 x-289\right )}{3887 (2 x+1) \sqrt{3 x^2-x+2}}+\frac{2 \tanh ^{-1}\left (\frac{9-8 x}{2 \sqrt{13} \sqrt{3 x^2-x+2}}\right )}{169 \sqrt{13}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.055, size = 109, normalized size = 1.3 \begin{align*}{\frac{-2+12\,x}{23}{\frac{1}{\sqrt{3\,{x}^{2}-x+2}}}}-{\frac{1}{169}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}-{\frac{-82+492\,x}{3887}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}}+{\frac{2\,\sqrt{13}}{2197}{\it Artanh} \left ({\frac{2\,\sqrt{13}}{13} \left ({\frac{9}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+1/2 \right ) ^{2}-16\,x+5}}}} \right ) }-{\frac{1}{26} \left ( x+{\frac{1}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+1/2 \right ) ^{2}-4\,x+{\frac{5}{4}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.50075, size = 130, normalized size = 1.49 \begin{align*} -\frac{2}{2197} \, \sqrt{13} \operatorname{arsinh}\left (\frac{8 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 1 \right |}} - \frac{9 \, \sqrt{23}}{23 \,{\left | 2 \, x + 1 \right |}}\right ) + \frac{1536 \, x}{3887 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{279}{3887 \, \sqrt{3 \, x^{2} - x + 2}} - \frac{1}{13 \,{\left (2 \, \sqrt{3 \, x^{2} - x + 2} x + \sqrt{3 \, x^{2} - x + 2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.15101, size = 285, normalized size = 3.28 \begin{align*} \frac{23 \, \sqrt{13}{\left (6 \, x^{3} + x^{2} + 3 \, x + 2\right )} \log \left (\frac{4 \, \sqrt{13} \sqrt{3 \, x^{2} - x + 2}{\left (8 \, x - 9\right )} - 220 \, x^{2} + 196 \, x - 185}{4 \, x^{2} + 4 \, x + 1}\right ) + 26 \,{\left (1536 \, x^{2} + 489 \, x - 289\right )} \sqrt{3 \, x^{2} - x + 2}}{50531 \,{\left (6 \, x^{3} + x^{2} + 3 \, x + 2\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{4 x^{2} + 3 x + 1}{\left (2 x + 1\right )^{2} \left (3 x^{2} - x + 2\right )^{\frac{3}{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{4 \, x^{2} + 3 \, x + 1}{{\left (3 \, x^{2} - x + 2\right )}^{\frac{3}{2}}{\left (2 \, x + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]