Optimal. Leaf size=119 \[ -\frac{6 (47 x+37)}{5 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}-\frac{864 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{166 \sqrt{3 x^2+5 x+2}}{5 (2 x+3)^2}+\frac{483 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.074717, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \[ -\frac{6 (47 x+37)}{5 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}-\frac{864 \sqrt{3 x^2+5 x+2}}{25 (2 x+3)}-\frac{166 \sqrt{3 x^2+5 x+2}}{5 (2 x+3)^2}+\frac{483 \tanh ^{-1}\left (\frac{8 x+7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 822
Rule 834
Rule 806
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{5-x}{(3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac{6 (37+47 x)}{5 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}-\frac{2}{5} \int \frac{431+564 x}{(3+2 x)^3 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}-\frac{166 \sqrt{2+5 x+3 x^2}}{5 (3+2 x)^2}+\frac{1}{25} \int \frac{-1575-2490 x}{(3+2 x)^2 \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}-\frac{166 \sqrt{2+5 x+3 x^2}}{5 (3+2 x)^2}-\frac{864 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{483}{25} \int \frac{1}{(3+2 x) \sqrt{2+5 x+3 x^2}} \, dx\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}-\frac{166 \sqrt{2+5 x+3 x^2}}{5 (3+2 x)^2}-\frac{864 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}-\frac{966}{25} \operatorname{Subst}\left (\int \frac{1}{20-x^2} \, dx,x,\frac{-7-8 x}{\sqrt{2+5 x+3 x^2}}\right )\\ &=-\frac{6 (37+47 x)}{5 (3+2 x)^2 \sqrt{2+5 x+3 x^2}}-\frac{166 \sqrt{2+5 x+3 x^2}}{5 (3+2 x)^2}-\frac{864 \sqrt{2+5 x+3 x^2}}{25 (3+2 x)}+\frac{483 \tanh ^{-1}\left (\frac{7+8 x}{2 \sqrt{5} \sqrt{2+5 x+3 x^2}}\right )}{25 \sqrt{5}}\\ \end{align*}
Mathematica [A] time = 0.0471767, size = 79, normalized size = 0.66 \[ -\frac{2 \left (2592 x^3+9453 x^2+10988 x+3977\right )}{25 (2 x+3)^2 \sqrt{3 x^2+5 x+2}}-\frac{483 \tanh ^{-1}\left (\frac{-8 x-7}{2 \sqrt{5} \sqrt{3 x^2+5 x+2}}\right )}{25 \sqrt{5}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.009, size = 111, normalized size = 0.9 \begin{align*} -{\frac{13}{40} \left ( x+{\frac{3}{2}} \right ) ^{-2}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{5}{2} \left ( x+{\frac{3}{2}} \right ) ^{-1}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}+{\frac{483}{50}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{1080+1296\,x}{25}{\frac{1}{\sqrt{3\, \left ( x+3/2 \right ) ^{2}-4\,x-{\frac{19}{4}}}}}}-{\frac{483\,\sqrt{5}}{125}{\it Artanh} \left ({\frac{2\,\sqrt{5}}{5} \left ( -{\frac{7}{2}}-4\,x \right ){\frac{1}{\sqrt{12\, \left ( x+3/2 \right ) ^{2}-16\,x-19}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.61597, size = 212, normalized size = 1.78 \begin{align*} -\frac{483}{125} \, \sqrt{5} \log \left (\frac{\sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac{5}{2 \,{\left | 2 \, x + 3 \right |}} - 2\right ) - \frac{1296 \, x}{25 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{1677}{50 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{13}{10 \,{\left (4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}} - \frac{5}{2 \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.77647, size = 346, normalized size = 2.91 \begin{align*} \frac{483 \, \sqrt{5}{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\right )} \log \left (\frac{4 \, \sqrt{5} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 20 \,{\left (2592 \, x^{3} + 9453 \, x^{2} + 10988 \, x + 3977\right )} \sqrt{3 \, x^{2} + 5 \, x + 2}}{250 \,{\left (12 \, x^{4} + 56 \, x^{3} + 95 \, x^{2} + 69 \, x + 18\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}{24 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 148 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 358 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 423 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 243 x \sqrt{3 x^{2} + 5 x + 2} + 54 \sqrt{3 x^{2} + 5 x + 2}}\, dx - \int - \frac{5}{24 x^{5} \sqrt{3 x^{2} + 5 x + 2} + 148 x^{4} \sqrt{3 x^{2} + 5 x + 2} + 358 x^{3} \sqrt{3 x^{2} + 5 x + 2} + 423 x^{2} \sqrt{3 x^{2} + 5 x + 2} + 243 x \sqrt{3 x^{2} + 5 x + 2} + 54 \sqrt{3 x^{2} + 5 x + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16764, size = 304, normalized size = 2.55 \begin{align*} \frac{483}{125} \, \sqrt{5} \log \left (\frac{{\left | -4 \, \sqrt{3} x - 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt{3} x + 2 \, \sqrt{5} - 6 \, \sqrt{3} + 4 \, \sqrt{3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac{6 \,{\left (903 \, x + 653\right )}}{125 \, \sqrt{3 \, x^{2} + 5 \, x + 2}} - \frac{2 \,{\left (2442 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 9999 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 35473 \, \sqrt{3} x + 12979 \, \sqrt{3} - 35473 \, \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}}{125 \,{\left (2 \,{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]