Optimal. Leaf size=101 \[ \frac{917 x+1191}{3312 \sqrt{2 x^2-x+3}}+\frac{5}{8} \sqrt{2 x^2-x+3}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1728 \sqrt{2}}+\frac{39 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.151246, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1646, 1653, 843, 619, 215, 724, 206} \[ \frac{917 x+1191}{3312 \sqrt{2 x^2-x+3}}+\frac{5}{8} \sqrt{2 x^2-x+3}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{1728 \sqrt{2}}+\frac{39 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1646
Rule 1653
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{2+x+3 x^2-x^3+5 x^4}{(5+2 x) \left (3-x+2 x^2\right )^{3/2}} \, dx &=\frac{1191+917 x}{3312 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{-\frac{6739}{576}+\frac{69 x}{8}+\frac{115 x^2}{4}}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{1191+917 x}{3312 \sqrt{3-x+2 x^2}}+\frac{5}{8} \sqrt{3-x+2 x^2}+\frac{1}{92} \int \frac{\frac{3611}{72}-\frac{897 x}{2}}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{1191+917 x}{3312 \sqrt{3-x+2 x^2}}+\frac{5}{8} \sqrt{3-x+2 x^2}-\frac{39}{16} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{3667}{288} \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx\\ &=\frac{1191+917 x}{3312 \sqrt{3-x+2 x^2}}+\frac{5}{8} \sqrt{3-x+2 x^2}-\frac{3667}{144} \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )-\frac{39 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{16 \sqrt{46}}\\ &=\frac{1191+917 x}{3312 \sqrt{3-x+2 x^2}}+\frac{5}{8} \sqrt{3-x+2 x^2}+\frac{39 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{2}}-\frac{3667 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{1728 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.349433, size = 86, normalized size = 0.85 \[ \frac{\frac{12 \left (4140 x^2-1153 x+7401\right )}{23 \sqrt{x^2-\frac{x}{2}+\frac{3}{2}}}-3667 \log \left (12 \sqrt{4 x^2-2 x+6}-22 x+17\right )+3667 \log (2 x+5)-4212 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{1728 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.061, size = 148, normalized size = 1.5 \begin{align*}{\frac{5\,{x}^{2}}{4}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{39\,x}{16}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{309}{64}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{-5507+22028\,x}{1472}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{39\,\sqrt{2}}{32}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{3667}{576}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}+{\frac{-40337+161348\,x}{13248}{\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}}-{\frac{3667\,\sqrt{2}}{3456}{\it Artanh} \left ({\frac{\sqrt{2}}{12} \left ({\frac{17}{2}}-11\,x \right ){\frac{1}{\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52708, size = 134, normalized size = 1.33 \begin{align*} \frac{5 \, x^{2}}{4 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{39}{32} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{3667}{3456} \, \sqrt{2} \operatorname{arsinh}\left (\frac{22 \, \sqrt{23} x}{23 \,{\left | 2 \, x + 5 \right |}} - \frac{17 \, \sqrt{23}}{23 \,{\left | 2 \, x + 5 \right |}}\right ) - \frac{1153 \, x}{3312 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{2467}{1104 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.38289, size = 413, normalized size = 4.09 \begin{align*} \frac{96876 \, \sqrt{2}{\left (2 \, x^{2} - x + 3\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 84341 \, \sqrt{2}{\left (2 \, x^{2} - x + 3\right )} \log \left (-\frac{24 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \,{\left (4140 \, x^{2} - 1153 \, x + 7401\right )} \sqrt{2 \, x^{2} - x + 3}}{158976 \,{\left (2 \, x^{2} - x + 3\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{5 x^{4} - x^{3} + 3 x^{2} + x + 2}{\left (2 x + 5\right ) \left (2 x^{2} - x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16943, size = 159, normalized size = 1.57 \begin{align*} \frac{39}{32} \, \sqrt{2} \log \left (-4 \, \sqrt{2} x + \sqrt{2} + 4 \, \sqrt{2 \, x^{2} - x + 3}\right ) - \frac{3667}{3456} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{3667}{3456} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{{\left (4140 \, x - 1153\right )} x + 7401}{3312 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]