Optimal. Leaf size=128 \[ \frac{92239 \sqrt{2 x^2-x+3}}{27648 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{1152 (2 x+5)^2}+\frac{5}{16} \sqrt{2 x^2-x+3}-\frac{1546507 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{331776 \sqrt{2}}+\frac{149 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.207631, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {1650, 1653, 843, 619, 215, 724, 206} \[ \frac{92239 \sqrt{2 x^2-x+3}}{27648 (2 x+5)}-\frac{3667 \sqrt{2 x^2-x+3}}{1152 (2 x+5)^2}+\frac{5}{16} \sqrt{2 x^2-x+3}-\frac{1546507 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{331776 \sqrt{2}}+\frac{149 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1650
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)^3 \sqrt{3-x+2 x^2}} \, dx &=-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}-\frac{1}{144} \int \frac{\frac{20347}{16}-\frac{6917 x}{4}+972 x^2-360 x^3}{(5+2 x)^2 \sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}+\frac{92239 \sqrt{3-x+2 x^2}}{27648 (5+2 x)}+\frac{\int \frac{\frac{647841}{16}-67392 x+12960 x^2}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{10368}\\ &=\frac{5}{16} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}+\frac{92239 \sqrt{3-x+2 x^2}}{27648 (5+2 x)}+\frac{\int \frac{\frac{777441}{2}-772416 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{82944}\\ &=\frac{5}{16} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}+\frac{92239 \sqrt{3-x+2 x^2}}{27648 (5+2 x)}-\frac{149}{32} \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx+\frac{1546507 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{55296}\\ &=\frac{5}{16} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}+\frac{92239 \sqrt{3-x+2 x^2}}{27648 (5+2 x)}-\frac{1546507 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{27648}-\frac{149 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{32 \sqrt{46}}\\ &=\frac{5}{16} \sqrt{3-x+2 x^2}-\frac{3667 \sqrt{3-x+2 x^2}}{1152 (5+2 x)^2}+\frac{92239 \sqrt{3-x+2 x^2}}{27648 (5+2 x)}+\frac{149 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{32 \sqrt{2}}-\frac{1546507 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{331776 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.129727, size = 88, normalized size = 0.69 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (34560 x^2+357278 x+589187\right )}{(2 x+5)^2}-1546507 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+1544832 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{663552} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.065, size = 102, normalized size = 0.8 \begin{align*}{\frac{5}{16}\sqrt{2\,{x}^{2}-x+3}}-{\frac{149\,\sqrt{2}}{64}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }-{\frac{3667}{4608}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}+{\frac{92239}{55296}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}-{\frac{1546507\,\sqrt{2}}{663552}{\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 = 1.50934, size = 154, normalized size = 1.2 \begin{align*} -\frac{149}{64} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{1546507}{663552} \, \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{5}{16} \, \sqrt{2 \, x^{2} - x + 3} - \frac{3667 \, \sqrt{2 \, x^{2} - x + 3}}{1152 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac{92239 \, \sqrt{2 \, x^{2} - x + 3}}{27648 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.42837, size = 443, normalized size = 3.46 \begin{align*} \frac{1544832 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 1546507 \, \sqrt{2}{\left (4 \, x^{2} + 20 \, x + 25\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 (34560 \, x^{2} + 357278 \, x + 589187\right )} \sqrt{2 \, x^{2} - x + 3}}{1327104 \,{\left (4 \, x^{2} + 20 \, x + 25\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 )^{3} \sqrt{2 x^{2} - x + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.20277, size = 335, normalized size = 2.62 \begin{align*} \frac{149}{64} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac{1546507}{663552} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x + \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{1546507}{663552} \, \sqrt{2} \log \left ({\left | -2 \, \sqrt{2} x - 11 \, \sqrt{2} + 2 \, \sqrt{2 \, x^{2} - x + 3} \right |}\right ) + \frac{5}{16} \, \sqrt{2 \, x^{2} - x + 3} + \frac{\sqrt{2}{\left (2381290 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{3} + 16628406 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} - 25697445 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 16720645\right )}}{55296 \,{\left (2 \,{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]