Optimal. Leaf size=166 \[ \frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.203946, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1660, 1661, 640, 619, 215} \[ \frac{625}{24} \sqrt{2 x^2-x+3} x^5+\frac{10075}{96} \sqrt{2 x^2-x+3} x^4+\frac{79425}{512} \sqrt{2 x^2-x+3} x^3-\frac{111315 \sqrt{2 x^2-x+3} x^2}{2048}-\frac{8992487 \sqrt{2 x^2-x+3} x}{16384}-\frac{31009685 \sqrt{2 x^2-x+3}}{65536}-\frac{14641 (79 x+101)}{1472 \sqrt{2 x^2-x+3}}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1660
Rule 1661
Rule 640
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \frac{\left (2+3 x+5 x^2\right )^4}{\left (3-x+2 x^2\right )^{3/2}} \, dx &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{2}{23} \int \frac{\frac{2821893}{256}-\frac{661181 x}{128}-\frac{488267 x^2}{64}+\frac{143635 x^3}{32}+\frac{213325 x^4}{16}+\frac{83375 x^5}{8}+\frac{14375 x^6}{4}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{1}{138} \int \frac{\frac{8465679}{64}-\frac{1983543 x}{32}-\frac{1464801 x^2}{16}+\frac{430905 x^3}{8}+\frac{212175 x^4}{2}+\frac{1158625 x^5}{8}}{\sqrt{3-x+2 x^2}} \, dx\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{42328395}{32}-\frac{9917715 x}{16}-\frac{7324005 x^2}{8}-\frac{4797225 x^3}{4}+\frac{27401625 x^4}{16}}{\sqrt{3-x+2 x^2}} \, dx}{1380}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{42328395}{4}-\frac{9917715 x}{2}-\frac{363798705 x^2}{16}-\frac{115211025 x^3}{32}}{\sqrt{3-x+2 x^2}} \, dx}{11040}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{126985185}{2}-\frac{130417245 x}{16}-\frac{9307224045 x^2}{64}}{\sqrt{3-x+2 x^2}} \, dx}{66240}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{\int \frac{\frac{44175775815}{64}-\frac{32095023975 x}{128}}{\sqrt{3-x+2 x^2}} \, dx}{264960}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{310445587 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{131072}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}+\frac{310445587 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{131072 \sqrt{46}}\\ &=-\frac{14641 (101+79 x)}{1472 \sqrt{3-x+2 x^2}}-\frac{31009685 \sqrt{3-x+2 x^2}}{65536}-\frac{8992487 x \sqrt{3-x+2 x^2}}{16384}-\frac{111315 x^2 \sqrt{3-x+2 x^2}}{2048}+\frac{79425}{512} x^3 \sqrt{3-x+2 x^2}+\frac{10075}{96} x^4 \sqrt{3-x+2 x^2}+\frac{625}{24} x^5 \sqrt{3-x+2 x^2}-\frac{310445587 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{131072 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.376634, size = 95, normalized size = 0.57 \[ \sqrt{2 x^2-x+3} \left (\frac{625 x^5}{24}+\frac{10075 x^4}{96}+\frac{79425 x^3}{512}-\frac{111315 x^2}{2048}-\frac{14641 (79 x+101)}{1472 \left (2 x^2-x+3\right )}-\frac{8992487 x}{16384}-\frac{31009685}{65536}\right )+\frac{310445587 \sinh ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{131072 \sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.066, size = 166, normalized size = 1. \begin{align*}{\frac{8825\,{x}^{6}}{48}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{217675\,{x}^{5}}{768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{52235\,{x}^{4}}{1024}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{310445587\,\sqrt{2}}{262144}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-1234044515+4936178060\,x}{12058624}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}+{\frac{625\,{x}^{7}}{12}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{310445587\,x}{131072}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{4734827\,{x}^{3}}{8192}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{18367831\,{x}^{2}}{32768}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}}-{\frac{1217267299}{524288}{\frac{1}{\sqrt{2\,{x}^{2}-x+3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.49527, size = 200, normalized size = 1.2 \begin{align*} \frac{625 \, x^{7}}{12 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{8825 \, x^{6}}{48 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{217675 \, x^{5}}{768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{52235 \, x^{4}}{1024 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{4734827 \, x^{3}}{8192 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{18367831 \, x^{2}}{32768 \, \sqrt{2 \, x^{2} - x + 3}} + \frac{310445587}{262144} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{2953101993 \, x}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} - \frac{3653899049}{1507328 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41592, size = 385, normalized size = 2.32 \begin{align*} \frac{21420745503 \, \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 ) + 8 \,{\left (235520000 \, x^{7} + 831385600 \, x^{6} + 1281670400 \, x^{5} + 230669760 \, x^{4} - 2613624504 \, x^{3} - 2534760678 \, x^{2} - 8859305979 \, x - 10961697147\right )} \sqrt{2 \, x^{2} - x + 3}}{36175872 \,{\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{\left (5 x^{2} + 3 x + 2\right )^{4}}{\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 = 1.1978, size = 111, normalized size = 0.67 \begin{align*} -\frac{310445587}{262144} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac{{\left (46 \,{\left (4 \,{\left (40 \,{\left (20 \,{\left (16 \,{\left (100 \, x + 353\right )} x + 8707\right )} x + 31341\right )} x - 14204481\right )} x - 55103493\right )} x - 8859305979\right )} x - 10961697147}{4521984 \, \sqrt{2 \, x^{2} - x + 3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]