Optimal. Leaf size=254 \[ \frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}+\frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]
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Rubi [A] time = 0.372556, antiderivative size = 254, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{625}{28} \left (2 x^2-x+3\right )^{7/2} x^7+\frac{13875}{208} \left (2 x^2-x+3\right )^{7/2} x^6+\frac{1046225 \left (2 x^2-x+3\right )^{7/2} x^5}{9984}+\frac{3684995 \left (2 x^2-x+3\right )^{7/2} x^4}{39936}+\frac{23460839 \left (2 x^2-x+3\right )^{7/2} x^3}{532480}+\frac{122595067 \left (2 x^2-x+3\right )^{7/2} x^2}{19169280}+\frac{112244125 \left (2 x^2-x+3\right )^{7/2} x}{122683392}+\frac{25250178739 \left (2 x^2-x+3\right )^{7/2}}{5725224960}-\frac{401135647 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{335544320}-\frac{9226119881 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2147483648}-\frac{636602271789 (1-4 x) \sqrt{2 x^2-x+3}}{34359738368}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{28} \int \left (3-x+2 x^2\right )^{5/2} \left (448+2688 x+10528 x^2+26208 x^3+49308 x^4+65520 x^5+52675 x^6+\frac{97125 x^7}{2}\right ) \, dx\\ &=\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{728} \int \left (3-x+2 x^2\right )^{5/2} \left (11648+69888 x+273728 x^2+681408 x^3+1282008 x^4+829395 x^5+\frac{7323575 x^6}{4}\right ) \, dx\\ &=\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (279552+1677312 x+6569472 x^2+16353792 x^3+\frac{13219143 x^4}{4}+\frac{283744615 x^5}{8}\right ) \, dx}{17472}\\ &=\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (6150144+36900864 x+144528384 x^2-\frac{131666997 x^3}{2}+\frac{5419453809 x^4}{16}\right ) \, dx}{384384}\\ &=\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (123002880+738017280 x-\frac{2526001401 x^2}{16}+\frac{28319460477 x^3}{32}\right ) \, dx}{7687680}\\ &=\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (2214051840+\frac{127590595209 x}{16}+\frac{129641964375 x^2}{64}\right ) \, dx}{138378240}\\ &=\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (\frac{1878263191035}{64}+\frac{17498373866127 x}{128}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{2214051840}\\ &=\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{1203406941 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{41943040}\\ &=-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{9226119881 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{134217728}\\ &=-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{636602271789 \int \sqrt{3-x+2 x^2} \, dx}{4294967296}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{14641852251147 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{68719476736}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}+\frac{\left (636602271789 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{68719476736}\\ &=-\frac{636602271789 (1-4 x) \sqrt{3-x+2 x^2}}{34359738368}-\frac{9226119881 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2147483648}-\frac{401135647 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{335544320}+\frac{25250178739 \left (3-x+2 x^2\right )^{7/2}}{5725224960}+\frac{112244125 x \left (3-x+2 x^2\right )^{7/2}}{122683392}+\frac{122595067 x^2 \left (3-x+2 x^2\right )^{7/2}}{19169280}+\frac{23460839 x^3 \left (3-x+2 x^2\right )^{7/2}}{532480}+\frac{3684995 x^4 \left (3-x+2 x^2\right )^{7/2}}{39936}+\frac{1046225 x^5 \left (3-x+2 x^2\right )^{7/2}}{9984}+\frac{13875}{208} x^6 \left (3-x+2 x^2\right )^{7/2}+\frac{625}{28} x^7 \left (3-x+2 x^2\right )^{7/2}-\frac{14641852251147 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{68719476736 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.453735, size = 105, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (25125558681600000 x^{13}+37398427729920000 x^{12}+137233466130432000 x^{11}+204932411660697600 x^{10}+363646430503501824 x^9+439064558846345216 x^8+530502956133122048 x^7+485091164642279424 x^6+405468382284161024 x^5+257786732552566784 x^4+142490931553577856 x^3+50064174038215008 x^2+12071614275862524 x+10820567498568669\right )-59958384968446965 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{562812514467840} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.08, size = 204, normalized size = 0.8 \begin{align*}{\frac{1046225\,{x}^{5}}{9984} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{3684995\,{x}^{4}}{39936} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{23460839\,{x}^{3}}{532480} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{122595067\,{x}^{2}}{19169280} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{112244125\,x}{122683392} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{-636602271789+2546409087156\,x}{34359738368}\sqrt{2\,{x}^{2}-x+3}}+{\frac{14641852251147\,\sqrt{2}}{137438953472}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-401135647+1604542588\,x}{335544320} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-9226119881+36904479524\,x}{2147483648} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{25250178739}{5725224960} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{13875\,{x}^{6}}{208} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{625\,{x}^{7}}{28} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.53405, size = 317, normalized size = 1.25 \begin{align*} \frac{625}{28} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{7} + \frac{13875}{208} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{6} + \frac{1046225}{9984} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{5} + \frac{3684995}{39936} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{4} + \frac{23460839}{532480} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{122595067}{19169280} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{112244125}{122683392} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x + \frac{25250178739}{5725224960} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{401135647}{83886080} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{401135647}{335544320} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{9226119881}{536870912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{9226119881}{2147483648} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{636602271789}{8589934592} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{14641852251147}{137438953472} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{636602271789}{34359738368} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40691, size = 657, normalized size = 2.59 \begin{align*} \frac{1}{140703128616960} \,{\left (25125558681600000 \, x^{13} + 37398427729920000 \, x^{12} + 137233466130432000 \, x^{11} + 204932411660697600 \, x^{10} + 363646430503501824 \, x^{9} + 439064558846345216 \, x^{8} + 530502956133122048 \, x^{7} + 485091164642279424 \, x^{6} + 405468382284161024 \, x^{5} + 257786732552566784 \, x^{4} + 142490931553577856 \, x^{3} + 50064174038215008 \, x^{2} + 12071614275862524 \, x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{14641852251147}{274877906944} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1553, size = 153, normalized size = 0.6 \begin{align*} \frac{1}{140703128616960} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (32 \,{\left (12 \,{\left (200 \,{\left (20 \,{\left (240 \,{\left (260 \, x + 387\right )} x + 340823\right )} x + 10179103\right )} x + 3612502719\right )} x + 52340574127\right )} x + 2023708176167\right )} x + 7401903757359\right )} x + 49495652134297\right )} x + 125872428004183\right )} x + 1113210402762327\right )} x + 1564505438694219\right )} x + 3017903568965631\right )} x + 10820567498568669\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{14641852251147}{137438953472} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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