Optimal. Leaf size=208 \[ \frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]
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Rubi [A] time = 0.309322, antiderivative size = 208, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{125}{4} \left (2 x^2-x+3\right )^{3/2} x^7+\frac{14125}{144} \left (2 x^2-x+3\right )^{3/2} x^6+\frac{233225 \left (2 x^2-x+3\right )^{3/2} x^5}{1536}+\frac{4796405 \left (2 x^2-x+3\right )^{3/2} x^4}{43008}+\frac{8325631 \left (2 x^2-x+3\right )^{3/2} x^3}{1032192}-\frac{83948353 \left (2 x^2-x+3\right )^{3/2} x^2}{2293760}+\frac{804243809 \left (2 x^2-x+3\right )^{3/2} x}{36700160}+\frac{27185733541 \left (2 x^2-x+3\right )^{3/2}}{440401920}-\frac{359471503 (1-4 x) \sqrt{2 x^2-x+3}}{67108864}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1661
Rule 640
Rule 612
Rule 619
Rule 215
Rubi steps
\begin{align*} \int \sqrt{3-x+2 x^2} \left (2+3 x+5 x^2\right )^4 \, dx &=\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{20} \int \sqrt{3-x+2 x^2} \left (320+1920 x+7520 x^2+18720 x^3+35220 x^4+46800 x^5+33875 x^6+\frac{70625 x^7}{2}\right ) \, dx\\ &=\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{1}{360} \int \sqrt{3-x+2 x^2} \left (5760+34560 x+135360 x^2+336960 x^3+633960 x^4+206775 x^5+\frac{3498375 x^6}{4}\right ) \, dx\\ &=\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (92160+552960 x+2165760 x^2+5391360 x^3-\frac{11902185 x^4}{4}+\frac{71946075 x^5}{8}\right ) \, dx}{5760}\\ &=\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (1290240+7741440 x+30320640 x^2-\frac{64880145 x^3}{2}+\frac{124884465 x^4}{16}\right ) \, dx}{80640}\\ &=\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (15482880+92897280 x+\frac{4697602695 x^2}{16}-\frac{11333027655 x^3}{32}\right ) \, dx}{967680}\\ &=-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \sqrt{3-x+2 x^2} \left (154828800+\frac{48862647765 x}{16}+\frac{108572914215 x^2}{64}\right ) \, dx}{9676800}\\ &=\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\int \left (-\frac{246446397045}{64}+\frac{3670074028035 x}{128}\right ) \sqrt{3-x+2 x^2} \, dx}{77414400}\\ &=\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{359471503 \int \sqrt{3-x+2 x^2} \, dx}{8388608}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{8267844569 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{134217728}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}+\frac{\left (359471503 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{134217728}\\ &=-\frac{359471503 (1-4 x) \sqrt{3-x+2 x^2}}{67108864}+\frac{27185733541 \left (3-x+2 x^2\right )^{3/2}}{440401920}+\frac{804243809 x \left (3-x+2 x^2\right )^{3/2}}{36700160}-\frac{83948353 x^2 \left (3-x+2 x^2\right )^{3/2}}{2293760}+\frac{8325631 x^3 \left (3-x+2 x^2\right )^{3/2}}{1032192}+\frac{4796405 x^4 \left (3-x+2 x^2\right )^{3/2}}{43008}+\frac{233225 x^5 \left (3-x+2 x^2\right )^{3/2}}{1536}+\frac{14125}{144} x^6 \left (3-x+2 x^2\right )^{3/2}+\frac{125}{4} x^7 \left (3-x+2 x^2\right )^{3/2}-\frac{8267844569 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{134217728 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.291108, size = 85, normalized size = 0.41 \[ \frac{4 \sqrt{2 x^2-x+3} \left (1321205760000 x^9+3486515200000 x^8+6327795712000 x^7+7725962035200 x^6+7612808028160 x^5+5354741991424 x^4+2211683657856 x^3-174418077792 x^2+537752185764 x+3801512106459\right )-2604371039235 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{84557168640} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 166, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{7}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{14125\,{x}^{6}}{144} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{233225\,{x}^{5}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{4796405\,{x}^{4}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{8325631\,{x}^{3}}{1032192} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{83948353\,{x}^{2}}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{804243809\,x}{36700160} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}+{\frac{-359471503+1437886012\,x}{67108864}\sqrt{2\,{x}^{2}-x+3}}+{\frac{8267844569\,\sqrt{2}}{268435456}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{27185733541}{440401920} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48403, size = 239, normalized size = 1.15 \begin{align*} \frac{125}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{7} + \frac{14125}{144} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{6} + \frac{233225}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} + \frac{4796405}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} + \frac{8325631}{1032192} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{83948353}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + \frac{804243809}{36700160} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{27185733541}{440401920} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{359471503}{16777216} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{8267844569}{268435456} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{359471503}{67108864} \, \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 = 1.40846, size = 439, normalized size = 2.11 \begin{align*} \frac{1}{21139292160} \,{\left (1321205760000 \, x^{9} + 3486515200000 \, x^{8} + 6327795712000 \, x^{7} + 7725962035200 \, x^{6} + 7612808028160 \, x^{5} + 5354741991424 \, x^{4} + 2211683657856 \, x^{3} - 174418077792 \, x^{2} + 537752185764 \, x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{8267844569}{536870912} \, \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 \sqrt{2 x^{2} - x + 3} \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 = 1.13665, size = 126, normalized size = 0.61 \begin{align*} \frac{1}{21139292160} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (40 \,{\left (140 \,{\left (160 \,{\left (36 \, x + 95\right )} x + 27587\right )} x + 4715553\right )} x + 185859571\right )} x + 2614620113\right )} x + 17278778577\right )} x - 5450564931\right )} x + 134438046441\right )} x + 3801512106459\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{8267844569}{268435456} \, \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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