Optimal. Leaf size=189 \[ \frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}+\frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]
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Rubi [A] time = 0.18971, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{25}{4} \left (2 x^2-x+3\right )^{5/2} x^5+\frac{725}{48} \left (2 x^2-x+3\right )^{5/2} x^4+\frac{27785 \left (2 x^2-x+3\right )^{5/2} x^3}{1536}+\frac{384739 \left (2 x^2-x+3\right )^{5/2} x^2}{43008}-\frac{81685 \left (2 x^2-x+3\right )^{5/2} x}{114688}-\frac{4625907 \left (2 x^2-x+3\right )^{5/2}}{2293760}-\frac{667795 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{2097152}-\frac{46077855 (1-4 x) \sqrt{2 x^2-x+3}}{33554432}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}} \]
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 )^{3/2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{20} \int \left (3-x+2 x^2\right )^{3/2} \left (160+720 x+2280 x^2+4140 x^3+3825 x^4+\frac{10875 x^5}{2}\right ) \, dx\\ &=\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1}{360} \int \left (3-x+2 x^2\right )^{3/2} \left (2880+12960 x+41040 x^2+9270 x^3+\frac{416775 x^4}{4}\right ) \, dx\\ &=\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (3-x+2 x^2\right )^{3/2} \left (46080+207360 x-\frac{1124415 x^2}{4}+\frac{5771085 x^3}{8}\right ) \, dx}{5760}\\ &=\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (645120-\frac{5701095 x}{4}-\frac{11027475 x^2}{16}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{80640}\\ &=-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\int \left (\frac{156945465}{16}-\frac{624497445 x}{32}\right ) \left (3-x+2 x^2\right )^{3/2} \, dx}{967680}\\ &=-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{667795 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{131072}\\ &=-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{46077855 \int \sqrt{3-x+2 x^2} \, dx}{4194304}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{1059790665 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{67108864}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}+\frac{\left (46077855 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{67108864}\\ &=-\frac{46077855 (1-4 x) \sqrt{3-x+2 x^2}}{33554432}-\frac{667795 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{2097152}-\frac{4625907 \left (3-x+2 x^2\right )^{5/2}}{2293760}-\frac{81685 x \left (3-x+2 x^2\right )^{5/2}}{114688}+\frac{384739 x^2 \left (3-x+2 x^2\right )^{5/2}}{43008}+\frac{27785 x^3 \left (3-x+2 x^2\right )^{5/2}}{1536}+\frac{725}{48} x^4 \left (3-x+2 x^2\right )^{5/2}+\frac{25}{4} x^5 \left (3-x+2 x^2\right )^{5/2}-\frac{1059790665 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{67108864 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.2339, size = 85, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (88080384000 x^9+124780544000 x^8+328328806400 x^7+430820229120 x^6+571298324480 x^5+487891884032 x^4+389257196928 x^3+199615064544 x^2+53985432012 x-72152399943\right )-111278019825 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{14092861440} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 151, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{5}}{4} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{725\,{x}^{4}}{48} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{27785\,{x}^{3}}{1536} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{384739\,{x}^{2}}{43008} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}-{\frac{81685\,x}{114688} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-46077855+184311420\,x}{33554432}\sqrt{2\,{x}^{2}-x+3}}+{\frac{1059790665\,\sqrt{2}}{134217728}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-667795+2671180\,x}{2097152} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{4625907}{2293760} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50344, size = 232, normalized size = 1.23 \begin{align*} \frac{25}{4} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{5} + \frac{725}{48} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{4} + \frac{27785}{1536} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{3} + \frac{384739}{43008} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x^{2} - \frac{81685}{114688} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{4625907}{2293760} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{667795}{524288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{667795}{2097152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{46077855}{8388608} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{1059790665}{134217728} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{46077855}{33554432} \, \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 = 1.59326, size = 423, normalized size = 2.24 \begin{align*} \frac{1}{3523215360} \,{\left (88080384000 \, x^{9} + 124780544000 \, x^{8} + 328328806400 \, x^{7} + 430820229120 \, x^{6} + 571298324480 \, x^{5} + 487891884032 \, x^{4} + 389257196928 \, x^{3} + 199615064544 \, x^{2} + 53985432012 \, x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{1059790665}{268435456} \, \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{3}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16042, size = 126, normalized size = 0.67 \begin{align*} \frac{1}{3523215360} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (20 \,{\left (8 \,{\left (140 \,{\left (160 \,{\left (12 \, x + 17\right )} x + 7157\right )} x + 1314759\right )} x + 13947713\right )} x + 238228459\right )} x + 3041071851\right )} x + 6237970767\right )} x + 13496358003\right )} x - 72152399943\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{1059790665}{134217728} \, \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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