Optimal. Leaf size=188 \[ -\frac{14477995 \left (2 x^2-x+3\right )^{5/2}}{23887872 (2 x+5)^2}+\frac{224815 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2304 (2 x+5)^4}+\frac{(67865260 x+762984903) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)}+\frac{(2339916063-389975609 x) \sqrt{2 x^2-x+3}}{31850496}-\frac{8969688643 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{21233664 \sqrt{2}}+\frac{432565 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]
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Rubi [A] time = 0.265479, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1650, 812, 814, 843, 619, 215, 724, 206} \[ -\frac{14477995 \left (2 x^2-x+3\right )^{5/2}}{23887872 (2 x+5)^2}+\frac{224815 \left (2 x^2-x+3\right )^{5/2}}{165888 (2 x+5)^3}-\frac{3667 \left (2 x^2-x+3\right )^{5/2}}{2304 (2 x+5)^4}+\frac{(67865260 x+762984903) \left (2 x^2-x+3\right )^{3/2}}{95551488 (2 x+5)}+\frac{(2339916063-389975609 x) \sqrt{2 x^2-x+3}}{31850496}-\frac{8969688643 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{2 x^2-x+3}}\right )}{21233664 \sqrt{2}}+\frac{432565 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1650
Rule 812
Rule 814
Rule 843
Rule 619
Rule 215
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^5} \, dx &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}-\frac{1}{288} \int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{51695}{16}-\frac{24835 x}{4}+1944 x^2-720 x^3\right )}{(5+2 x)^4} \, dx\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}+\frac{\int \frac{\left (3-x+2 x^2\right )^{3/2} \left (\frac{5995005}{16}-\frac{1483149 x}{2}+77760 x^2\right )}{(5+2 x)^3} \, dx}{62208}\\ &=-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac{\int \frac{\left (\frac{252996909}{16}-\frac{152696835 x}{4}\right ) \left (3-x+2 x^2\right )^{3/2}}{(5+2 x)^2} \, dx}{8957952}\\ &=\frac{(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}+\frac{\int \frac{\left (\frac{10531588167}{8}-3509780481 x\right ) \sqrt{3-x+2 x^2}}{5+2 x} \, dx}{71663616}\\ &=\frac{(2339916063-389975609 x) \sqrt{3-x+2 x^2}}{31850496}+\frac{(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac{\int \frac{-968737607064+1937448253440 x}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{2293235712}\\ &=\frac{(2339916063-389975609 x) \sqrt{3-x+2 x^2}}{31850496}+\frac{(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac{432565 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{1024}+\frac{8969688643 \int \frac{1}{(5+2 x) \sqrt{3-x+2 x^2}} \, dx}{3538944}\\ &=\frac{(2339916063-389975609 x) \sqrt{3-x+2 x^2}}{31850496}+\frac{(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}-\frac{8969688643 \operatorname{Subst}\left (\int \frac{1}{288-x^2} \, dx,x,\frac{17-22 x}{\sqrt{3-x+2 x^2}}\right )}{1769472}-\frac{432565 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{1024 \sqrt{46}}\\ &=\frac{(2339916063-389975609 x) \sqrt{3-x+2 x^2}}{31850496}+\frac{(762984903+67865260 x) \left (3-x+2 x^2\right )^{3/2}}{95551488 (5+2 x)}-\frac{3667 \left (3-x+2 x^2\right )^{5/2}}{2304 (5+2 x)^4}+\frac{224815 \left (3-x+2 x^2\right )^{5/2}}{165888 (5+2 x)^3}-\frac{14477995 \left (3-x+2 x^2\right )^{5/2}}{23887872 (5+2 x)^2}+\frac{432565 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{1024 \sqrt{2}}-\frac{8969688643 \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{2} \sqrt{3-x+2 x^2}}\right )}{21233664 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.239099, size = 108, normalized size = 0.57 \[ \frac{\frac{24 \sqrt{2 x^2-x+3} \left (2949120 x^6-29270016 x^5+468043776 x^4+11761910072 x^3+60528581892 x^2+121473790266 x+86386856771\right )}{(2 x+5)^4}-8969688643 \sqrt{2} \tanh ^{-1}\left (\frac{17-22 x}{12 \sqrt{4 x^2-2 x+6}}\right )+8969667840 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{42467328} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 204, normalized size = 1.1 \begin{align*} -{\frac{-389975609+1559902436\,x}{127401984}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}-{\frac{14477995}{95551488} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-2}}-{\frac{8969688643\,\sqrt{2}}{42467328}{\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 ) }-{\frac{432565\,\sqrt{2}}{2048}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{8969688643}{6879707136} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{8969688643}{127401984}\sqrt{2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}}}}+{\frac{224815}{1327104} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-3}}-{\frac{-593321753+2373287012\,x}{6879707136} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{3}{2}}}}+{\frac{593321753}{3439853568} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-1}}-{\frac{3667}{36864} \left ( 2\, \left ( x+5/2 \right ) ^{2}-11\,x-{\frac{19}{2}} \right ) ^{{\frac{5}{2}}} \left ( x+{\frac{5}{2}} \right ) ^{-4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.56479, size = 284, normalized size = 1.51 \begin{align*} \frac{16966315}{47775744} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{3667 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{2304 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac{224815 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{165888 \,{\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac{14477995 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{23887872 \,{\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac{389975609}{31850496} \, \sqrt{2 \, x^{2} - x + 3} x - \frac{432565}{2048} \, \sqrt{2} \operatorname{arsinh}\left (\frac{4}{23} \, \sqrt{23} x - \frac{1}{23} \, \sqrt{23}\right ) + \frac{8969688643}{42467328} \, \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{779972021}{10616832} \, \sqrt{2 \, x^{2} - x + 3} + \frac{593321753 \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{95551488 \,{\left (2 \, x + 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49125, size = 657, normalized size = 3.49 \begin{align*} \frac{8969667840 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )} \log \left (4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 8969688643 \, \sqrt{2}{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 (2949120 \, x^{6} - 29270016 \, x^{5} + 468043776 \, x^{4} + 11761910072 \, x^{3} + 60528581892 \, x^{2} + 121473790266 \, x + 86386856771\right )} \sqrt{2 \, x^{2} - x + 3}}{84934656 \,{\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\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 \frac{\left (2 x^{2} - x + 3\right )^{\frac{3}{2}} \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{5}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.33648, size = 679, normalized size = 3.61 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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