Optimal. Leaf size=261 \[ \frac{62005241 \sqrt{-3 x^2-5 x-2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right ),-\frac{2}{3}\right )}{297729432 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{1166 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2295}+\frac{\sqrt{2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}}{984555}-\frac{\sqrt{2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}}{8270262}+\frac{(12174838-22593339 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{744323580}-\frac{34355693 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{212663880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
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Rubi [A] time = 0.202637, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {832, 814, 843, 718, 424, 419} \[ -\frac{2}{51} (2 x+3)^{3/2} \left (3 x^2+5 x+2\right )^{7/2}+\frac{1166 \sqrt{2 x+3} \left (3 x^2+5 x+2\right )^{7/2}}{2295}+\frac{\sqrt{2 x+3} (1253571 x+1063774) \left (3 x^2+5 x+2\right )^{5/2}}{984555}-\frac{\sqrt{2 x+3} (1332121 x+949997) \left (3 x^2+5 x+2\right )^{3/2}}{8270262}+\frac{(12174838-22593339 x) \sqrt{2 x+3} \sqrt{3 x^2+5 x+2}}{744323580}+\frac{62005241 \sqrt{-3 x^2-5 x-2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{297729432 \sqrt{3} \sqrt{3 x^2+5 x+2}}-\frac{34355693 \sqrt{-3 x^2-5 x-2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{x+1}\right )|-\frac{2}{3}\right )}{212663880 \sqrt{3} \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 814
Rule 843
Rule 718
Rule 424
Rule 419
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac{2}{51} \int \sqrt{3+2 x} \left (441+\frac{583 x}{2}\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac{4 \int \frac{\left (\frac{55523}{4}+\frac{37987 x}{4}\right ) \left (2+5 x+3 x^2\right )^{5/2}}{\sqrt{3+2 x}} \, dx}{2295}\\ &=\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac{2 \int \frac{\left (\frac{380353}{2}+\frac{570909 x}{4}\right ) \left (2+5 x+3 x^2\right )^{3/2}}{\sqrt{3+2 x}} \, dx}{196911}\\ &=-\frac{\sqrt{3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}+\frac{\int \frac{\left (-\frac{4091997}{4}-\frac{7531113 x}{4}\right ) \sqrt{2+5 x+3 x^2}}{\sqrt{3+2 x}} \, dx}{12405393}\\ &=\frac{(12174838-22593339 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{744323580}-\frac{\sqrt{3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac{\int \frac{\frac{308582511}{2}+\frac{721469553 x}{4}}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{1116485370}\\ &=\frac{(12174838-22593339 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{744323580}-\frac{\sqrt{3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac{34355693 \int \frac{\sqrt{3+2 x}}{\sqrt{2+5 x+3 x^2}} \, dx}{425327760}+\frac{62005241 \int \frac{1}{\sqrt{3+2 x} \sqrt{2+5 x+3 x^2}} \, dx}{595458864}\\ &=\frac{(12174838-22593339 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{744323580}-\frac{\sqrt{3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac{\left (34355693 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 x^2}{3}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{212663880 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{\left (62005241 \sqrt{-2-5 x-3 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 x^2}{3}}} \, dx,x,\frac{\sqrt{6+6 x}}{\sqrt{2}}\right )}{297729432 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ &=\frac{(12174838-22593339 x) \sqrt{3+2 x} \sqrt{2+5 x+3 x^2}}{744323580}-\frac{\sqrt{3+2 x} (949997+1332121 x) \left (2+5 x+3 x^2\right )^{3/2}}{8270262}+\frac{\sqrt{3+2 x} (1063774+1253571 x) \left (2+5 x+3 x^2\right )^{5/2}}{984555}+\frac{1166 \sqrt{3+2 x} \left (2+5 x+3 x^2\right )^{7/2}}{2295}-\frac{2}{51} (3+2 x)^{3/2} \left (2+5 x+3 x^2\right )^{7/2}-\frac{34355693 \sqrt{-2-5 x-3 x^2} E\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{212663880 \sqrt{3} \sqrt{2+5 x+3 x^2}}+\frac{62005241 \sqrt{-2-5 x-3 x^2} F\left (\sin ^{-1}\left (\sqrt{3} \sqrt{1+x}\right )|-\frac{2}{3}\right )}{297729432 \sqrt{3} \sqrt{2+5 x+3 x^2}}\\ \end{align*}
Mathematica [A] time = 0.388094, size = 223, normalized size = 0.85 \[ -\frac{-54474128 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right ),\frac{3}{5}\right )+2 \left (28371863520 x^{10}+90474720336 x^9-474654810168 x^8-3544442250300 x^7-9891695193912 x^6-15763726088406 x^5-15827034250764 x^4-10225439632143 x^3-4138096653600 x^2-956384897657 x-96409103246\right ) \sqrt{2 x+3}+240489851 \sqrt{5} \sqrt{\frac{x+1}{2 x+3}} \sqrt{\frac{3 x+2}{2 x+3}} (2 x+3)^2 E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{5}{3}}}{\sqrt{2 x+3}}\right )|\frac{3}{5}\right )}{4465941480 (2 x+3) \sqrt{3 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 171, normalized size = 0.7 \begin{align*}{\frac{1}{267956488800\,{x}^{3}+848528881200\,{x}^{2}+848528881200\,x+267956488800}\sqrt{3+2\,x}\sqrt{3\,{x}^{2}+5\,x+2} \left ( -567437270400\,{x}^{10}-1809494406720\,{x}^{9}+9493096203360\,{x}^{8}+70888845006000\,{x}^{7}+197833903878240\,{x}^{6}+315274521768120\,{x}^{5}+69536354\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticF} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +240489851\,\sqrt{3+2\,x}\sqrt{15}\sqrt{-2-2\,x}\sqrt{-20-30\,x}{\it EllipticE} \left ( 1/5\,\sqrt{30\,x+45},1/3\,\sqrt{15} \right ) +316540685015280\,{x}^{4}+204508792642860\,{x}^{3}+82776362463060\,{x}^{2}+19151746938240\,x+1937801658960 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (18 \, x^{6} - 3 \, x^{5} - 271 \, x^{4} - 669 \, x^{3} - 687 \, x^{2} - 328 \, x - 60\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} \sqrt{2 \, x + 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}}{\left (2 \, x + 3\right )}^{\frac{3}{2}}{\left (x - 5\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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