Optimal. Leaf size=229 \[ -\frac{1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac{439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac{205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac{74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac{519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac{2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac{2595845 (6 x+5) \sqrt{3 x^2+5 x+2}}{5159780352}+\frac{2595845 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10319560704 \sqrt{3}} \]
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Rubi [A] time = 0.140771, antiderivative size = 229, 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 = {832, 779, 612, 621, 206} \[ -\frac{1}{39} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+\frac{439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}}{1404}+\frac{205}{351} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{(389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}}{227448}+\frac{74167 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{186624}-\frac{519169 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{13436928}+\frac{2595845 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{644972544}-\frac{2595845 (6 x+5) \sqrt{3 x^2+5 x+2}}{5159780352}+\frac{2595845 \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{3} \sqrt{3 x^2+5 x+2}}\right )}{10319560704 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 832
Rule 779
Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2} \, dx &=-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{1}{39} \int (3+2 x)^3 \left (\frac{1337}{2}+439 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{\int (3+2 x)^2 \left (\frac{74595}{2}+27060 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{1404}\\ &=\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{\int (3+2 x) \left (\frac{3298845}{2}+1189815 x\right ) \left (2+5 x+3 x^2\right )^{7/2} \, dx}{46332}\\ &=\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac{74167 \int \left (2+5 x+3 x^2\right )^{7/2} \, dx}{3888}\\ &=\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}-\frac{519169 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{373248}\\ &=-\frac{519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac{2595845 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{26873856}\\ &=\frac{2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac{519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}-\frac{2595845 \int \sqrt{2+5 x+3 x^2} \, dx}{429981696}\\ &=-\frac{2595845 (5+6 x) \sqrt{2+5 x+3 x^2}}{5159780352}+\frac{2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac{519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac{2595845 \int \frac{1}{\sqrt{2+5 x+3 x^2}} \, dx}{10319560704}\\ &=-\frac{2595845 (5+6 x) \sqrt{2+5 x+3 x^2}}{5159780352}+\frac{2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac{519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac{2595845 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{5+6 x}{\sqrt{2+5 x+3 x^2}}\right )}{5159780352}\\ &=-\frac{2595845 (5+6 x) \sqrt{2+5 x+3 x^2}}{5159780352}+\frac{2595845 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{644972544}-\frac{519169 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{13436928}+\frac{74167 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{186624}+\frac{205}{351} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{9/2}+\frac{439 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{9/2}}{1404}-\frac{1}{39} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{9/2}+\frac{(852175+389394 x) \left (2+5 x+3 x^2\right )^{9/2}}{227448}+\frac{2595845 \tanh ^{-1}\left (\frac{5+6 x}{2 \sqrt{3} \sqrt{2+5 x+3 x^2}}\right )}{10319560704 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.235414, size = 184, normalized size = 0.8 \[ \frac{-36 (2 x+3)^4 \left (3 x^2+5 x+2\right )^{9/2}+439 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{9/2}+820 (2 x+3)^2 \left (3 x^2+5 x+2\right )^{9/2}+\frac{1}{162} (389394 x+852175) \left (3 x^2+5 x+2\right )^{9/2}+\frac{964171 \left (6 \sqrt{3 x^2+5 x+2} \left (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )+35 \sqrt{3} \tanh ^{-1}\left (\frac{6 x+5}{2 \sqrt{9 x^2+15 x+6}}\right )\right )}{286654464}}{1404} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 187, normalized size = 0.8 \begin{align*} -{\frac{16\,{x}^{4}}{39} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{14\,{x}^{3}}{351} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{2827\,{x}^{2}}{351} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{84521\,x}{4212} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}}+{\frac{370835+445002\,x}{186624} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{7}{2}}}}-{\frac{2595845+3115014\,x}{13436928} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{5}{2}}}}+{\frac{12979225+15575070\,x}{644972544} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{3}{2}}}}+{\frac{2595845\,\sqrt{3}}{30958682112}\ln \left ({\frac{\sqrt{3}}{3} \left ({\frac{5}{2}}+3\,x \right ) }+\sqrt{3\,{x}^{2}+5\,x+2} \right ) }-{\frac{12979225+15575070\,x}{5159780352}\sqrt{3\,{x}^{2}+5\,x+2}}+{\frac{3495529}{227448} \left ( 3\,{x}^{2}+5\,x+2 \right ) ^{{\frac{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.64023, size = 304, normalized size = 1.33 \begin{align*} -\frac{16}{39} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{4} + \frac{14}{351} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{3} + \frac{2827}{351} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x^{2} + \frac{84521}{4212} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} x + \frac{3495529}{227448} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{9}{2}} + \frac{74167}{31104} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} x + \frac{370835}{186624} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{7}{2}} - \frac{519169}{2239488} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} x - \frac{2595845}{13436928} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{5}{2}} + \frac{2595845}{107495424} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} x + \frac{12979225}{644972544} \,{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac{3}{2}} - \frac{2595845}{859963392} \, \sqrt{3 \, x^{2} + 5 \, x + 2} x + \frac{2595845}{30958682112} \, \sqrt{3} \log \left (2 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac{12979225}{5159780352} \, \sqrt{3 \, x^{2} + 5 \, x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4304, size = 567, normalized size = 2.48 \begin{align*} -\frac{1}{67077144576} \,{\left (2229025112064 \, x^{12} + 14643456638976 \, x^{11} - 2110350163968 \, x^{10} - 333952593887232 \, x^{9} - 1590604366381056 \, x^{8} - 4022427759003648 \, x^{7} - 6524509131334656 \, x^{6} - 7203650864723712 \, x^{5} - 5499074981552256 \, x^{4} - 2865856228323984 \, x^{3} - 975104480077800 \, x^{2} - 195441229635490 \, x - 17510968283403\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} + \frac{2595845}{61917364224} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{3 \, x^{2} + 5 \, x + 2}{\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\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 - 32292 x \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 142182 x^{2} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 363291 x^{3} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 594106 x^{4} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 644932 x^{5} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 463440 x^{6} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 209413 x^{7} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 49624 x^{8} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 504 x^{9} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 2592 x^{10} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int 432 x^{11} \sqrt{3 x^{2} + 5 x + 2}\, dx - \int - 3240 \sqrt{3 x^{2} + 5 x + 2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18854, size = 147, normalized size = 0.64 \begin{align*} -\frac{1}{67077144576} \,{\left (2 \,{\left (12 \,{\left (6 \,{\left (8 \,{\left (6 \,{\left (36 \,{\left (2 \,{\left (48 \,{\left (54 \,{\left (4 \,{\left (6 \,{\left (72 \, x + 473\right )} x - 409\right )} x - 258889\right )} x - 66586273\right )} x - 8082617507\right )} x - 26220538883\right )} x - 1042194858901\right )} x - 4773502588153\right )} x - 19901779363361\right )} x - 40629353336575\right )} x - 97720614817745\right )} x - 17510968283403\right )} \sqrt{3 \, x^{2} + 5 \, x + 2} - \frac{2595845}{30958682112} \, \sqrt{3} \log \left ({\left | -2 \, \sqrt{3}{\left (\sqrt{3} x - \sqrt{3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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