Optimal. Leaf size=181 \[ -\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac {169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {182917 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {182917 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}} \]
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Rubi [A] time = 0.10, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \begin {gather*} -\frac {1}{30} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}+\frac {169}{405} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}+\frac {(213878 x+477101) \left (3 x^2+5 x+2\right )^{7/2}}{136080}+\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {182917 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {182917 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {182917 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 832
Rubi steps
\begin {align*} \int (5-x) (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{30} \int (3+2 x)^2 \left (\frac {1029}{2}+338 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{810} \int (3+2 x) \left (\frac {42451}{2}+15277 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac {182917 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960}\\ &=\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624}\\ &=-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}+\frac {182917 \int \sqrt {2+5 x+3 x^2} \, dx}{2985984}\\ &=\frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{71663616}\\ &=\frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{35831808}\\ &=\frac {182917 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {182917 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {169}{405} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{30} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(477101+213878 x) \left (2+5 x+3 x^2\right )^{7/2}}{136080}-\frac {182917 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 97, normalized size = 0.54 \begin {gather*} \frac {-6402095 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-6 \sqrt {3 x^2+5 x+2} \left (9029615616 x^9+29262643200 x^8-147947046912 x^7-1086687912960 x^6-2893044950784 x^5-4253933381760 x^4-3762746217360 x^3-1995914277480 x^2-585749416130 x-73178684475\right )}{7524679680} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.93, size = 99, normalized size = 0.55 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-9029615616 x^9-29262643200 x^8+147947046912 x^7+1086687912960 x^6+2893044950784 x^5+4253933381760 x^4+3762746217360 x^3+1995914277480 x^2+585749416130 x+73178684475\right )}{1254113280}-\frac {182917 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{35831808 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 98, normalized size = 0.54 \begin {gather*} -\frac {1}{1254113280} \, {\left (9029615616 \, x^{9} + 29262643200 \, x^{8} - 147947046912 \, x^{7} - 1086687912960 \, x^{6} - 2893044950784 \, x^{5} - 4253933381760 \, x^{4} - 3762746217360 \, x^{3} - 1995914277480 \, x^{2} - 585749416130 \, x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{429981696} \, \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 94, normalized size = 0.52 \begin {gather*} -\frac {1}{1254113280} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, x + 175\right )} x - 42469\right )} x - 4367155\right )} x - 418553957\right )} x - 3692650505\right )} x - 26130182065\right )} x - 83163094895\right )} x - 292874708065\right )} x - 73178684475\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {182917}{214990848} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 151, normalized size = 0.83 \begin {gather*} -\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{3}}{15}+\frac {38 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{2}}{81}+\frac {46453 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{9720}-\frac {182917 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{214990848}+\frac {182917 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {182917 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {182917 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}+\frac {173137 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{27216} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 179, normalized size = 0.99 \begin {gather*} -\frac {4}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {38}{81} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {46453}{9720} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {173137}{27216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {182917}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {182917}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {182917}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {914585}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {182917}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {182917}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {914585}{35831808} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int {\left (2\,x+3\right )}^3\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- 3672 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 10359 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 15577 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 13215 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5955 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 958 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 204 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 72 x^{8} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 540 \sqrt {3 x^{2} + 5 x + 2}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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