Optimal. Leaf size=183 \[ -\frac {1}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac {299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac {487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac {(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac {454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac {454969 (6 x+5) \sqrt {3 x^2+5 x+2}}{4478976}+\frac {454969 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{8957952 \sqrt {3}} \]
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Rubi [A] time = 0.11, antiderivative size = 183, 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}{27} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^4+\frac {299}{648} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^3+\frac {487}{486} \left (3 x^2+5 x+2\right )^{5/2} (2 x+3)^2+\frac {(188910 x+420721) \left (3 x^2+5 x+2\right )^{5/2}}{58320}+\frac {454969 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{559872}-\frac {454969 (6 x+5) \sqrt {3 x^2+5 x+2}}{4478976}+\frac {454969 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{8957952 \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)^4 \left (2+5 x+3 x^2\right )^{3/2} \, dx &=-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{27} \int (3+2 x)^3 \left (\frac {917}{2}+299 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {1}{648} \int (3+2 x)^2 \left (\frac {36423}{2}+13636 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx\\ &=\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {\int (3+2 x) \left (\frac {1053773}{2}+396711 x\right ) \left (2+5 x+3 x^2\right )^{3/2} \, dx}{13608}\\ &=\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac {454969 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{23328}\\ &=\frac {454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}-\frac {454969 \int \sqrt {2+5 x+3 x^2} \, dx}{373248}\\ &=-\frac {454969 (5+6 x) \sqrt {2+5 x+3 x^2}}{4478976}+\frac {454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac {454969 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{8957952}\\ &=-\frac {454969 (5+6 x) \sqrt {2+5 x+3 x^2}}{4478976}+\frac {454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac {454969 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{4478976}\\ &=-\frac {454969 (5+6 x) \sqrt {2+5 x+3 x^2}}{4478976}+\frac {454969 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{559872}+\frac {487}{486} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {299}{648} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{27} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(420721+188910 x) \left (2+5 x+3 x^2\right )^{5/2}}{58320}+\frac {454969 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{8957952 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 92, normalized size = 0.50 \begin {gather*} \frac {2274845 \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 (119439360 x^8+370759680 x^7-2143687680 x^6-14811482880 x^5-37262745216 x^4-49917376080 x^3-37650690888 x^2-15049298650 x-2471988351\right )}{134369280} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.87, size = 94, normalized size = 0.51 \begin {gather*} \frac {454969 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{4478976 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-119439360 x^8-370759680 x^7+2143687680 x^6+14811482880 x^5+37262745216 x^4+49917376080 x^3+37650690888 x^2+15049298650 x+2471988351\right )}{22394880} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 93, normalized size = 0.51 \begin {gather*} -\frac {1}{22394880} \, {\left (119439360 \, x^{8} + 370759680 \, x^{7} - 2143687680 \, x^{6} - 14811482880 \, x^{5} - 37262745216 \, x^{4} - 49917376080 \, x^{3} - 37650690888 \, x^{2} - 15049298650 \, x - 2471988351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {454969}{53747712} \, \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 = 0.21, size = 89, normalized size = 0.49 \begin {gather*} -\frac {1}{22394880} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (30 \, {\left (36 \, {\left (2 \, {\left (48 \, x + 149\right )} x - 1723\right )} x - 428573\right )} x - 32346133\right )} x - 346648445\right )} x - 1568778787\right )} x - 7524649325\right )} x - 2471988351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {454969}{26873856} \, \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 = 0.06, size = 149, normalized size = 0.81 \begin {gather*} -\frac {16 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{4}}{27}+\frac {11 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{3}}{81}+\frac {6133 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x^{2}}{486}+\frac {2317 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}} x}{72}+\frac {454969 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{26873856}-\frac {454969 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{4478976}+\frac {454969 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{559872}+\frac {1498291 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{58320} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 167, normalized size = 0.91 \begin {gather*} -\frac {16}{27} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{4} + \frac {11}{81} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{3} + \frac {6133}{486} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {2317}{72} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {1498291}{58320} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {454969}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {2274845}{559872} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {454969}{746496} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {454969}{26873856} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {2274845}{4478976} \, \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 )}^4\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,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 (- 4023 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7938 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 7845 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3880 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 680 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 128 x^{6} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 48 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 810 \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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