Optimal. Leaf size=206 \[ -\frac {1}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac {41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac {3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {249299 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {249299 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{71663616 \sqrt {3}} \]
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Rubi [A] time = 0.12, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 9, 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}{33} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^4+\frac {41}{110} \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^3+\frac {3298 \left (3 x^2+5 x+2\right )^{7/2} (2 x+3)^2}{4455}+\frac {(3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}}{1496880}+\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{466560}-\frac {249299 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{4478976}+\frac {249299 (6 x+5) \sqrt {3 x^2+5 x+2}}{35831808}-\frac {249299 \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)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx &=-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{33} \int (3+2 x)^3 \left (\frac {1127}{2}+369 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {1}{990} \int (3+2 x)^2 \left (\frac {53829}{2}+19788 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx\\ &=\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {\int (3+2 x) \left (\frac {1965801}{2}+721227 x\right ) \left (2+5 x+3 x^2\right )^{5/2} \, dx}{26730}\\ &=\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac {249299 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{12960}\\ &=\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{186624}\\ &=-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}+\frac {249299 \int \sqrt {2+5 x+3 x^2} \, dx}{2985984}\\ &=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{71663616}\\ &=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \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 {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \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 = 0.07, size = 102, normalized size = 0.50 \begin {gather*} \frac {-95980115 \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 (180592312320 x^{10}+875872714752 x^9-1932170526720 x^8-25759323039744 x^7-90095929758720 x^6-172473366866688 x^5-204855126595200 x^4-155155370878800 x^3-73069860056520 x^2-19521700361210 x-2261297826735\right )}{82771476480} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.31, size = 104, normalized size = 0.50 \begin {gather*} \frac {\sqrt {3 x^2+5 x+2} \left (-180592312320 x^{10}-875872714752 x^9+1932170526720 x^8+25759323039744 x^7+90095929758720 x^6+172473366866688 x^5+204855126595200 x^4+155155370878800 x^3+73069860056520 x^2+19521700361210 x+2261297826735\right )}{13795246080}-\frac {249299 \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 = 103, normalized size = 0.50 \begin {gather*} -\frac {1}{13795246080} \, {\left (180592312320 \, x^{10} + 875872714752 \, x^{9} - 1932170526720 \, x^{8} - 25759323039744 \, x^{7} - 90095929758720 \, x^{6} - 172473366866688 \, x^{5} - 204855126595200 \, x^{4} - 155155370878800 \, x^{3} - 73069860056520 \, x^{2} - 19521700361210 \, x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{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 = 0.22, size = 99, normalized size = 0.48 \begin {gather*} -\frac {1}{13795246080} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, {\left (20 \, x + 97\right )} x - 11555\right )} x - 7394353\right )} x - 362075335\right )} x - 24952744049\right )} x - 177825630725\right )} x - 1077467853325\right )} x - 3044577502355\right )} x - 9760850180605\right )} x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{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 = 0.06, size = 168, normalized size = 0.82 \begin {gather*} -\frac {16 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{4}}{33}+\frac {4 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{3}}{55}+\frac {8762 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x^{2}}{891}+\frac {2642401 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}} x}{106920}-\frac {249299 \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 {249299 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {249299 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {249299 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}+\frac {5753773 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{299376} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 196, normalized size = 0.95 \begin {gather*} -\frac {16}{33} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{4} + \frac {4}{55} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {8762}{891} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {2642401}{106920} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {5753773}{299376} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {249299}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {249299}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {249299}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {1246495}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {249299}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {249299}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {1246495}{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 = -0.00 \begin {gather*} -\int {\left (2\,x+3\right )}^4\,\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 (- 12096 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 38421 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 67449 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 70799 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 44295 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 14784 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 1304 x^{7} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 624 x^{8} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 144 x^{9} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 1620 \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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