Optimal. Leaf size=154 \[ \frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac {1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac {9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac {9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac {9793 (6 x+5) \sqrt {3 x^2+5 x+2}}{47775744}+\frac {9793 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{95551488 \sqrt {3}} \]
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Rubi [A] time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {779, 612, 621, 206} \begin {gather*} \frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}+\frac {1399 (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}}{8640}-\frac {9793 (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}}{622080}+\frac {9793 (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}}{5971968}-\frac {9793 (6 x+5) \sqrt {3 x^2+5 x+2}}{47775744}+\frac {9793 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{95551488 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rubi steps
\begin {align*} \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx &=\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {1399}{180} \int \left (2+5 x+3 x^2\right )^{7/2} \, dx\\ &=\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}-\frac {9793 \int \left (2+5 x+3 x^2\right )^{5/2} \, dx}{17280}\\ &=-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {9793 \int \left (2+5 x+3 x^2\right )^{3/2} \, dx}{248832}\\ &=\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}-\frac {9793 \int \sqrt {2+5 x+3 x^2} \, dx}{3981312}\\ &=-\frac {9793 (5+6 x) \sqrt {2+5 x+3 x^2}}{47775744}+\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {9793 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{95551488}\\ &=-\frac {9793 (5+6 x) \sqrt {2+5 x+3 x^2}}{47775744}+\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {9793 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{47775744}\\ &=-\frac {9793 (5+6 x) \sqrt {2+5 x+3 x^2}}{47775744}+\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {9793 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{95551488 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 111, normalized size = 0.72 \begin {gather*} \frac {1399 \left (35 \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 (4478976 x^7+26127360 x^6+64800000 x^5+88560000 x^4+72023472 x^3+34858680 x^2+9298342 x+1054785\right )\right )}{1433272320}-\frac {1}{810} (54 x-265) \left (3 x^2+5 x+2\right )^{9/2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.92, size = 99, normalized size = 0.64 \begin {gather*} \frac {9793 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{47775744 \sqrt {3}}+\frac {\sqrt {3 x^2+5 x+2} \left (-1289945088 x^9-2269347840 x^8+23529056256 x^7+117850567680 x^6+250227954432 x^5+302902600320 x^4+224097754320 x^3+100612822920 x^2+25257845290 x+2726071095\right )}{238878720} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 98, normalized size = 0.64 \begin {gather*} -\frac {1}{238878720} \, {\left (1289945088 \, x^{9} + 2269347840 \, x^{8} - 23529056256 \, x^{7} - 117850567680 \, x^{6} - 250227954432 \, x^{5} - 302902600320 \, x^{4} - 224097754320 \, x^{3} - 100612822920 \, x^{2} - 25257845290 \, x - 2726071095\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {9793}{573308928} \, \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 = 0.23, size = 94, normalized size = 0.61 \begin {gather*} -\frac {1}{238878720} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, x + 95\right )} x - 47279\right )} x - 473615\right )} x - 36201961\right )} x - 262936285\right )} x - 1556234405\right )} x - 4192200955\right )} x - 12628922645\right )} x - 2726071095\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {9793}{286654464} \, \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 = 136, normalized size = 0.88 \begin {gather*} -\frac {\left (3 x^{2}+5 x +2\right )^{\frac {9}{2}} x}{15}+\frac {9793 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right )}{286654464}+\frac {53 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{162}+\frac {1399 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{8640}-\frac {9793 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{622080}+\frac {9793 \left (6 x +5\right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{5971968}-\frac {9793 \left (6 x +5\right ) \sqrt {3 x^{2}+5 x +2}}{47775744} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 174, normalized size = 1.13 \begin {gather*} -\frac {1}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {53}{162} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {1399}{1440} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {1399}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {9793}{103680} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {9793}{124416} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {9793}{995328} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {48965}{5971968} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {9793}{7962624} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {9793}{286654464} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {48965}{47775744} \, \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 )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,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 (- 956 x \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3194 x^{2} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5757 x^{3} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 5948 x^{4} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 3368 x^{5} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int \left (- 792 x^{6} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 81 x^{7} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int 54 x^{8} \sqrt {3 x^{2} + 5 x + 2}\, dx - \int \left (- 120 \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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