Optimal. Leaf size=149 \[ -\frac {167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac {1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac {1141 (8 x+7) \sqrt {3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac {1141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320000 \sqrt {5}} \]
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Rubi [A] time = 0.08, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {834, 806, 720, 724, 206} \begin {gather*} -\frac {167 \left (3 x^2+5 x+2\right )^{5/2}}{375 (2 x+3)^5}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{30 (2 x+3)^6}+\frac {1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{12000 (2 x+3)^4}-\frac {1141 (8 x+7) \sqrt {3 x^2+5 x+2}}{160000 (2 x+3)^2}+\frac {1141 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{320000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 720
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^7} \, dx &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {1}{30} \int \frac {\left (-\frac {217}{2}+39 x\right ) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^6} \, dx\\ &=-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac {1141}{300} \int \frac {\left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx\\ &=\frac {1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}-\frac {1141 \int \frac {\sqrt {2+5 x+3 x^2}}{(3+2 x)^3} \, dx}{8000}\\ &=-\frac {1141 (7+8 x) \sqrt {2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac {1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac {1141 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{320000}\\ &=-\frac {1141 (7+8 x) \sqrt {2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac {1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}-\frac {1141 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{160000}\\ &=-\frac {1141 (7+8 x) \sqrt {2+5 x+3 x^2}}{160000 (3+2 x)^2}+\frac {1141 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{12000 (3+2 x)^4}-\frac {13 \left (2+5 x+3 x^2\right )^{5/2}}{30 (3+2 x)^6}-\frac {167 \left (2+5 x+3 x^2\right )^{5/2}}{375 (3+2 x)^5}+\frac {1141 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{320000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 151, normalized size = 1.01 \begin {gather*} \frac {1}{30} \left (-\frac {334 \left (3 x^2+5 x+2\right )^{5/2}}{25 (2 x+3)^5}-\frac {13 \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^6}+\frac {1141 (8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{400 (2 x+3)^4}-\frac {3423 \left (\frac {10 \sqrt {3 x^2+5 x+2} (8 x+7)}{(2 x+3)^2}+\sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )}{160000}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.62, size = 86, normalized size = 0.58 \begin {gather*} \frac {1141 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{160000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (95616 x^5+799120 x^4+3065440 x^3+4479600 x^2+2526920 x+412679\right )}{480000 (2 x+3)^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 155, normalized size = 1.04 \begin {gather*} \frac {3423 \, \sqrt {5} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (95616 \, x^{5} + 799120 \, x^{4} + 3065440 \, x^{3} + 4479600 \, x^{2} + 2526920 \, x + 412679\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{9600000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 410, normalized size = 2.75 \begin {gather*} \frac {1141}{1600000} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {109536 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6127344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 70129360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 83080800 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 3334681440 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 9802137888 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 47432214576 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 48106882440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 94851959950 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 39436262415 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 28403540997 \, \sqrt {3} x - 3009604608 \, \sqrt {3} + 28403540997 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{480000 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 232, normalized size = 1.56 \begin {gather*} -\frac {1141 \sqrt {5}\, \arctanh \left (\frac {2 \left (-4 x -\frac {7}{2}\right ) \sqrt {5}}{5 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{1600000}-\frac {167 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{12000 \left (x +\frac {3}{2}\right )^{5}}-\frac {1141 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{48000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1141 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{30000 \left (x +\frac {3}{2}\right )^{3}}-\frac {35371 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{600000 \left (x +\frac {3}{2}\right )^{2}}-\frac {33089 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{375000 \left (x +\frac {3}{2}\right )}+\frac {1141 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{3000000}-\frac {1141 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{200000}+\frac {1141 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{1600000}+\frac {33089 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{750000}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{1920 \left (x +\frac {3}{2}\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.20, size = 287, normalized size = 1.93 \begin {gather*} \frac {35371}{200000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{30 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{375 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {1141 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1141 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{3750 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {35371 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{150000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3423}{100000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1141}{1600000} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {21679}{800000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {33089 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{150000 \, {\left (2 \, x + 3\right )}} \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 \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^7} \,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 (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{128 x^{7} + 1344 x^{6} + 6048 x^{5} + 15120 x^{4} + 22680 x^{3} + 20412 x^{2} + 10206 x + 2187}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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