Optimal. Leaf size=195 \[ -\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac {7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac {35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac {35 (2701 x+5795) \sqrt {3 x^2+5 x+2}}{1024 (2 x+3)}-\frac {744275 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{4096} \]
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Rubi [A] time = 0.13, antiderivative size = 195, 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 = {812, 843, 621, 206, 724} \begin {gather*} -\frac {(x+8) \left (3 x^2+5 x+2\right )^{7/2}}{8 (2 x+3)^4}+\frac {7 (43 x+93) \left (3 x^2+5 x+2\right )^{5/2}}{96 (2 x+3)^3}-\frac {35 (343 x+736) \left (3 x^2+5 x+2\right )^{3/2}}{768 (2 x+3)^2}+\frac {35 (2701 x+5795) \sqrt {3 x^2+5 x+2}}{1024 (2 x+3)}-\frac {744275 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{4096} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 724
Rule 812
Rule 843
Rubi steps
\begin {align*} \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^5} \, dx &=-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {7}{128} \int \frac {(-288-344 x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^4} \, dx\\ &=\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac {35 \int \frac {(-14064-16464 x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx}{9216}\\ &=-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {35 \int \frac {(-443136-518592 x) \sqrt {2+5 x+3 x^2}}{(3+2 x)^2} \, dx}{98304}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}+\frac {35 \int \frac {-6977664-8165760 x}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{786432}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{4096}+\frac {960855 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{4096}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )}{2048}-\frac {960855 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{2048}\\ &=\frac {35 (5795+2701 x) \sqrt {2+5 x+3 x^2}}{1024 (3+2 x)}-\frac {35 (736+343 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)^2}+\frac {7 (93+43 x) \left (2+5 x+3 x^2\right )^{5/2}}{96 (3+2 x)^3}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{7/2}}{8 (3+2 x)^4}-\frac {744275 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{4096 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{4096}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 130, normalized size = 0.67 \begin {gather*} \frac {-576513 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-744275 \sqrt {3} \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {9 x^2+15 x+6}}\right )-\frac {12 \sqrt {3 x^2+5 x+2} \left (3456 x^7-12864 x^6-38288 x^5-253688 x^4-2869312 x^3-9107922 x^2-11295211 x-4933171\right )}{(2 x+3)^4}}{12288} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.93, size = 131, normalized size = 0.67 \begin {gather*} -\frac {744275 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {3} (x+1)}\right )}{2048 \sqrt {3}}+\frac {192171 \sqrt {5} \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{2048}+\frac {\sqrt {3 x^2+5 x+2} \left (-3456 x^7+12864 x^6+38288 x^5+253688 x^4+2869312 x^3+9107922 x^2+11295211 x+4933171\right )}{1024 (2 x+3)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 203, normalized size = 1.04 \begin {gather*} \frac {744275 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 576513 \, \sqrt {5} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\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 ) - 24 \, {\left (3456 \, x^{7} - 12864 \, x^{6} - 38288 \, x^{5} - 253688 \, x^{4} - 2869312 \, x^{3} - 9107922 \, x^{2} - 11295211 \, x - 4933171\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{24576 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.31, size = 636, normalized size = 3.26 \begin {gather*} \frac {744275}{12288} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {5}}{2 \, x + 3} \right |}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {192171}{4096} \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{4096} \, {\left (\frac {5 \, {\left (\frac {50 \, {\left (\frac {13 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 88 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 14343 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 181996 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} - \frac {479709 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 499296 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 3133183 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 3365712 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 7550211 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 8139744 \, \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 6574257 \, {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 6966000 \, \sqrt {5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2048 \, {\left ({\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )}^{2} - 3\right )}^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 295, normalized size = 1.51 \begin {gather*} -\frac {192171 \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 )}{4096}-\frac {744275 \sqrt {3}\, \ln \left (\frac {\left (3 x +\frac {5}{2}\right ) \sqrt {3}}{3}+\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\right )}{12288}-\frac {13 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{320 \left (x +\frac {3}{2}\right )^{4}}-\frac {1263 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{4000 \left (x +\frac {3}{2}\right )^{2}}+\frac {3 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{100 \left (x +\frac {3}{2}\right )^{3}}-\frac {1479 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{1000}-\frac {10101 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{4000}+\frac {1479 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {9}{2}}}{500 \left (x +\frac {3}{2}\right )}-\frac {6069 \left (6 x +5\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{1280}-\frac {24409 \left (6 x +5\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{2048}+\frac {192171 \sqrt {-16 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{4096}+\frac {64057 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{2560}+\frac {192171 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {5}{2}}}{16000}+\frac {27453 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {7}{2}}}{4000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.18, size = 285, normalized size = 1.46 \begin {gather*} \frac {3789}{4000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} + \frac {6 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{25 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {1263 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{1000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {30303}{2000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {9849}{16000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1479 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{200 \, {\left (2 \, x + 3\right )}} - \frac {18207}{640} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {3367}{2560} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {73227}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {744275}{12288} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {192171}{4096} \, \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 {35063}{1024} \, \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 \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^5} \,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 {40 \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{32 x^{5} + 240 x^{4} + 720 x^{3} + 1080 x^{2} + 810 x + 243}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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