Optimal. Leaf size=148 \[ -\frac {1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac {111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac {281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac {1017 (4-9 x) \sqrt {3 x^2+2}}{7503125 (2 x+3)^2}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \]
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Rubi [A] time = 0.09, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {835, 807, 721, 725, 206} \begin {gather*} -\frac {1207 \left (3 x^2+2\right )^{3/2}}{857500 (2 x+3)^3}-\frac {111 \left (3 x^2+2\right )^{3/2}}{17500 (2 x+3)^4}-\frac {281 \left (3 x^2+2\right )^{3/2}}{12250 (2 x+3)^5}-\frac {13 \left (3 x^2+2\right )^{3/2}}{210 (2 x+3)^6}-\frac {1017 (4-9 x) \sqrt {3 x^2+2}}{7503125 (2 x+3)^2}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{7503125 \sqrt {35}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 721
Rule 725
Rule 807
Rule 835
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^7} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {1}{210} \int \frac {(-246+117 x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}+\frac {\int \frac {(8730-5058 x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx}{36750}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {\int \frac {(-233352+97902 x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx}{5145000}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac {2034 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{214375}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}+\frac {6102 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{7503125}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac {6102 \operatorname {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{7503125}\\ &=-\frac {1017 (4-9 x) \sqrt {2+3 x^2}}{7503125 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{210 (3+2 x)^6}-\frac {281 \left (2+3 x^2\right )^{3/2}}{12250 (3+2 x)^5}-\frac {111 \left (2+3 x^2\right )^{3/2}}{17500 (3+2 x)^4}-\frac {1207 \left (2+3 x^2\right )^{3/2}}{857500 (3+2 x)^3}-\frac {6102 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{7503125 \sqrt {35}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 80, normalized size = 0.54 \begin {gather*} \frac {-36612 \sqrt {35} \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )-\frac {35 \sqrt {3 x^2+2} \left (642132 x^5+5388660 x^4+18236055 x^3+30753930 x^2+18651300 x+22308548\right )}{(2 x+3)^6}}{1575656250} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.61, size = 96, normalized size = 0.65 \begin {gather*} \frac {12204 \tanh ^{-1}\left (-\frac {2 \sqrt {3 x^2+2}}{\sqrt {35}}+2 \sqrt {\frac {3}{35}} x+3 \sqrt {\frac {3}{35}}\right )}{7503125 \sqrt {35}}+\frac {\sqrt {3 x^2+2} \left (-642132 x^5-5388660 x^4-18236055 x^3-30753930 x^2-18651300 x-22308548\right )}{45018750 (2 x+3)^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 149, normalized size = 1.01 \begin {gather*} \frac {18306 \, \sqrt {35} {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (642132 \, x^{5} + 5388660 \, x^{4} + 18236055 \, x^{3} + 30753930 \, x^{2} + 18651300 \, x + 22308548\right )} \sqrt {3 \, x^{2} + 2}}{1575656250 \, {\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 = 0.27, size = 367, normalized size = 2.48 \begin {gather*} \frac {6102}{262609375} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {3 \, \sqrt {3} {\left (21696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} + 1073952 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} + 6978880 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 87678735 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 66333990 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 258582989 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 426764436 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 755892540 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 355133440 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 207134880 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 19853952 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 2283136\right )}}{240100000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 191, normalized size = 1.29 \begin {gather*} \frac {27459 \sqrt {-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}\, x}{262609375}-\frac {6102 \sqrt {35}\, \arctanh \left (\frac {2 \left (-9 x +4\right ) \sqrt {35}}{35 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}\right )}{262609375}-\frac {13 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{13440 \left (x +\frac {3}{2}\right )^{6}}-\frac {281 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{392000 \left (x +\frac {3}{2}\right )^{5}}-\frac {111 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{280000 \left (x +\frac {3}{2}\right )^{4}}-\frac {1207 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{6860000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1017 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )^{2}}-\frac {9153 \left (-9 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}{262609375 \left (x +\frac {3}{2}\right )}+\frac {6102 \sqrt {-36 x +12 \left (x +\frac {3}{2}\right )^{2}-19}}{262609375} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 229, normalized size = 1.55 \begin {gather*} \frac {6102}{262609375} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {3051}{15006250} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{210 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {281 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{12250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {111 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{17500 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {1207 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{857500 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {2034 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{7503125 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {9153 \, \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 223, normalized size = 1.51 \begin {gather*} \frac {6102\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{262609375}-\frac {6102\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{262609375}+\frac {127\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1568000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {109\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{44800\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {53511\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{240100000\,\left (x+\frac {3}{2}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}-\frac {2727\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{13720000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {479\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3920000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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