Optimal. Leaf size=126 \[ -\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{214375 \sqrt {35}} \]
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Rubi [A]
time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {849, 821, 735,
739, 212} \begin {gather*} -\frac {43 \left (3 x^2+2\right )^{3/2}}{6125 (2 x+3)^3}-\frac {23 \left (3 x^2+2\right )^{3/2}}{875 (2 x+3)^4}-\frac {13 \left (3 x^2+2\right )^{3/2}}{175 (2 x+3)^5}-\frac {339 (4-9 x) \sqrt {3 x^2+2}}{428750 (2 x+3)^2}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{214375 \sqrt {35}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 735
Rule 739
Rule 821
Rule 849
Rubi steps
\begin {align*} \int \frac {(5-x) \sqrt {2+3 x^2}}{(3+2 x)^6} \, dx &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {1}{175} \int \frac {(-205+78 x) \sqrt {2+3 x^2}}{(3+2 x)^5} \, dx\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}+\frac {\int \frac {(6132-1932 x) \sqrt {2+3 x^2}}{(3+2 x)^4} \, dx}{24500}\\ &=-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}+\frac {339 \int \frac {\sqrt {2+3 x^2}}{(3+2 x)^3} \, dx}{6125}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}+\frac {1017 \int \frac {1}{(3+2 x) \sqrt {2+3 x^2}} \, dx}{214375}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}-\frac {1017 \text {Subst}\left (\int \frac {1}{35-x^2} \, dx,x,\frac {4-9 x}{\sqrt {2+3 x^2}}\right )}{214375}\\ &=-\frac {339 (4-9 x) \sqrt {2+3 x^2}}{428750 (3+2 x)^2}-\frac {13 \left (2+3 x^2\right )^{3/2}}{175 (3+2 x)^5}-\frac {23 \left (2+3 x^2\right )^{3/2}}{875 (3+2 x)^4}-\frac {43 \left (2+3 x^2\right )^{3/2}}{6125 (3+2 x)^3}-\frac {1017 \tanh ^{-1}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{214375 \sqrt {35}}\\ \end {align*}
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Mathematica [A]
time = 0.92, size = 88, normalized size = 0.70 \begin {gather*} \frac {-\frac {35 \sqrt {2+3 x^2} \left (222112+108167 x+186392 x^2+76992 x^3+11712 x^4\right )}{(3+2 x)^5}+4068 \sqrt {35} \tanh ^{-1}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{15006250} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.74, size = 170, normalized size = 1.35
method | result | size |
risch | \(-\frac {35136 x^{6}+230976 x^{5}+582600 x^{4}+478485 x^{3}+1039120 x^{2}+216334 x +444224}{428750 \left (2 x +3\right )^{5} \sqrt {3 x^{2}+2}}-\frac {1017 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7503125}\) | \(80\) |
trager | \(-\frac {\left (11712 x^{4}+76992 x^{3}+186392 x^{2}+108167 x +222112\right ) \sqrt {3 x^{2}+2}}{428750 \left (2 x +3\right )^{5}}-\frac {1017 \RootOf \left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \RootOf \left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \RootOf \left (\textit {\_Z}^{2}-35\right )}{2 x +3}\right )}{7503125}\) | \(86\) |
default | \(-\frac {23 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{14000 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{49000 \left (x +\frac {3}{2}\right )^{3}}-\frac {339 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{857500 \left (x +\frac {3}{2}\right )^{2}}-\frac {3051 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{15006250 \left (x +\frac {3}{2}\right )}+\frac {1017 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{7503125}-\frac {1017 \sqrt {35}\, \arctanh \left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{7503125}+\frac {9153 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{15006250}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{5600 \left (x +\frac {3}{2}\right )^{5}}\) | \(170\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 186, normalized size = 1.48 \begin {gather*} \frac {1017}{7503125} \, \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 {1017}{857500} \, \sqrt {3 \, x^{2} + 2} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{175 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {23 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{875 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {43 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{6125 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {339 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{214375 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {3051 \, \sqrt {3 \, x^{2} + 2}}{857500 \, {\left (2 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.32, size = 134, normalized size = 1.06 \begin {gather*} \frac {1017 \, \sqrt {35} {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\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 (11712 \, x^{4} + 76992 \, x^{3} + 186392 \, x^{2} + 108167 \, x + 222112\right )} \sqrt {3 \, x^{2} + 2}}{15006250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 322 vs.
\(2 (103) = 206\).
time = 1.16, size = 322, normalized size = 2.56 \begin {gather*} \frac {1017}{7503125} \, \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 (904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 36612 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 254217 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 142464 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 338184 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 4315808 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} + 1676892 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 1737184 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 219776 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 31232\right )}}{1715000 \, {\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 )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.80, size = 178, normalized size = 1.41 \begin {gather*} \frac {1017\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{7503125}-\frac {1017\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{7503125}+\frac {73\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{11200\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {13\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{640\,\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 {183\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{214375\,\left (x+\frac {3}{2}\right )}-\frac {3\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{6125\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{7000\,\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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