Optimal. Leaf size=169 \[ -\frac {6 (47 x+37)}{5 (2 x+3)^4 \sqrt {3 x^2+5 x+2}}-\frac {25458 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}-\frac {973 \sqrt {3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac {11596 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac {817 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^4}+\frac {82039 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2500 \sqrt {5}} \]
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Rubi [A] time = 0.12, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {822, 834, 806, 724, 206} \begin {gather*} -\frac {6 (47 x+37)}{5 (2 x+3)^4 \sqrt {3 x^2+5 x+2}}-\frac {25458 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}-\frac {973 \sqrt {3 x^2+5 x+2}}{30 (2 x+3)^2}-\frac {11596 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}-\frac {817 \sqrt {3 x^2+5 x+2}}{25 (2 x+3)^4}+\frac {82039 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2500 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 822
Rule 834
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^5 \left (2+5 x+3 x^2\right )^{3/2}} \, dx &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {2}{5} \int \frac {875+1128 x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}+\frac {1}{50} \int \frac {-10463-14706 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {1}{750} \int \frac {87103+139152 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}+\frac {\int \frac {-330885-729750 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{7500}\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2500}\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}-\frac {82039 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1250}\\ &=-\frac {6 (37+47 x)}{5 (3+2 x)^4 \sqrt {2+5 x+3 x^2}}-\frac {817 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)^4}-\frac {11596 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {973 \sqrt {2+5 x+3 x^2}}{30 (3+2 x)^2}-\frac {25458 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {82039 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2500 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 89, normalized size = 0.53 \begin {gather*} \frac {-246117 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \left (3665952 x^5+24066204 x^4+62190544 x^3+78737669 x^2+48537379 x+11545002\right )}{(2 x+3)^4 \sqrt {3 x^2+5 x+2}}}{37500} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.67, size = 98, normalized size = 0.58 \begin {gather*} \frac {82039 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{1250 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-3665952 x^5-24066204 x^4-62190544 x^3-78737669 x^2-48537379 x-11545002\right )}{3750 (x+1) (2 x+3)^4 (3 x+2)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 155, normalized size = 0.92 \begin {gather*} \frac {246117 \, \sqrt {5} {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\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 (3665952 \, x^{5} + 24066204 \, x^{4} + 62190544 \, x^{3} + 78737669 \, x^{2} + 48537379 \, x + 11545002\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{75000 \, {\left (48 \, x^{6} + 368 \, x^{5} + 1160 \, x^{4} + 1920 \, x^{3} + 1755 \, x^{2} + 837 \, x + 162\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 235, normalized size = 1.39 \begin {gather*} \frac {1}{12500} \, \sqrt {5} {\left (50916 \, \sqrt {5} \sqrt {3} + 82039 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {\frac {\frac {5 \, {\left (\frac {\frac {10 \, {\left (\frac {448}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {195}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {9619}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {27724}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {857109}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {458244}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{7500 \, \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} - \frac {82039 \, \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 )}{12500 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 153, normalized size = 0.91 \begin {gather*} -\frac {82039 \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 )}{12500}-\frac {13}{320 \left (x +\frac {3}{2}\right )^{4} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {14}{75 \left (x +\frac {3}{2}\right )^{3} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {9619}{12000 \left (x +\frac {3}{2}\right )^{2} \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {6931}{1500 \left (x +\frac {3}{2}\right ) \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {82039}{5000 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {12729 \left (6 x +5\right )}{1250 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.25, size = 310, normalized size = 1.83 \begin {gather*} -\frac {82039}{12500} \, \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 {38187 \, x}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {172541}{5000 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13}{20 \, {\left (16 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{4} + 96 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 216 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {112}{75 \, {\left (8 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + 36 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 54 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {9619}{3000 \, {\left (4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + 12 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}} - \frac {6931}{750 \, {\left (2 \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + 3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\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 {x-5}{{\left (2\,x+3\right )}^5\,{\left (3\,x^2+5\,x+2\right )}^{3/2}} \,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 \frac {x}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{96 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 880 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 3424 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 7320 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 9270 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 6939 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2835 x \sqrt {3 x^{2} + 5 x + 2} + 486 \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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