Optimal. Leaf size=174 \[ -\frac {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {9696 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}+\frac {1048 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac {47552 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac {12 (638 x+603)}{25 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}+\frac {46108 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{625 \sqrt {5}} \]
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Rubi [A] time = 0.12, antiderivative size = 174, 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 {2 (47 x+37)}{5 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}}+\frac {9696 \sqrt {3 x^2+5 x+2}}{625 (2 x+3)}+\frac {1048 \sqrt {3 x^2+5 x+2}}{15 (2 x+3)^2}+\frac {47552 \sqrt {3 x^2+5 x+2}}{375 (2 x+3)^3}+\frac {12 (638 x+603)}{25 (2 x+3)^3 \sqrt {3 x^2+5 x+2}}+\frac {46108 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{625 \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)^4 \left (2+5 x+3 x^2\right )^{5/2}} \, dx &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}-\frac {2}{15} \int \frac {1473+1410 x}{(3+2 x)^4 \left (2+5 x+3 x^2\right )^{3/2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {4}{75} \int \frac {33846+34452 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}-\frac {4 \int \frac {-222726-213984 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx}{1125}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac {1048 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac {2 \int \frac {775170+589500 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{5625}\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac {1048 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac {9696 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {46108}{625} \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac {1048 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac {9696 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}-\frac {92216}{625} \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=-\frac {2 (37+47 x)}{5 (3+2 x)^3 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {12 (603+638 x)}{25 (3+2 x)^3 \sqrt {2+5 x+3 x^2}}+\frac {47552 \sqrt {2+5 x+3 x^2}}{375 (3+2 x)^3}+\frac {1048 \sqrt {2+5 x+3 x^2}}{15 (3+2 x)^2}+\frac {9696 \sqrt {2+5 x+3 x^2}}{625 (3+2 x)}+\frac {46108 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{625 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 150, normalized size = 0.86 \begin {gather*} \frac {2 \left (594400 \left (3 x^2+5 x+2\right )^2+2250 (638 x+603) \left (3 x^2+5 x+2\right )+2 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \left (10 (7272 x+27283) \sqrt {3 x^2+5 x+2}-34581 \sqrt {5} (2 x+3)^2 \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-1875 (47 x+37)\right )}{9375 (2 x+3)^3 \left (3 x^2+5 x+2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.60, size = 103, normalized size = 0.59 \begin {gather*} \frac {92216 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{625 \sqrt {5}}+\frac {2 \sqrt {3 x^2+5 x+2} \left (523584 x^6+4495032 x^5+15334836 x^4+26717636 x^3+25105026 x^2+12060957 x+2313929\right )}{1875 (x+1)^2 (2 x+3)^3 (3 x+2)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 170, normalized size = 0.98 \begin {gather*} \frac {2 \, {\left (34581 \, \sqrt {5} {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\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 ) + 5 \, {\left (523584 \, x^{6} + 4495032 \, x^{5} + 15334836 \, x^{4} + 26717636 \, x^{3} + 25105026 \, x^{2} + 12060957 \, x + 2313929\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \, {\left (72 \, x^{7} + 564 \, x^{6} + 1862 \, x^{5} + 3355 \, x^{4} + 3560 \, x^{3} + 2223 \, x^{2} + 756 \, x + 108\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 285, normalized size = 1.64 \begin {gather*} \frac {46108}{3125} \, \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 {2 \, {\left ({\left (12 \, {\left (19992 \, x + 58207\right )} x + 636631\right )} x + 184301\right )}}{3125 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {8 \, {\left (296724 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2103870 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 16891990 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 21246975 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 38063715 \, \sqrt {3} x + 8723544 \, \sqrt {3} - 38063715 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{9375 \, {\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 )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 169, normalized size = 0.97 \begin {gather*} -\frac {46108 \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 )}{3125}-\frac {151}{200 \left (x +\frac {3}{2}\right )^{2} \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {862}{125 \left (x +\frac {3}{2}\right ) \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {11527}{750 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {2366 \left (6 x +5\right )}{375 \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {14544 x}{625}+\frac {2424}{125}}{\sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}+\frac {23054}{625 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}-\frac {13}{120 \left (x +\frac {3}{2}\right )^{3} \left (-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 254, normalized size = 1.46 \begin {gather*} -\frac {46108}{3125} \, \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 {14544 \, x}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {35174}{625 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {4732 \, x}{125 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{15 \, {\left (8 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{3} + 36 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 54 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {151}{50 \, {\left (4 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x^{2} + 12 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {1724}{125 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {12133}{750 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{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 {x-5}{{\left (2\,x+3\right )}^4\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,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}{144 x^{8} \sqrt {3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2484 x \sqrt {3 x^{2} + 5 x + 2} + 324 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{144 x^{8} \sqrt {3 x^{2} + 5 x + 2} + 1344 x^{7} \sqrt {3 x^{2} + 5 x + 2} + 5416 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 12296 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 17185 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 15126 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 8181 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 2484 x \sqrt {3 x^{2} + 5 x + 2} + 324 \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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