Optimal. Leaf size=139 \[ -\frac {681 \sqrt {3 x^2+5 x+2}}{250 (2 x+3)}-\frac {41 \sqrt {3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac {86 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac {5771 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2000 \sqrt {5}} \]
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Rubi [A] time = 0.10, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {834, 806, 724, 206} \begin {gather*} -\frac {681 \sqrt {3 x^2+5 x+2}}{250 (2 x+3)}-\frac {41 \sqrt {3 x^2+5 x+2}}{24 (2 x+3)^2}-\frac {86 \sqrt {3 x^2+5 x+2}}{75 (2 x+3)^3}-\frac {13 \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^4}+\frac {5771 \tanh ^{-1}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2000 \sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 724
Rule 806
Rule 834
Rubi steps
\begin {align*} \int \frac {5-x}{(3+2 x)^5 \sqrt {2+5 x+3 x^2}} \, dx &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {1}{20} \int \frac {\frac {7}{2}+117 x}{(3+2 x)^4 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}+\frac {1}{300} \int \frac {-\frac {1067}{2}-2064 x}{(3+2 x)^3 \sqrt {2+5 x+3 x^2}} \, dx\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {\int \frac {\frac {5265}{2}+15375 x}{(3+2 x)^2 \sqrt {2+5 x+3 x^2}} \, dx}{3000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}+\frac {5771 \int \frac {1}{(3+2 x) \sqrt {2+5 x+3 x^2}} \, dx}{2000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}-\frac {5771 \operatorname {Subst}\left (\int \frac {1}{20-x^2} \, dx,x,\frac {-7-8 x}{\sqrt {2+5 x+3 x^2}}\right )}{1000}\\ &=-\frac {13 \sqrt {2+5 x+3 x^2}}{20 (3+2 x)^4}-\frac {86 \sqrt {2+5 x+3 x^2}}{75 (3+2 x)^3}-\frac {41 \sqrt {2+5 x+3 x^2}}{24 (3+2 x)^2}-\frac {681 \sqrt {2+5 x+3 x^2}}{250 (3+2 x)}+\frac {5771 \tanh ^{-1}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{2000 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 79, normalized size = 0.57 \begin {gather*} \frac {-17313 \sqrt {5} \tanh ^{-1}\left (\frac {-8 x-7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )-\frac {10 \sqrt {3 x^2+5 x+2} \left (65376 x^3+314692 x^2+509668 x+279039\right )}{(2 x+3)^4}}{30000} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.64, size = 76, normalized size = 0.55 \begin {gather*} \frac {5771 \tanh ^{-1}\left (\frac {\sqrt {3 x^2+5 x+2}}{\sqrt {5} (x+1)}\right )}{1000 \sqrt {5}}+\frac {\sqrt {3 x^2+5 x+2} \left (-65376 x^3-314692 x^2-509668 x-279039\right )}{3000 (2 x+3)^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 125, normalized size = 0.90 \begin {gather*} \frac {17313 \, \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 ) - 20 \, {\left (65376 \, x^{3} + 314692 \, x^{2} + 509668 \, x + 279039\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{60000 \, {\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 [A] time = 0.37, size = 193, normalized size = 1.39 \begin {gather*} \frac {1}{10000} \, \sqrt {5} {\left (2724 \, \sqrt {5} \sqrt {3} + 5771 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{6000} \, {\left (\frac {5 \, {\left (\frac {2 \, {\left (\frac {344}{\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 {1025}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} + \frac {8172}{\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 {5771 \, \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 )}{10000 \, \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 = 116, normalized size = 0.83 \begin {gather*} -\frac {5771 \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 )}{10000}-\frac {13 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{320 \left (x +\frac {3}{2}\right )^{4}}-\frac {43 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{300 \left (x +\frac {3}{2}\right )^{3}}-\frac {41 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{96 \left (x +\frac {3}{2}\right )^{2}}-\frac {681 \sqrt {-4 x +3 \left (x +\frac {3}{2}\right )^{2}-\frac {19}{4}}}{500 \left (x +\frac {3}{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 157, normalized size = 1.13 \begin {gather*} -\frac {5771}{10000} \, \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 {13 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{20 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {86 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{75 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {41 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{24 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {681 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{250 \, {\left (2 \, x + 3\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\,\sqrt {3\,x^2+5\,x+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}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{32 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 240 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 720 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 1080 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 810 x \sqrt {3 x^{2} + 5 x + 2} + 243 \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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