Optimal. Leaf size=137 \[ \frac {391}{135} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{648} (27519+9650 x) \sqrt {2+5 x+3 x^2}+\frac {28051 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1296 \sqrt {3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 137, 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 = {846, 793, 635,
212} \begin {gather*} -\frac {1}{15} \sqrt {3 x^2+5 x+2} (2 x+3)^4+\frac {53}{60} \sqrt {3 x^2+5 x+2} (2 x+3)^3+\frac {391}{135} \sqrt {3 x^2+5 x+2} (2 x+3)^2+\frac {1}{648} (9650 x+27519) \sqrt {3 x^2+5 x+2}+\frac {28051 \tanh ^{-1}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{1296 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 846
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^4}{\sqrt {2+5 x+3 x^2}} \, dx &=-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{15} \int \frac {(3+2 x)^3 \left (\frac {497}{2}+159 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{180} \int \frac {(3+2 x)^2 \left (\frac {11691}{2}+4692 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx\\ &=\frac {391}{135} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {\int \frac {(3+2 x) \left (\frac {170205}{2}+72375 x\right )}{\sqrt {2+5 x+3 x^2}} \, dx}{1620}\\ &=\frac {391}{135} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{648} (27519+9650 x) \sqrt {2+5 x+3 x^2}+\frac {28051 \int \frac {1}{\sqrt {2+5 x+3 x^2}} \, dx}{1296}\\ &=\frac {391}{135} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{648} (27519+9650 x) \sqrt {2+5 x+3 x^2}+\frac {28051}{648} \text {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {5+6 x}{\sqrt {2+5 x+3 x^2}}\right )\\ &=\frac {391}{135} (3+2 x)^2 \sqrt {2+5 x+3 x^2}+\frac {53}{60} (3+2 x)^3 \sqrt {2+5 x+3 x^2}-\frac {1}{15} (3+2 x)^4 \sqrt {2+5 x+3 x^2}+\frac {1}{648} (27519+9650 x) \sqrt {2+5 x+3 x^2}+\frac {28051 \tanh ^{-1}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1296 \sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 71, normalized size = 0.52 \begin {gather*} \frac {-3 \sqrt {2+5 x+3 x^2} \left (-281829-268750 x-93912 x^2-2160 x^3+3456 x^4\right )+140255 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{9720} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 111, normalized size = 0.81
method | result | size |
risch | \(-\frac {\left (3456 x^{4}-2160 x^{3}-93912 x^{2}-268750 x -281829\right ) \sqrt {3 x^{2}+5 x +2}}{3240}+\frac {28051 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3888}\) | \(65\) |
trager | \(\left (-\frac {16}{15} x^{4}+\frac {2}{3} x^{3}+\frac {3913}{135} x^{2}+\frac {26875}{324} x +\frac {93943}{1080}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {28051 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \RootOf \left (\textit {\_Z}^{2}-3\right ) x +5 \RootOf \left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{3888}\) | \(76\) |
default | \(-\frac {16 x^{4} \sqrt {3 x^{2}+5 x +2}}{15}+\frac {2 x^{3} \sqrt {3 x^{2}+5 x +2}}{3}+\frac {3913 x^{2} \sqrt {3 x^{2}+5 x +2}}{135}+\frac {26875 x \sqrt {3 x^{2}+5 x +2}}{324}+\frac {93943 \sqrt {3 x^{2}+5 x +2}}{1080}+\frac {28051 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3888}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 109, normalized size = 0.80 \begin {gather*} -\frac {16}{15} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{4} + \frac {2}{3} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{3} + \frac {3913}{135} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x^{2} + \frac {26875}{324} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {28051}{3888} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {93943}{1080} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.49, size = 73, normalized size = 0.53 \begin {gather*} -\frac {1}{3240} \, {\left (3456 \, x^{4} - 2160 \, x^{3} - 93912 \, x^{2} - 268750 \, x - 281829\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {28051}{7776} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \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 \left (- \frac {999 x}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{\sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{\sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{\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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Giac [A]
time = 2.17, size = 69, normalized size = 0.50 \begin {gather*} -\frac {1}{3240} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, x - 5\right )} x - 3913\right )} x - 134375\right )} x - 281829\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {28051}{3888} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\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 {{\left (2\,x+3\right )}^4\,\left (x-5\right )}{\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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