Optimal. Leaf size=98 \[ \frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-154 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {2800}{27} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 98, 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 = {832, 838, 840,
1180, 213} \begin {gather*} -\frac {(139 x+121) (2 x+3)^{5/2}}{3 \left (3 x^2+5 x+2\right )}+\frac {826}{27} (2 x+3)^{3/2}+\frac {1358}{27} \sqrt {2 x+3}-154 \tanh ^{-1}\left (\sqrt {2 x+3}\right )+\frac {2800}{27} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {2 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 832
Rule 838
Rule 840
Rule 1180
Rubi steps
\begin {align*} \int \frac {(5-x) (3+2 x)^{7/2}}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{3} \int \frac {(3+2 x)^{3/2} (182+413 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{9} \int \frac {\sqrt {3+2 x} (-14+679 x)}{2+5 x+3 x^2} \, dx\\ &=\frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {1}{27} \int \frac {-2842-763 x}{\sqrt {3+2 x} \left (2+5 x+3 x^2\right )} \, dx\\ &=\frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+\frac {2}{27} \text {Subst}\left (\int \frac {-3395-763 x^2}{5-8 x^2+3 x^4} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}+462 \text {Subst}\left (\int \frac {1}{-3+3 x^2} \, dx,x,\sqrt {3+2 x}\right )-\frac {14000}{27} \text {Subst}\left (\int \frac {1}{-5+3 x^2} \, dx,x,\sqrt {3+2 x}\right )\\ &=\frac {1358}{27} \sqrt {3+2 x}+\frac {826}{27} (3+2 x)^{3/2}-\frac {(3+2 x)^{5/2} (121+139 x)}{3 \left (2+5 x+3 x^2\right )}-154 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {2800}{27} \sqrt {\frac {5}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 81, normalized size = 0.83 \begin {gather*} -154 \tanh ^{-1}\left (\sqrt {3+2 x}\right )+\frac {1}{81} \left (-\frac {3 \sqrt {3+2 x} \left (2129+1843 x-400 x^2+48 x^3\right )}{2+5 x+3 x^2}+2800 \sqrt {15} \tanh ^{-1}\left (\sqrt {\frac {3}{5}} \sqrt {3+2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 104, normalized size = 1.06
method | result | size |
risch | \(-\frac {\left (48 x^{3}-400 x^{2}+1843 x +2129\right ) \sqrt {3+2 x}}{27 \left (3 x^{2}+5 x +2\right )}-77 \ln \left (\sqrt {3+2 x}+1\right )+\frac {2800 \arctanh \left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}+77 \ln \left (\sqrt {3+2 x}-1\right )\) | \(80\) |
trager | \(-\frac {\left (48 x^{3}-400 x^{2}+1843 x +2129\right ) \sqrt {3+2 x}}{27 \left (3 x^{2}+5 x +2\right )}-77 \ln \left (\frac {\sqrt {3+2 x}+2+x}{1+x}\right )+\frac {1400 \RootOf \left (\textit {\_Z}^{2}-15\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-15\right ) x +15 \sqrt {3+2 x}+7 \RootOf \left (\textit {\_Z}^{2}-15\right )}{2+3 x}\right )}{81}\) | \(101\) |
derivativedivides | \(-\frac {8 \left (3+2 x \right )^{\frac {3}{2}}}{27}+\frac {184 \sqrt {3+2 x}}{27}-\frac {4250 \sqrt {3+2 x}}{81 \left (\frac {4}{3}+2 x \right )}+\frac {2800 \arctanh \left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}-\frac {6}{\sqrt {3+2 x}+1}-77 \ln \left (\sqrt {3+2 x}+1\right )-\frac {6}{\sqrt {3+2 x}-1}+77 \ln \left (\sqrt {3+2 x}-1\right )\) | \(104\) |
default | \(-\frac {8 \left (3+2 x \right )^{\frac {3}{2}}}{27}+\frac {184 \sqrt {3+2 x}}{27}-\frac {4250 \sqrt {3+2 x}}{81 \left (\frac {4}{3}+2 x \right )}+\frac {2800 \arctanh \left (\frac {\sqrt {15}\, \sqrt {3+2 x}}{5}\right ) \sqrt {15}}{81}-\frac {6}{\sqrt {3+2 x}+1}-77 \ln \left (\sqrt {3+2 x}+1\right )-\frac {6}{\sqrt {3+2 x}-1}+77 \ln \left (\sqrt {3+2 x}-1\right )\) | \(104\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 116, normalized size = 1.18 \begin {gather*} -\frac {8}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1400}{81} \, \sqrt {15} \log \left (-\frac {\sqrt {15} - 3 \, \sqrt {2 \, x + 3}}{\sqrt {15} + 3 \, \sqrt {2 \, x + 3}}\right ) + \frac {184}{27} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (2611 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 2935 \, \sqrt {2 \, x + 3}\right )}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 77 \, \log \left (\sqrt {2 \, x + 3} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.96, size = 129, normalized size = 1.32 \begin {gather*} \frac {1400 \, \sqrt {5} \sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\frac {\sqrt {5} \sqrt {3} \sqrt {2 \, x + 3} + 3 \, x + 7}{3 \, x + 2}\right ) - 6237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} + 1\right ) + 6237 \, {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (\sqrt {2 \, x + 3} - 1\right ) - 3 \, {\left (48 \, x^{3} - 400 \, x^{2} + 1843 \, x + 2129\right )} \sqrt {2 \, x + 3}}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\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 [A]
time = 1.08, size = 120, normalized size = 1.22 \begin {gather*} -\frac {8}{27} \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - \frac {1400}{81} \, \sqrt {15} \log \left (\frac {{\left | -2 \, \sqrt {15} + 6 \, \sqrt {2 \, x + 3} \right |}}{2 \, {\left (\sqrt {15} + 3 \, \sqrt {2 \, x + 3}\right )}}\right ) + \frac {184}{27} \, \sqrt {2 \, x + 3} - \frac {2 \, {\left (2611 \, {\left (2 \, x + 3\right )}^{\frac {3}{2}} - 2935 \, \sqrt {2 \, x + 3}\right )}}{27 \, {\left (3 \, {\left (2 \, x + 3\right )}^{2} - 16 \, x - 19\right )}} - 77 \, \log \left (\sqrt {2 \, x + 3} + 1\right ) + 77 \, \log \left ({\left | \sqrt {2 \, x + 3} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 90, normalized size = 0.92 \begin {gather*} \frac {184\,\sqrt {2\,x+3}}{27}-\frac {\frac {5870\,\sqrt {2\,x+3}}{81}-\frac {5222\,{\left (2\,x+3\right )}^{3/2}}{81}}{\frac {16\,x}{3}-{\left (2\,x+3\right )}^2+\frac {19}{3}}-\frac {8\,{\left (2\,x+3\right )}^{3/2}}{27}+\mathrm {atan}\left (\sqrt {2\,x+3}\,1{}\mathrm {i}\right )\,154{}\mathrm {i}-\frac {\sqrt {15}\,\mathrm {atan}\left (\frac {\sqrt {15}\,\sqrt {2\,x+3}\,1{}\mathrm {i}}{5}\right )\,2800{}\mathrm {i}}{81} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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