Optimal. Leaf size=125 \[ \frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \]
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Rubi [A]
time = 0.03, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {99, 154, 152,
65, 212} \begin {gather*} \frac {2 \sqrt {1-2 x} (5 x+3)^3}{(3 x+2)^2}-\frac {(1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {251 \sqrt {1-2 x} (5 x+3)^2}{63 (3 x+2)}-\frac {5}{567} \sqrt {1-2 x} (7265 x+2323)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 99
Rule 152
Rule 154
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{9} \int \frac {(6-45 x) \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {1}{54} \int \frac {(306-1800 x) (3+5 x)^2}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {\int \frac {(20934-130770 x) (3+5 x)}{\sqrt {1-2 x} (2+3 x)} \, dx}{1134}\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)+\frac {18019}{567} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)-\frac {18019}{567} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {251 \sqrt {1-2 x} (3+5 x)^2}{63 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {2 \sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^2}-\frac {5}{567} \sqrt {1-2 x} (2323+7265 x)-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 68, normalized size = 0.54 \begin {gather*} \frac {\sqrt {1-2 x} \left (47939+199243 x+259614 x^2+81900 x^3-31500 x^4\right )}{567 (2+3 x)^3}-\frac {36038 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{567 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 75, normalized size = 0.60
method | result | size |
risch | \(\frac {63000 x^{5}-195300 x^{4}-437328 x^{3}-138872 x^{2}+103365 x +47939}{567 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {36038 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(61\) |
derivativedivides | \(\frac {250 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {2050 \sqrt {1-2 x}}{243}+\frac {-\frac {7876 \left (1-2 x \right )^{\frac {5}{2}}}{189}+\frac {46612 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {53648 \sqrt {1-2 x}}{243}}{\left (-4-6 x \right )^{3}}-\frac {36038 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(75\) |
default | \(\frac {250 \left (1-2 x \right )^{\frac {3}{2}}}{243}+\frac {2050 \sqrt {1-2 x}}{243}+\frac {-\frac {7876 \left (1-2 x \right )^{\frac {5}{2}}}{189}+\frac {46612 \left (1-2 x \right )^{\frac {3}{2}}}{243}-\frac {53648 \sqrt {1-2 x}}{243}}{\left (-4-6 x \right )^{3}}-\frac {36038 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{11907}\) | \(75\) |
trager | \(-\frac {\left (31500 x^{4}-81900 x^{3}-259614 x^{2}-199243 x -47939\right ) \sqrt {1-2 x}}{567 \left (2+3 x \right )^{3}}+\frac {18019 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{11907}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 110, normalized size = 0.88 \begin {gather*} \frac {250}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {18019}{11907} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2050}{243} \, \sqrt {-2 \, x + 1} + \frac {4 \, {\left (17721 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 81571 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 93884 \, \sqrt {-2 \, x + 1}\right )}}{1701 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.15, size = 94, normalized size = 0.75 \begin {gather*} \frac {18019 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (31500 \, x^{4} - 81900 \, x^{3} - 259614 \, x^{2} - 199243 \, x - 47939\right )} \sqrt {-2 \, x + 1}}{11907 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 = 0.83, size = 102, normalized size = 0.82 \begin {gather*} \frac {250}{243} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {18019}{11907} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {2050}{243} \, \sqrt {-2 \, x + 1} + \frac {17721 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 81571 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 93884 \, \sqrt {-2 \, x + 1}}{3402 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 91, normalized size = 0.73 \begin {gather*} \frac {2050\,\sqrt {1-2\,x}}{243}+\frac {250\,{\left (1-2\,x\right )}^{3/2}}{243}+\frac {\frac {53648\,\sqrt {1-2\,x}}{6561}-\frac {46612\,{\left (1-2\,x\right )}^{3/2}}{6561}+\frac {7876\,{\left (1-2\,x\right )}^{5/2}}{5103}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,36038{}\mathrm {i}}{11907} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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