Optimal. Leaf size=121 \[ \frac {1540}{729} \sqrt {1-2 x}+\frac {220}{729} (1-2 x)^{3/2}+\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)-\frac {1540}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {99, 12, 158,
152, 52, 65, 212} \begin {gather*} -\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac {55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac {220}{729} (1-2 x)^{3/2}-\frac {22}{567} (1-2 x)^{5/2} (100 x+69)+\frac {1540}{729} \sqrt {1-2 x}-\frac {1540}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^2} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}+\frac {1}{3} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{2+3 x} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {55}{3} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{2+3 x} \, dx\\ &=\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}+\frac {55}{81} \int \frac {(1-2 x)^{3/2} (3+5 x) (10+24 x)}{2+3 x} \, dx\\ &=\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)+\frac {110}{81} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {220}{729} (1-2 x)^{3/2}+\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)+\frac {770}{243} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {1540}{729} \sqrt {1-2 x}+\frac {220}{729} (1-2 x)^{3/2}+\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)+\frac {5390}{729} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1540}{729} \sqrt {1-2 x}+\frac {220}{729} (1-2 x)^{3/2}+\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)-\frac {5390}{729} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1540}{729} \sqrt {1-2 x}+\frac {220}{729} (1-2 x)^{3/2}+\frac {55}{81} (1-2 x)^{5/2} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{3 (2+3 x)}-\frac {22}{567} (1-2 x)^{5/2} (69+100 x)-\frac {1540}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 73, normalized size = 0.60 \begin {gather*} \frac {\frac {3 \sqrt {1-2 x} \left (13759+65558 x+25275 x^2-159714 x^3-17100 x^4+189000 x^5\right )}{2+3 x}-10780 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{15309} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 81, normalized size = 0.67
method | result | size |
risch | \(-\frac {378000 x^{6}-223200 x^{5}-302328 x^{4}+210264 x^{3}+105841 x^{2}-38040 x -13759}{5103 \left (2+3 x \right ) \sqrt {1-2 x}}-\frac {1540 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}\) | \(66\) |
derivativedivides | \(\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{162}-\frac {725 \left (1-2 x \right )^{\frac {7}{2}}}{378}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {214 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {1526 \sqrt {1-2 x}}{729}-\frac {98 \sqrt {1-2 x}}{2187 \left (-\frac {4}{3}-2 x \right )}-\frac {1540 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}\) | \(81\) |
default | \(\frac {125 \left (1-2 x \right )^{\frac {9}{2}}}{162}-\frac {725 \left (1-2 x \right )^{\frac {7}{2}}}{378}+\frac {2 \left (1-2 x \right )^{\frac {5}{2}}}{27}+\frac {214 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {1526 \sqrt {1-2 x}}{729}-\frac {98 \sqrt {1-2 x}}{2187 \left (-\frac {4}{3}-2 x \right )}-\frac {1540 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{2187}\) | \(81\) |
trager | \(\frac {\left (189000 x^{5}-17100 x^{4}-159714 x^{3}+25275 x^{2}+65558 x +13759\right ) \sqrt {1-2 x}}{10206+15309 x}-\frac {770 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \RootOf \left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{2187}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 98, normalized size = 0.81 \begin {gather*} \frac {125}{162} \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - \frac {725}{378} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {2}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {214}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {770}{2187} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1526}{729} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{729 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.55, size = 85, normalized size = 0.70 \begin {gather*} \frac {5390 \, \sqrt {7} \sqrt {3} {\left (3 \, x + 2\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (189000 \, x^{5} - 17100 \, x^{4} - 159714 \, x^{3} + 25275 \, x^{2} + 65558 \, x + 13759\right )} \sqrt {-2 \, x + 1}}{15309 \, {\left (3 \, 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 = 0.64, size = 122, normalized size = 1.01 \begin {gather*} \frac {125}{162} \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + \frac {725}{378} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {2}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {214}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {770}{2187} \, \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 {1526}{729} \, \sqrt {-2 \, x + 1} + \frac {49 \, \sqrt {-2 \, x + 1}}{729 \, {\left (3 \, x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 82, normalized size = 0.68 \begin {gather*} \frac {98\,\sqrt {1-2\,x}}{2187\,\left (2\,x+\frac {4}{3}\right )}+\frac {1526\,\sqrt {1-2\,x}}{729}+\frac {214\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {2\,{\left (1-2\,x\right )}^{5/2}}{27}-\frac {725\,{\left (1-2\,x\right )}^{7/2}}{378}+\frac {125\,{\left (1-2\,x\right )}^{9/2}}{162}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1540{}\mathrm {i}}{2187} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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