Optimal. Leaf size=147 \[ \frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {22}{243} \sqrt {1-2 x} (578+1885 x)-\frac {41360 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \]
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Rubi [A]
time = 0.04, antiderivative size = 147, 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, 154,
158, 152, 65, 212} \begin {gather*} -\frac {275 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {1441}{27} \sqrt {1-2 x} (5 x+3)^2-\frac {22}{243} \sqrt {1-2 x} (1885 x+578)-\frac {41360 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 99
Rule 152
Rule 154
Rule 158
Rule 212
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^4} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {1}{9} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}-\frac {55}{9} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^3} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}+\frac {55}{54} \int \frac {\sqrt {1-2 x} (3+5 x)^2 (6+54 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {55}{162} \int \frac {(3+5 x)^2 (-684+2358 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}+\frac {11}{486} \int \frac {(3+5 x) (-2232+13572 x)}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {22}{243} \sqrt {1-2 x} (578+1885 x)+\frac {20680}{243} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {22}{243} \sqrt {1-2 x} (578+1885 x)-\frac {20680}{243} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1441}{27} \sqrt {1-2 x} (3+5 x)^2-\frac {(1-2 x)^{5/2} (3+5 x)^3}{9 (2+3 x)^3}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^2}-\frac {275 \sqrt {1-2 x} (3+5 x)^3}{9 (2+3 x)}-\frac {22}{243} \sqrt {1-2 x} (578+1885 x)-\frac {41360 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 73, normalized size = 0.50 \begin {gather*} \frac {\frac {21 \sqrt {1-2 x} \left (56141+229336 x+289719 x^2+87030 x^3-20700 x^4+16200 x^5\right )}{(2+3 x)^3}-41360 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{5103} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 84, normalized size = 0.57
method | result | size |
risch | \(-\frac {32400 x^{6}-57600 x^{5}+194760 x^{4}+492408 x^{3}+168953 x^{2}-117054 x -56141}{243 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {41360 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(66\) |
derivativedivides | \(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(84\) |
default | \(\frac {50 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {2050 \left (1-2 x \right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {1-2 x}}{729}+\frac {-\frac {8306 \left (1-2 x \right )^{\frac {5}{2}}}{81}+\frac {344260 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {1-2 x}}{729}}{\left (-4-6 x \right )^{3}}-\frac {41360 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{5103}\) | \(84\) |
trager | \(\frac {\left (16200 x^{5}-20700 x^{4}+87030 x^{3}+289719 x^{2}+229336 x +56141\right ) \sqrt {1-2 x}}{243 \left (2+3 x \right )^{3}}+\frac {20680 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{5103}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 119, normalized size = 0.81 \begin {gather*} \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (37377 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\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 = 1.01, size = 99, normalized size = 0.67 \begin {gather*} \frac {20680 \, \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 (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt {-2 \, x + 1}}{5103 \, {\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 = 0.58, size = 118, normalized size = 0.80 \begin {gather*} \frac {50}{81} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \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 {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {37377 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}}{2916 \, {\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 = 0.05, size = 100, normalized size = 0.68 \begin {gather*} \frac {16570\,\sqrt {1-2\,x}}{729}+\frac {2050\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {50\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {\frac {396410\,\sqrt {1-2\,x}}{19683}-\frac {344260\,{\left (1-2\,x\right )}^{3/2}}{19683}+\frac {8306\,{\left (1-2\,x\right )}^{5/2}}{2187}}{\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 )\,41360{}\mathrm {i}}{5103} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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