Optimal. Leaf size=147 \[ -\frac {275 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {1441}{27} \sqrt {1-2 x} (5 x+3)^2-\frac {22}{243} \sqrt {1-2 x} (1885 x+578)-\frac {41360 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \]
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Rubi [A] time = 0.06, 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 = {97, 12, 149, 153, 147, 63, 206} \[ -\frac {275 \sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)}+\frac {55 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^2}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{9 (3 x+2)^3}+\frac {1441}{27} \sqrt {1-2 x} (5 x+3)^2-\frac {22}{243} \sqrt {1-2 x} (1885 x+578)-\frac {41360 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{243 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 97
Rule 147
Rule 149
Rule 153
Rule 206
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} \operatorname {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 [C] time = 0.03, size = 59, normalized size = 0.40 \[ \frac {(1-2 x)^{7/2} \left (49632 (3 x+2)^3 \, _2F_1\left (2,\frac {7}{2};\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )-343 \left (11025 x^2+14858 x+5003\right )\right )}{453789 (3 x+2)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 99, normalized size = 0.67 \[ \frac {20680 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (16200 \, x^{5} - 20700 \, x^{4} + 87030 \, x^{3} + 289719 \, x^{2} + 229336 \, x + 56141\right )} \sqrt {-2 \, x + 1}}{5103 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 118, normalized size = 0.80 \[ \frac {50}{81} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {37377 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}}{2916 \, {\left (3 \, x + 2\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.57 \[ -\frac {41360 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{5103}+\frac {50 \left (-2 x +1\right )^{\frac {5}{2}}}{81}+\frac {2050 \left (-2 x +1\right )^{\frac {3}{2}}}{729}+\frac {16570 \sqrt {-2 x +1}}{729}+\frac {-\frac {8306 \left (-2 x +1\right )^{\frac {5}{2}}}{81}+\frac {344260 \left (-2 x +1\right )^{\frac {3}{2}}}{729}-\frac {396410 \sqrt {-2 x +1}}{729}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 119, normalized size = 0.81 \[ \frac {50}{81} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {2050}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {20680}{5103} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {16570}{729} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (37377 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 172130 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 198205 \, \sqrt {-2 \, x + 1}\right )}}{729 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \]
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 \[ \frac {16570\,\sqrt {1-2\,x}}{729}+\frac {2050\,{\left (1-2\,x\right )}^{3/2}}{729}+\frac {50\,{\left (1-2\,x\right )}^{5/2}}{81}+\frac {\frac {396410\,\sqrt {1-2\,x}}{19683}-\frac {344260\,{\left (1-2\,x\right )}^{3/2}}{19683}+\frac {8306\,{\left (1-2\,x\right )}^{5/2}}{2187}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,41360{}\mathrm {i}}{5103} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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