Optimal. Leaf size=121 \[ -\frac {(1-2 x)^{5/2} (5 x+3)^3}{3 (3 x+2)}+\frac {55}{81} (1-2 x)^{5/2} (5 x+3)^2+\frac {220}{729} (1-2 x)^{3/2}-\frac {22}{567} (1-2 x)^{5/2} (100 x+69)+\frac {1540}{729} \sqrt {1-2 x}-\frac {1540}{729} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.04, 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 = {97, 12, 153, 147, 50, 63, 206} \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 50
Rule 63
Rule 97
Rule 147
Rule 153
Rule 206
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} \operatorname {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 = 0.06, size = 73, normalized size = 0.60 \begin {gather*} \frac {\frac {3 \sqrt {1-2 x} \left (189000 x^5-17100 x^4-159714 x^3+25275 x^2+65558 x+13759\right )}{3 x+2}-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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IntegrateAlgebraic [A] time = 0.13, size = 99, normalized size = 0.82 \begin {gather*} \frac {\left (23625 (1-2 x)^5-113850 (1-2 x)^4+139293 (1-2 x)^3+3696 (1-2 x)^2+43120 (1-2 x)-150920\right ) \sqrt {1-2 x}}{10206 (3 (1-2 x)-7)}-\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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fricas [A] time = 1.25, 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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giac [A] time = 1.12, 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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maple [A] time = 0.01, size = 81, normalized size = 0.67 \begin {gather*} -\frac {1540 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{2187}+\frac {125 \left (-2 x +1\right )^{\frac {9}{2}}}{162}-\frac {725 \left (-2 x +1\right )^{\frac {7}{2}}}{378}+\frac {2 \left (-2 x +1\right )^{\frac {5}{2}}}{27}+\frac {214 \left (-2 x +1\right )^{\frac {3}{2}}}{729}+\frac {1526 \sqrt {-2 x +1}}{729}-\frac {98 \sqrt {-2 x +1}}{2187 \left (-2 x -\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, 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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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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sympy [F(-1)] 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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