Optimal. Leaf size=109 \[ \frac {143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac {211}{441} (1-2 x)^{5/2}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {1055}{81} \sqrt {1-2 x}-\frac {1055}{81} \sqrt {\frac {7}{3}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \begin {gather*} \frac {143 (1-2 x)^{7/2}}{882 (3 x+2)}-\frac {(1-2 x)^{7/2}}{126 (3 x+2)^2}+\frac {211}{441} (1-2 x)^{5/2}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {1055}{81} \sqrt {1-2 x}-\frac {1055}{81} \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 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {(1-2 x)^{5/2} (557+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{294} \int \frac {(1-2 x)^{5/2}}{2+3 x} \, dx\\ &=\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{126} \int \frac {(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {1055}{54} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}+\frac {7385}{162} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {7385}{162} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {1055}{81} \sqrt {1-2 x}+\frac {1055}{567} (1-2 x)^{3/2}+\frac {211}{441} (1-2 x)^{5/2}-\frac {(1-2 x)^{7/2}}{126 (2+3 x)^2}+\frac {143 (1-2 x)^{7/2}}{882 (2+3 x)}-\frac {1055}{81} \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 = 0.05, size = 68, normalized size = 0.62 \begin {gather*} \frac {1}{486} \left (\frac {3 \sqrt {1-2 x} \left (2160 x^4-3960 x^3+12828 x^2+25987 x+10007\right )}{(3 x+2)^2}-2110 \sqrt {21} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 90, normalized size = 0.83 \begin {gather*} \frac {\left (270 (1-2 x)^4-90 (1-2 x)^3+5064 (1-2 x)^2-36925 (1-2 x)+51695\right ) \sqrt {1-2 x}}{81 (3 (1-2 x)-7)^2}-\frac {1055}{81} \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 = 0.92, size = 90, normalized size = 0.83 \begin {gather*} \frac {1055 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) + 3 \, {\left (2160 \, x^{4} - 3960 \, x^{3} + 12828 \, x^{2} + 25987 \, x + 10007\right )} \sqrt {-2 \, x + 1}}{486 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.15, size = 102, normalized size = 0.94 \begin {gather*} \frac {10}{27} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \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 {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{324 \, {\left (3 \, x + 2\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 75, normalized size = 0.69 \begin {gather*} -\frac {1055 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{243}+\frac {10 \left (-2 x +1\right )^{\frac {5}{2}}}{27}+\frac {130 \left (-2 x +1\right )^{\frac {3}{2}}}{81}+\frac {1006 \sqrt {-2 x +1}}{81}+\frac {-\frac {1043 \left (-2 x +1\right )^{\frac {3}{2}}}{81}+\frac {2401 \sqrt {-2 x +1}}{81}}{\left (-6 x -4\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 101, normalized size = 0.93 \begin {gather*} \frac {10}{27} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {130}{81} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {1055}{486} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {1006}{81} \, \sqrt {-2 \, x + 1} - \frac {7 \, {\left (149 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 343 \, \sqrt {-2 \, x + 1}\right )}}{81 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 82, normalized size = 0.75 \begin {gather*} \frac {1006\,\sqrt {1-2\,x}}{81}+\frac {130\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {10\,{\left (1-2\,x\right )}^{5/2}}{27}+\frac {\frac {2401\,\sqrt {1-2\,x}}{729}-\frac {1043\,{\left (1-2\,x\right )}^{3/2}}{729}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}}+\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,1055{}\mathrm {i}}{243} \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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