Optimal. Leaf size=81 \[ \frac {139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac {863}{441} \sqrt {1-2 x}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]
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Rubi [A] time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {139 (1-2 x)^{3/2}}{882 (3 x+2)}-\frac {(1-2 x)^{3/2}}{126 (3 x+2)^2}+\frac {863}{441} \sqrt {1-2 x}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 78
Rule 89
Rule 206
Rubi steps
\begin {align*} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {1}{126} \int \frac {\sqrt {1-2 x} (561+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{294} \int \frac {\sqrt {1-2 x}}{2+3 x} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}+\frac {863}{126} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863}{126} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {863}{441} \sqrt {1-2 x}-\frac {(1-2 x)^{3/2}}{126 (2+3 x)^2}+\frac {139 (1-2 x)^{3/2}}{882 (2+3 x)}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 58, normalized size = 0.72 \[ \frac {\sqrt {1-2 x} \left (2100 x^2+2941 x+1025\right )}{126 (3 x+2)^2}-\frac {863 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{63 \sqrt {21}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 74, normalized size = 0.91 \[ \frac {863 \, \sqrt {21} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (2100 \, x^{2} + 2941 \, x + 1025\right )} \sqrt {-2 \, x + 1}}{2646 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.22, size = 77, normalized size = 0.95 \[ \frac {863}{2646} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{756 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 57, normalized size = 0.70 \[ -\frac {863 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{1323}+\frac {50 \sqrt {-2 x +1}}{27}+\frac {-\frac {47 \left (-2 x +1\right )^{\frac {3}{2}}}{21}+\frac {139 \sqrt {-2 x +1}}{27}}{\left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 83, normalized size = 1.02 \[ \frac {863}{2646} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {50}{27} \, \sqrt {-2 \, x + 1} - \frac {423 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 973 \, \sqrt {-2 \, x + 1}}{189 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 62, normalized size = 0.77 \[ \frac {50\,\sqrt {1-2\,x}}{27}-\frac {863\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{1323}+\frac {\frac {139\,\sqrt {1-2\,x}}{243}-\frac {47\,{\left (1-2\,x\right )}^{3/2}}{189}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]
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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