Optimal. Leaf size=100 \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {720 x+487}{294 \sqrt {1-2 x} (3 x+2)^2}+\frac {905 \sqrt {1-2 x}}{2058 (3 x+2)}+\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
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Rubi [A] time = 0.03, antiderivative size = 100, 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 = {98, 144, 51, 63, 206} \[ \frac {11 (5 x+3)^2}{21 (1-2 x)^{3/2} (3 x+2)^2}-\frac {720 x+487}{294 \sqrt {1-2 x} (3 x+2)^2}+\frac {905 \sqrt {1-2 x}}{2058 (3 x+2)}+\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 98
Rule 144
Rule 206
Rubi steps
\begin {align*} \int \frac {(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)^3} \, dx &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {1}{21} \int \frac {(3+5 x) (31+15 x)}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}-\frac {905}{294} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}-\frac {905 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2058}\\ &=\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2058}\\ &=\frac {905 \sqrt {1-2 x}}{2058 (2+3 x)}+\frac {11 (3+5 x)^2}{21 (1-2 x)^{3/2} (2+3 x)^2}-\frac {487+720 x}{294 \sqrt {1-2 x} (2+3 x)^2}+\frac {905 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{1029 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 62, normalized size = 0.62 \[ -\frac {-7240 \left (6 x^2+x-2\right )^2 \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {3}{7}-\frac {6 x}{7}\right )-343 \left (875 x^2+1303 x+128\right )}{21609 (1-2 x)^{3/2} (3 x+2)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 100, normalized size = 1.00 \[ \frac {905 \, \sqrt {21} {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (10860 \, x^{3} + 33410 \, x^{2} + 29593 \, x + 8103\right )} \sqrt {-2 \, x + 1}}{43218 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 89, normalized size = 0.89 \[ -\frac {905}{43218} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {121 \, {\left (36 \, x + 59\right )}}{7203 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {597 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1379 \, \sqrt {-2 \, x + 1}}{28812 \, {\left (3 \, x + 2\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 66, normalized size = 0.66 \[ \frac {905 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{21609}+\frac {1331}{1029 \left (-2 x +1\right )^{\frac {3}{2}}}-\frac {726}{2401 \sqrt {-2 x +1}}-\frac {18 \left (-\frac {199 \left (-2 x +1\right )^{\frac {3}{2}}}{18}+\frac {1379 \sqrt {-2 x +1}}{54}\right )}{2401 \left (-6 x -4\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.29, size = 92, normalized size = 0.92 \[ -\frac {905}{43218} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2715 \, {\left (2 \, x - 1\right )}^{3} + 24850 \, {\left (2 \, x - 1\right )}^{2} + 142296 \, x - 5929}{1029 \, {\left (9 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 42 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 49 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 71, normalized size = 0.71 \[ \frac {905\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{21609}+\frac {\frac {968\,x}{63}+\frac {3550\,{\left (2\,x-1\right )}^2}{1323}+\frac {905\,{\left (2\,x-1\right )}^3}{3087}-\frac {121}{189}}{\frac {49\,{\left (1-2\,x\right )}^{3/2}}{9}-\frac {14\,{\left (1-2\,x\right )}^{5/2}}{3}+{\left (1-2\,x\right )}^{7/2}} \]
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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