Optimal. Leaf size=140 \[ -\frac {129 \sqrt {1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac {2643 \sqrt {1-2 x} (3 x+2)^3}{1750}+\frac {1404 \sqrt {1-2 x} (3 x+2)^2}{3125}+\frac {9 \sqrt {1-2 x} (1375 x+32)}{31250}-\frac {12279 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]
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Rubi [A] time = 0.05, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {97, 149, 153, 147, 63, 206} \[ -\frac {129 \sqrt {1-2 x} (3 x+2)^4}{50 (5 x+3)}-\frac {(1-2 x)^{3/2} (3 x+2)^4}{10 (5 x+3)^2}+\frac {2643 \sqrt {1-2 x} (3 x+2)^3}{1750}+\frac {1404 \sqrt {1-2 x} (3 x+2)^2}{3125}+\frac {9 \sqrt {1-2 x} (1375 x+32)}{31250}-\frac {12279 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 97
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^4}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(6-33 x) \sqrt {1-2 x} (2+3 x)^3}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {1}{50} \int \frac {(870-2643 x) (2+3 x)^3}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}-\frac {\int \frac {(2+3 x)^2 (-5397+19656 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{1750}\\ &=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {\int \frac {(33978-86625 x) (2+3 x)}{\sqrt {1-2 x} (3+5 x)} \, dx}{43750}\\ &=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}+\frac {12279 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{31250}\\ &=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31250}\\ &=\frac {1404 \sqrt {1-2 x} (2+3 x)^2}{3125}+\frac {2643 \sqrt {1-2 x} (2+3 x)^3}{1750}-\frac {(1-2 x)^{3/2} (2+3 x)^4}{10 (3+5 x)^2}-\frac {129 \sqrt {1-2 x} (2+3 x)^4}{50 (3+5 x)}+\frac {9 \sqrt {1-2 x} (32+1375 x)}{31250}-\frac {12279 \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625 \sqrt {55}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 73, normalized size = 0.52 \[ \frac {-\frac {55 \sqrt {1-2 x} \left (2025000 x^5+3267000 x^4-496350 x^3-2120880 x^2-489445 x+96776\right )}{(5 x+3)^2}-171906 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{12031250} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 89, normalized size = 0.64 \[ \frac {85953 \, \sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \, {\left (2025000 \, x^{5} + 3267000 \, x^{4} - 496350 \, x^{3} - 2120880 \, x^{2} - 489445 \, x + 96776\right )} \sqrt {-2 \, x + 1}}{12031250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.08, size = 118, normalized size = 0.84 \[ -\frac {81}{1750} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} - \frac {1107}{6250} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{62500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.60 \[ -\frac {12279 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{859375}+\frac {81 \left (-2 x +1\right )^{\frac {7}{2}}}{1750}-\frac {1107 \left (-2 x +1\right )^{\frac {5}{2}}}{6250}+\frac {36 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}+\frac {228 \sqrt {-2 x +1}}{3125}+\frac {\frac {259 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}-\frac {2871 \sqrt {-2 x +1}}{15625}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 110, normalized size = 0.79 \[ \frac {81}{1750} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - \frac {1107}{6250} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {36}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {12279}{1718750} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {228}{3125} \, \sqrt {-2 \, x + 1} + \frac {1295 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2871 \, \sqrt {-2 \, x + 1}}{15625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 92, normalized size = 0.66 \[ \frac {228\,\sqrt {1-2\,x}}{3125}+\frac {36\,{\left (1-2\,x\right )}^{3/2}}{3125}-\frac {1107\,{\left (1-2\,x\right )}^{5/2}}{6250}+\frac {81\,{\left (1-2\,x\right )}^{7/2}}{1750}-\frac {\frac {2871\,\sqrt {1-2\,x}}{390625}-\frac {259\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,12279{}\mathrm {i}}{859375} \]
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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