Optimal. Leaf size=135 \[ -\frac {47 (1-2 x)^{3/2} (3 x+2)^3}{25 (5 x+3)}-\frac {(1-2 x)^{5/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac {954}{875} (1-2 x)^{3/2} (3 x+2)^2+\frac {3 (1-2 x)^{3/2} (2403 x+1618)}{6250}+\frac {5559 \sqrt {1-2 x}}{15625}-\frac {5559 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
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Rubi [A] time = 0.04, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {97, 149, 153, 147, 50, 63, 206} \[ -\frac {47 (1-2 x)^{3/2} (3 x+2)^3}{25 (5 x+3)}-\frac {(1-2 x)^{5/2} (3 x+2)^3}{10 (5 x+3)^2}+\frac {954}{875} (1-2 x)^{3/2} (3 x+2)^2+\frac {3 (1-2 x)^{3/2} (2403 x+1618)}{6250}+\frac {5559 \sqrt {1-2 x}}{15625}-\frac {5559 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 97
Rule 147
Rule 149
Rule 153
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{(3+5 x)^3} \, dx &=-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}+\frac {1}{10} \int \frac {(-1-33 x) (1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^2} \, dx\\ &=-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}+\frac {1}{50} \int \frac {(-33-1908 x) \sqrt {1-2 x} (2+3 x)^2}{3+5 x} \, dx\\ &=\frac {954}{875} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}-\frac {\int \frac {\sqrt {1-2 x} (2+3 x) (2310+16821 x)}{3+5 x} \, dx}{1750}\\ &=\frac {954}{875} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}+\frac {3 (1-2 x)^{3/2} (1618+2403 x)}{6250}+\frac {5559 \int \frac {\sqrt {1-2 x}}{3+5 x} \, dx}{6250}\\ &=\frac {5559 \sqrt {1-2 x}}{15625}+\frac {954}{875} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}+\frac {3 (1-2 x)^{3/2} (1618+2403 x)}{6250}+\frac {61149 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{31250}\\ &=\frac {5559 \sqrt {1-2 x}}{15625}+\frac {954}{875} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}+\frac {3 (1-2 x)^{3/2} (1618+2403 x)}{6250}-\frac {61149 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{31250}\\ &=\frac {5559 \sqrt {1-2 x}}{15625}+\frac {954}{875} (1-2 x)^{3/2} (2+3 x)^2-\frac {(1-2 x)^{5/2} (2+3 x)^3}{10 (3+5 x)^2}-\frac {47 (1-2 x)^{3/2} (2+3 x)^3}{25 (3+5 x)}+\frac {3 (1-2 x)^{3/2} (1618+2403 x)}{6250}-\frac {5559 \sqrt {\frac {11}{5}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{15625}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 73, normalized size = 0.54 \[ \frac {\frac {5 \sqrt {1-2 x} \left (1350000 x^5-27000 x^4-1506900 x^3+1651030 x^2+2637795 x+770444\right )}{(5 x+3)^2}-77826 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{1093750} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 95, normalized size = 0.70 \[ \frac {38913 \, \sqrt {11} \sqrt {5} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 5 \, {\left (1350000 \, x^{5} - 27000 \, x^{4} - 1506900 \, x^{3} + 1651030 \, x^{2} + 2637795 \, x + 770444\right )} \sqrt {-2 \, x + 1}}{1093750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 118, normalized size = 0.87 \[ \frac {27}{875} \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + \frac {54}{3125} \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} + \frac {186}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5559}{156250} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {46}{125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (945 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2101 \, \sqrt {-2 \, x + 1}\right )}}{62500 \, {\left (5 \, x + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.62 \[ -\frac {5559 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{78125}-\frac {27 \left (-2 x +1\right )^{\frac {7}{2}}}{875}+\frac {54 \left (-2 x +1\right )^{\frac {5}{2}}}{3125}+\frac {186 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}+\frac {46 \sqrt {-2 x +1}}{125}+\frac {\frac {2079 \left (-2 x +1\right )^{\frac {3}{2}}}{3125}-\frac {23111 \sqrt {-2 x +1}}{15625}}{\left (-10 x -6\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.11, size = 110, normalized size = 0.81 \[ -\frac {27}{875} \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + \frac {54}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + \frac {186}{3125} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + \frac {5559}{156250} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {46}{125} \, \sqrt {-2 \, x + 1} + \frac {11 \, {\left (945 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 2101 \, \sqrt {-2 \, x + 1}\right )}}{15625 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 92, normalized size = 0.68 \[ \frac {46\,\sqrt {1-2\,x}}{125}+\frac {186\,{\left (1-2\,x\right )}^{3/2}}{3125}+\frac {54\,{\left (1-2\,x\right )}^{5/2}}{3125}-\frac {27\,{\left (1-2\,x\right )}^{7/2}}{875}-\frac {\frac {23111\,\sqrt {1-2\,x}}{390625}-\frac {2079\,{\left (1-2\,x\right )}^{3/2}}{78125}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}}+\frac {\sqrt {55}\,\mathrm {atan}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{11}\right )\,5559{}\mathrm {i}}{78125} \]
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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