Optimal. Leaf size=116 \[ \frac {160}{2401 \sqrt {1-2 x}}-\frac {16}{147 (1-2 x)^{3/2} (3 x+2)}+\frac {160}{3087 (1-2 x)^{3/2}}-\frac {16}{147 (1-2 x)^{3/2} (3 x+2)^2}+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac {160 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401} \]
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Rubi [A] time = 0.03, antiderivative size = 130, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {78, 51, 63, 206} \[ -\frac {240 \sqrt {1-2 x}}{2401 (3 x+2)}-\frac {80 \sqrt {1-2 x}}{343 (3 x+2)^2}+\frac {64}{147 \sqrt {1-2 x} (3 x+2)^2}+\frac {64}{441 (1-2 x)^{3/2} (3 x+2)^2}+\frac {1}{63 (1-2 x)^{3/2} (3 x+2)^3}-\frac {160 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 78
Rule 206
Rubi steps
\begin {align*} \int \frac {3+5 x}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {32}{21} \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^3} \, dx\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {32}{21} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^3} \, dx\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {64}{147 \sqrt {1-2 x} (2+3 x)^2}+\frac {160}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^3} \, dx\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {64}{147 \sqrt {1-2 x} (2+3 x)^2}-\frac {80 \sqrt {1-2 x}}{343 (2+3 x)^2}+\frac {240}{343} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {64}{147 \sqrt {1-2 x} (2+3 x)^2}-\frac {80 \sqrt {1-2 x}}{343 (2+3 x)^2}-\frac {240 \sqrt {1-2 x}}{2401 (2+3 x)}+\frac {240 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{2401}\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {64}{147 \sqrt {1-2 x} (2+3 x)^2}-\frac {80 \sqrt {1-2 x}}{343 (2+3 x)^2}-\frac {240 \sqrt {1-2 x}}{2401 (2+3 x)}-\frac {240 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{2401}\\ &=\frac {1}{63 (1-2 x)^{3/2} (2+3 x)^3}+\frac {64}{441 (1-2 x)^{3/2} (2+3 x)^2}+\frac {64}{147 \sqrt {1-2 x} (2+3 x)^2}-\frac {80 \sqrt {1-2 x}}{343 (2+3 x)^2}-\frac {240 \sqrt {1-2 x}}{2401 (2+3 x)}-\frac {160 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{2401}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 42, normalized size = 0.36 \[ \frac {256 \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+\frac {343}{(3 x+2)^3}}{21609 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 120, normalized size = 1.03 \[ \frac {240 \, \sqrt {7} \sqrt {3} {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 7 \, {\left (25920 \, x^{4} + 28800 \, x^{3} - 4464 \, x^{2} - 11280 \, x - 2237\right )} \sqrt {-2 \, x + 1}}{50421 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.24, size = 95, normalized size = 0.82 \[ \frac {80}{16807} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {8 \, {\left (1620 \, {\left (2 \, x - 1\right )}^{4} + 10080 \, {\left (2 \, x - 1\right )}^{3} + 19404 \, {\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 75, normalized size = 0.65 \[ -\frac {160 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{16807}+\frac {88}{7203 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {776}{16807 \sqrt {-2 x +1}}+\frac {\frac {9288 \left (-2 x +1\right )^{\frac {5}{2}}}{16807}-\frac {960 \left (-2 x +1\right )^{\frac {3}{2}}}{343}+\frac {1200 \sqrt {-2 x +1}}{343}}{\left (-6 x -4\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, size = 110, normalized size = 0.95 \[ \frac {80}{16807} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {8 \, {\left (1620 \, {\left (2 \, x - 1\right )}^{4} + 10080 \, {\left (2 \, x - 1\right )}^{3} + 19404 \, {\left (2 \, x - 1\right )}^{2} + 18816 \, x - 13181\right )}}{7203 \, {\left (27 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} - 189 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} + 441 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 343 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 92, normalized size = 0.79 \[ -\frac {\frac {1024\,x}{1323}+\frac {352\,{\left (2\,x-1\right )}^2}{441}+\frac {1280\,{\left (2\,x-1\right )}^3}{3087}+\frac {160\,{\left (2\,x-1\right )}^4}{2401}-\frac {2152}{3969}}{\frac {343\,{\left (1-2\,x\right )}^{3/2}}{27}-\frac {49\,{\left (1-2\,x\right )}^{5/2}}{3}+7\,{\left (1-2\,x\right )}^{7/2}-{\left (1-2\,x\right )}^{9/2}}-\frac {160\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{16807} \]
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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