Optimal. Leaf size=162 \[ \frac {11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac {559625 \sqrt {1-2 x}}{1333584 (3 x+2)}-\frac {559625 \sqrt {1-2 x}}{190512 (3 x+2)^2}+\frac {33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac {121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {559625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{666792 \sqrt {21}} \]
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Rubi [A] time = 0.07, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {97, 12, 149, 89, 78, 47, 51, 63, 206} \begin {gather*} \frac {11 (1-2 x)^{3/2} (5 x+3)^3}{27 (3 x+2)^5}-\frac {(1-2 x)^{5/2} (5 x+3)^3}{18 (3 x+2)^6}+\frac {559625 \sqrt {1-2 x}}{1333584 (3 x+2)}-\frac {559625 \sqrt {1-2 x}}{190512 (3 x+2)^2}+\frac {33275 (1-2 x)^{3/2}}{95256 (3 x+2)^3}-\frac {121 (1-2 x)^{3/2}}{4536 (3 x+2)^4}+\frac {559625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{666792 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 47
Rule 51
Rule 63
Rule 78
Rule 89
Rule 97
Rule 149
Rule 206
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^7} \, dx &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {1}{18} \int -\frac {55 (1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}-\frac {55}{18} \int \frac {(1-2 x)^{3/2} x (3+5 x)^2}{(2+3 x)^6} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {11}{54} \int \frac {33 \sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {121}{18} \int \frac {\sqrt {1-2 x} (3+5 x)^2}{(2+3 x)^5} \, dx\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {121 \int \frac {\sqrt {1-2 x} (1125+2100 x)}{(2+3 x)^4} \, dx}{4536}\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {559625 \int \frac {\sqrt {1-2 x}}{(2+3 x)^3} \, dx}{31752}\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {559625 \sqrt {1-2 x}}{190512 (2+3 x)^2}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}-\frac {559625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)^2} \, dx}{190512}\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {559625 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {559625 \sqrt {1-2 x}}{1333584 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}-\frac {559625 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx}{1333584}\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {559625 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {559625 \sqrt {1-2 x}}{1333584 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {559625 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{1333584}\\ &=-\frac {121 (1-2 x)^{3/2}}{4536 (2+3 x)^4}+\frac {33275 (1-2 x)^{3/2}}{95256 (2+3 x)^3}-\frac {559625 \sqrt {1-2 x}}{190512 (2+3 x)^2}+\frac {559625 \sqrt {1-2 x}}{1333584 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{18 (2+3 x)^6}+\frac {11 (1-2 x)^{3/2} (3+5 x)^3}{27 (2+3 x)^5}+\frac {559625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{666792 \sqrt {21}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 52, normalized size = 0.32 \begin {gather*} \frac {(1-2 x)^{7/2} \left (\frac {1764735 \left (110250 x^2+146875 x+48919\right )}{(3 x+2)^6}-161172000 \, _2F_1\left (\frac {7}{2},5;\frac {9}{2};\frac {3}{7}-\frac {6 x}{7}\right )\right )}{4669488810} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.50, size = 97, normalized size = 0.60 \begin {gather*} \frac {\sqrt {1-2 x} \left (308539125 (1-2 x)^5-2983070475 (1-2 x)^4+11598653682 (1-2 x)^3-22803823350 (1-2 x)^2+22842213625 (1-2 x)-9405617375\right )}{666792 (3 (1-2 x)-7)^6}+\frac {559625 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{666792 \sqrt {21}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.41, size = 130, normalized size = 0.80 \begin {gather*} \frac {559625 \, \sqrt {21} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (308539125 \, x^{5} + 720187425 \, x^{4} + 687940758 \, x^{3} + 352611738 \, x^{2} + 102558856 \, x + 13847024\right )} \sqrt {-2 \, x + 1}}{28005264 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.98, size = 132, normalized size = 0.81 \begin {gather*} -\frac {559625}{28005264} \, \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 {308539125 \, {\left (2 \, x - 1\right )}^{5} \sqrt {-2 \, x + 1} + 2983070475 \, {\left (2 \, x - 1\right )}^{4} \sqrt {-2 \, x + 1} + 11598653682 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 22803823350 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 22842213625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 9405617375 \, \sqrt {-2 \, x + 1}}{42674688 \, {\left (3 \, x + 2\right )}^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 84, normalized size = 0.52 \begin {gather*} \frac {559625 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{14002632}-\frac {11664 \left (-\frac {3809125 \left (-2 x +1\right )^{\frac {11}{2}}}{96018048}+\frac {47350325 \left (-2 x +1\right )^{\frac {9}{2}}}{123451776}-\frac {4383467 \left (-2 x +1\right )^{\frac {7}{2}}}{2939328}+\frac {1231175 \left (-2 x +1\right )^{\frac {5}{2}}}{419904}-\frac {66595375 \left (-2 x +1\right )^{\frac {3}{2}}}{22674816}+\frac {27421625 \sqrt {-2 x +1}}{22674816}\right )}{\left (-6 x -4\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 146, normalized size = 0.90 \begin {gather*} -\frac {559625}{28005264} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {308539125 \, {\left (-2 \, x + 1\right )}^{\frac {11}{2}} - 2983070475 \, {\left (-2 \, x + 1\right )}^{\frac {9}{2}} + 11598653682 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 22803823350 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 22842213625 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 9405617375 \, \sqrt {-2 \, x + 1}}{666792 \, {\left (729 \, {\left (2 \, x - 1\right )}^{6} + 10206 \, {\left (2 \, x - 1\right )}^{5} + 59535 \, {\left (2 \, x - 1\right )}^{4} + 185220 \, {\left (2 \, x - 1\right )}^{3} + 324135 \, {\left (2 \, x - 1\right )}^{2} + 605052 \, x - 184877\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 126, normalized size = 0.78 \begin {gather*} \frac {559625\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{14002632}-\frac {\frac {27421625\,\sqrt {1-2\,x}}{1417176}-\frac {66595375\,{\left (1-2\,x\right )}^{3/2}}{1417176}+\frac {1231175\,{\left (1-2\,x\right )}^{5/2}}{26244}-\frac {4383467\,{\left (1-2\,x\right )}^{7/2}}{183708}+\frac {47350325\,{\left (1-2\,x\right )}^{9/2}}{7715736}-\frac {3809125\,{\left (1-2\,x\right )}^{11/2}}{6001128}}{\frac {67228\,x}{81}+\frac {12005\,{\left (2\,x-1\right )}^2}{27}+\frac {6860\,{\left (2\,x-1\right )}^3}{27}+\frac {245\,{\left (2\,x-1\right )}^4}{3}+14\,{\left (2\,x-1\right )}^5+{\left (2\,x-1\right )}^6-\frac {184877}{729}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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