Optimal. Leaf size=209 \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)}+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{16595712 (3 x+2)^2}-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^3}-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{1481760 (3 x+2)^4}+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{26460 (3 x+2)^5}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8605184 \sqrt {7}} \]
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Rubi [A] time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {98, 149, 151, 12, 93, 204} \[ \frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{126 (3 x+2)^6}+\frac {122343637 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)}+\frac {958171 \sqrt {1-2 x} \sqrt {5 x+3}}{16595712 (3 x+2)^2}-\frac {71369 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^3}-\frac {149951 \sqrt {1-2 x} \sqrt {5 x+3}}{1481760 (3 x+2)^4}+\frac {503 \sqrt {1-2 x} \sqrt {5 x+3}}{26460 (3 x+2)^5}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{8605184 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 98
Rule 149
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx &=\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {1}{126} \int \frac {\left (-\frac {1179}{2}-1010 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {394523}{4}-166690 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{13230}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {5498457}{8}-\frac {2249265 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{370440}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {73502625}{16}-\frac {7493745 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{7779240}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int \frac {-\frac {2940587895}{32}+\frac {503039775 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{108909360}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {\int -\frac {149044934115}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{762365520}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac {52573169 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{17210368}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}+\frac {52573169 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{8605184}\\ &=\frac {503 \sqrt {1-2 x} \sqrt {3+5 x}}{26460 (2+3 x)^5}-\frac {149951 \sqrt {1-2 x} \sqrt {3+5 x}}{1481760 (2+3 x)^4}-\frac {71369 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^3}+\frac {958171 \sqrt {1-2 x} \sqrt {3+5 x}}{16595712 (2+3 x)^2}+\frac {122343637 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)}+\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{126 (2+3 x)^6}-\frac {52573169 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{8605184 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 193, normalized size = 0.92 \[ \frac {1}{42} \left (\frac {591 \sqrt {1-2 x} (5 x+3)^{7/2}}{70 (3 x+2)^5}+\frac {3 \sqrt {1-2 x} (5 x+3)^{7/2}}{(3 x+2)^6}+\frac {352839984 \sqrt {1-2 x} (5 x+3)^{7/2}-39499 (3 x+2) \left (2744 \sqrt {1-2 x} (5 x+3)^{5/2}+55 (3 x+2) \left (7 \sqrt {1-2 x} \sqrt {5 x+3} (169 x+108)+363 \sqrt {7} (3 x+2)^2 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )\right )}{21512960 (3 x+2)^4}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 146, normalized size = 0.70 \[ -\frac {788597535 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (16516390995 \, x^{5} + 55658284380 \, x^{4} + 74931979536 \, x^{3} + 50261760608 \, x^{2} + 16771747280 \, x + 2225100096\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1807088640 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.64, size = 484, normalized size = 2.32 \[ \frac {52573169}{1204725760} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (118497 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 188015240 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 122630175360 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 17238395059200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 3670540357120000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {197895383347200000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {791581533388800000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{12907776 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 346, normalized size = 1.66 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (574887603015 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2299550412060 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+231229473930 \sqrt {-10 x^{2}-x +3}\, x^{5}+3832584020100 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+779215981320 \sqrt {-10 x^{2}-x +3}\, x^{4}+3406741351200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1049047713504 \sqrt {-10 x^{2}-x +3}\, x^{3}+1703370675600 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+703664648512 \sqrt {-10 x^{2}-x +3}\, x^{2}+454232180160 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+234804461920 \sqrt {-10 x^{2}-x +3}\, x +50470242240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+31151401344 \sqrt {-10 x^{2}-x +3}\right )}{1807088640 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 230, normalized size = 1.10 \[ \frac {52573169}{120472576} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{378 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {853 \, \sqrt {-10 \, x^{2} - x + 3}}{26460 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac {149951 \, \sqrt {-10 \, x^{2} - x + 3}}{1481760 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac {71369 \, \sqrt {-10 \, x^{2} - x + 3}}{2963520 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {958171 \, \sqrt {-10 \, x^{2} - x + 3}}{16595712 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {122343637 \, \sqrt {-10 \, x^{2} - x + 3}}{232339968 \, {\left (3 \, x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^7} \,d x \]
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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