Optimal. Leaf size=238 \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}+\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{464679936 (3 x+2)}+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{33191424 (3 x+2)^2}+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{5927040 (3 x+2)^3}+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^4}-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{52920 (3 x+2)^5}-\frac {6219452877 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{17210368 \sqrt {7}} \]
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Rubi [A] time = 0.10, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {97, 149, 151, 12, 93, 204} \[ \frac {37 \sqrt {1-2 x} (5 x+3)^{3/2}}{252 (3 x+2)^6}-\frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{21 (3 x+2)^7}+\frac {14677525921 \sqrt {1-2 x} \sqrt {5 x+3}}{464679936 (3 x+2)}+\frac {140331343 \sqrt {1-2 x} \sqrt {5 x+3}}{33191424 (3 x+2)^2}+\frac {4014523 \sqrt {1-2 x} \sqrt {5 x+3}}{5927040 (3 x+2)^3}+\frac {341917 \sqrt {1-2 x} \sqrt {5 x+3}}{2963520 (3 x+2)^4}-\frac {9901 \sqrt {1-2 x} \sqrt {5 x+3}}{52920 (3 x+2)^5}-\frac {6219452877 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{17210368 \sqrt {7}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 97
Rule 149
Rule 151
Rule 204
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^8} \, dx &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^7} \, dx\\ &=-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {1}{378} \int \frac {\sqrt {3+5 x} \left (-\frac {4941}{4}+1860 x\right )}{\sqrt {1-2 x} (2+3 x)^6} \, dx\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {\int \frac {-\frac {190077}{8}+28470 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{39690}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {\int \frac {-\frac {43274943}{16}+\frac {15386265 x}{4}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{1111320}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {\int \frac {-\frac {7990392375}{32}+\frac {1264574745 x}{4}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{23337720}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {\int \frac {-\frac {951748581105}{64}+\frac {221021865225 x}{16}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{326728080}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}+\frac {14677525921 \sqrt {1-2 x} \sqrt {3+5 x}}{464679936 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {\int -\frac {52896446718885}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2287096560}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}+\frac {14677525921 \sqrt {1-2 x} \sqrt {3+5 x}}{464679936 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}+\frac {6219452877 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{34420736}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}+\frac {14677525921 \sqrt {1-2 x} \sqrt {3+5 x}}{464679936 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}+\frac {6219452877 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{17210368}\\ &=-\frac {9901 \sqrt {1-2 x} \sqrt {3+5 x}}{52920 (2+3 x)^5}+\frac {341917 \sqrt {1-2 x} \sqrt {3+5 x}}{2963520 (2+3 x)^4}+\frac {4014523 \sqrt {1-2 x} \sqrt {3+5 x}}{5927040 (2+3 x)^3}+\frac {140331343 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)^2}+\frac {14677525921 \sqrt {1-2 x} \sqrt {3+5 x}}{464679936 (2+3 x)}-\frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{21 (2+3 x)^7}+\frac {37 \sqrt {1-2 x} (3+5 x)^{3/2}}{252 (2+3 x)^6}-\frac {6219452877 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{17210368 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 176, normalized size = 0.74 \[ \frac {1}{49} \left (\frac {141599 \left (7 \sqrt {1-2 x} \sqrt {5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )-43923 \sqrt {7} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )\right )}{2458624 (3 x+2)^4}+\frac {11841 (1-2 x)^{5/2} (5 x+3)^{5/2}}{280 (3 x+2)^5}+\frac {333 (1-2 x)^{5/2} (5 x+3)^{5/2}}{28 (3 x+2)^6}+\frac {3 (1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^7}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.27, size = 161, normalized size = 0.68 \[ -\frac {31097264385 \, \sqrt {7} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (1981465999335 \, x^{6} + 8014272743430 \, x^{5} + 13509190228248 \, x^{4} + 12147806104256 \, x^{3} + 6146173476816 \, x^{2} + 1658923773088 \, x + 186609267072\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1204725760 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.95, size = 542, normalized size = 2.28 \[ \frac {6219452877}{2409451520} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {14641 \, \sqrt {10} {\left (424797 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{13} + 792954400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} - 748492373440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 270037116518400 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 49241484970496000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 4873941796864000000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {204705555468288000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {818822221873152000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{8605184 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 394, normalized size = 1.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (68009717209995 \sqrt {7}\, x^{7} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+317378680313310 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+27740523990690 \sqrt {-10 x^{2}-x +3}\, x^{6}+634757360626620 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+112199818408020 \sqrt {-10 x^{2}-x +3}\, x^{5}+705285956251800 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+189128663195472 \sqrt {-10 x^{2}-x +3}\, x^{4}+470190637501200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+170069285459584 \sqrt {-10 x^{2}-x +3}\, x^{3}+188076255000480 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+86046428675424 \sqrt {-10 x^{2}-x +3}\, x^{2}+41794723333440 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+23224932823232 \sqrt {-10 x^{2}-x +3}\, x +3980449841280 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+2612529739008 \sqrt {-10 x^{2}-x +3}\right )}{1204725760 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 324, normalized size = 1.36 \[ \frac {1167483755}{90354432} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{49 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {333 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1372 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {11841 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{13720 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {424797 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{153664 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {15717489 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2151296 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {700490253 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {9509080845}{60236288} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {6219452877}{240945152} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {8378271231}{120472576} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {2771517227 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{361417728 \, {\left (3 \, x + 2\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^8} \,d x \]
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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