Optimal. Leaf size=238 \[ -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{1764 (3 x+2)^6}+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{3252759552 (3 x+2)}+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)^2}+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{41489280 (3 x+2)^3}+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{20744640 (3 x+2)^4}-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{370440 (3 x+2)^5}-\frac {3735929329 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{120472576 \sqrt {7}} \]
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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} \begin {gather*} -\frac {\sqrt {1-2 x} (5 x+3)^{5/2}}{21 (3 x+2)^7}-\frac {59 \sqrt {1-2 x} (5 x+3)^{3/2}}{1764 (3 x+2)^6}+\frac {8818415317 \sqrt {1-2 x} \sqrt {5 x+3}}{3252759552 (3 x+2)}+\frac {84539611 \sqrt {1-2 x} \sqrt {5 x+3}}{232339968 (3 x+2)^2}+\frac {2524471 \sqrt {1-2 x} \sqrt {5 x+3}}{41489280 (3 x+2)^3}+\frac {369409 \sqrt {1-2 x} \sqrt {5 x+3}}{20744640 (3 x+2)^4}-\frac {6577 \sqrt {1-2 x} \sqrt {5 x+3}}{370440 (3 x+2)^5}-\frac {3735929329 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{120472576 \sqrt {7}} \end {gather*}
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 {\sqrt {1-2 x} (3+5 x)^{5/2}}{(2+3 x)^8} \, dx &=-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {1}{21} \int \frac {\left (\frac {19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^7} \, dx\\ &=-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\left (-\frac {783}{4}-2760 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx}{2646}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {-\frac {1154271}{8}-285690 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{277830}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {8684811}{16}-\frac {16623405 x}{4}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{7779240}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {4635547875}{32}-\frac {795208365 x}{4}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{163364040}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {\frac {570867242085}{64}-\frac {133149887325 x}{16}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{2287096560}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {\int \frac {31774078943145}{128 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{16009675920}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {3735929329 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{240945152}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}+\frac {3735929329 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{120472576}\\ &=-\frac {6577 \sqrt {1-2 x} \sqrt {3+5 x}}{370440 (2+3 x)^5}+\frac {369409 \sqrt {1-2 x} \sqrt {3+5 x}}{20744640 (2+3 x)^4}+\frac {2524471 \sqrt {1-2 x} \sqrt {3+5 x}}{41489280 (2+3 x)^3}+\frac {84539611 \sqrt {1-2 x} \sqrt {3+5 x}}{232339968 (2+3 x)^2}+\frac {8818415317 \sqrt {1-2 x} \sqrt {3+5 x}}{3252759552 (2+3 x)}-\frac {59 \sqrt {1-2 x} (3+5 x)^{3/2}}{1764 (2+3 x)^6}-\frac {\sqrt {1-2 x} (3+5 x)^{5/2}}{21 (2+3 x)^7}-\frac {3735929329 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{120472576 \sqrt {7}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 221, normalized size = 0.93 \begin {gather*} \frac {1}{49} \left (\frac {267 (1-2 x)^{3/2} (5 x+3)^{7/2}}{28 (3 x+2)^6}+\frac {3 (1-2 x)^{3/2} (5 x+3)^{7/2}}{(3 x+2)^7}+\frac {6344698752 (1-2 x)^{3/2} (5 x+3)^{7/2}+255169 (3 x+2) \left (115248 \sqrt {1-2 x} (5 x+3)^{7/2}-11 (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 )\right )}{258155520 (3 x+2)^5}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 170, normalized size = 0.71 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {3827535 (1-2 x)^6}{(5 x+3)^6}+\frac {178618300 (1-2 x)^5}{(5 x+3)^5}+\frac {3538428523 (1-2 x)^4}{(5 x+3)^4}-\frac {26128800768 (1-2 x)^3}{(5 x+3)^3}-\frac {166754547323 (1-2 x)^2}{(5 x+3)^2}-\frac {427485708860 (1-2 x)}{5 x+3}-450305665215\right )}{1807088640 \sqrt {5 x+3} \left (\frac {1-2 x}{5 x+3}+7\right )^7}-\frac {3735929329 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{120472576 \sqrt {7}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 161, normalized size = 0.68 \begin {gather*} -\frac {56038939935 \, \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 (3571458203385 \, x^{6} + 14445612678330 \, x^{5} + 24351227238888 \, x^{4} + 21898948566336 \, x^{3} + 11077661454896 \, x^{2} + 2987299350368 \, x + 335335888512\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{25299240960 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.99, size = 542, normalized size = 2.28 \begin {gather*} \frac {3735929329}{16866160640} \, \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 (765507 \, {\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} + 1428946400 \, {\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} + 1132297127360 \, {\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} - 334448649830400 \, {\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} - 85378328229376000 \, {\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} - 8754907317452800000 \, {\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 {368890400944128000000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1475561603776512000000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{180708864 \, {\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}} \end {gather*}
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 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (122557161637845 \sqrt {7}\, x^{7} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+571933420976610 \sqrt {7}\, x^{6} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+50000414847390 \sqrt {-10 x^{2}-x +3}\, x^{6}+1143866841953220 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+202238577496620 \sqrt {-10 x^{2}-x +3}\, x^{5}+1270963157725800 \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+340917181344432 \sqrt {-10 x^{2}-x +3}\, x^{4}+847308771817200 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+306585279928704 \sqrt {-10 x^{2}-x +3}\, x^{3}+338923508726880 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+155087260368544 \sqrt {-10 x^{2}-x +3}\, x^{2}+75316335272640 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+41822190905152 \sqrt {-10 x^{2}-x +3}\, x +7172984311680 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+4694702439168 \sqrt {-10 x^{2}-x +3}\right )}{25299240960 \sqrt {-10 x^{2}-x +3}\, \left (3 x +2\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 295, normalized size = 1.24 \begin {gather*} \frac {3735929329}{1686616064} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {154377245}{90354432} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{147 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} - \frac {191 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4116 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {919 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{96040 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {72203 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{768320 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {2612695 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6453888 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {92626347 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{60236288 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {1142391613 \, \sqrt {-10 \, x^{2} - x + 3}}{361417728 \, {\left (3 \, x + 2\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^8} \,d x \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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