Optimal. Leaf size=201 \[ -\frac {3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac {1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac {3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac {135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac {1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {5 x+3}}{131072000}+\frac {180773237579 \sqrt {1-2 x} \sqrt {5 x+3}}{1310720000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1310720000 \sqrt {10}} \]
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Rubi [A] time = 0.07, antiderivative size = 201, 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 = {100, 153, 147, 50, 54, 216} \[ -\frac {3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac {1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac {3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac {135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac {1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {5 x+3}}{131072000}+\frac {180773237579 \sqrt {1-2 x} \sqrt {5 x+3}}{1310720000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1310720000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {1}{80} \int \left (-452-\frac {1419 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\\ &=-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}+\frac {\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{5/2} \left (\frac {176225}{2}+\frac {548541 x}{4}\right ) \, dx}{5600}\\ &=-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {135817609 \int \sqrt {1-2 x} (3+5 x)^{5/2} \, dx}{2560000}\\ &=-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1493993699 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{8192000}\\ &=-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {16433930689 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{32768000}\\ &=-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {180773237579 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{262144000}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2621440000}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1310720000 \sqrt {5}}\\ &=\frac {180773237579 \sqrt {1-2 x} \sqrt {3+5 x}}{1310720000}-\frac {16433930689 (1-2 x)^{3/2} \sqrt {3+5 x}}{131072000}-\frac {1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac {135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac {1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac {3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac {1988505613369 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1310720000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 94, normalized size = 0.47 \[ \frac {41758617880749 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (13934592000000 x^8+54709862400000 x^7+84655134720000 x^6+58945337856000 x^5+6568746067200 x^4-18691732680640 x^3-14866250752760 x^2-7954905832042 x+5973304472091\right )}{275251200000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 92, normalized size = 0.46 \[ \frac {1}{27525120000} \, {\left (6967296000000 \, x^{7} + 30838579200000 \, x^{6} + 57746856960000 \, x^{5} + 58346097408000 \, x^{4} + 32457421737600 \, x^{3} + 6882844528480 \, x^{2} - 3991703112140 \, x - 5973304472091\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1988505613369}{26214400000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.61, size = 545, normalized size = 2.71 \[ \frac {27}{458752000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {603}{71680000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {5769}{2560000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {30649}{320000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1831}{300000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {933}{5000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {297}{125} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {216}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 172, normalized size = 0.86 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (139345920000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+616771584000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1154937139200000 \sqrt {-10 x^{2}-x +3}\, x^{5}+1166921948160000 \sqrt {-10 x^{2}-x +3}\, x^{4}+649148434752000 \sqrt {-10 x^{2}-x +3}\, x^{3}+137656890569600 \sqrt {-10 x^{2}-x +3}\, x^{2}-79834062242800 \sqrt {-10 x^{2}-x +3}\, x +41758617880749 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-119466089441820 \sqrt {-10 x^{2}-x +3}\right )}{550502400000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 138, normalized size = 0.69 \[ -\frac {405}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{5} - \frac {49059}{448} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {739881}{3584} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {80346831}{358400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {4513921183}{28672000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {26326737569}{344064000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {16433930689}{65536000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1988505613369}{26214400000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {16433930689}{1310720000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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