Optimal. Leaf size=209 \[ -\frac {3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac {12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac {132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac {1456961 \sqrt {5 x+3} (1-2 x)^{7/2}}{819200}+\frac {16026571 \sqrt {5 x+3} (1-2 x)^{5/2}}{24576000}+\frac {176292281 \sqrt {5 x+3} (1-2 x)^{3/2}}{98304000}+\frac {1939215091 \sqrt {5 x+3} \sqrt {1-2 x}}{327680000}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{327680000 \sqrt {10}} \]
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Rubi [A] time = 0.08, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \[ -\frac {3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac {12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac {132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac {1456961 \sqrt {5 x+3} (1-2 x)^{7/2}}{819200}+\frac {16026571 \sqrt {5 x+3} (1-2 x)^{5/2}}{24576000}+\frac {176292281 \sqrt {5 x+3} (1-2 x)^{3/2}}{98304000}+\frac {1939215091 \sqrt {5 x+3} \sqrt {1-2 x}}{327680000}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{327680000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}-\frac {1}{80} \int \left (-326-\frac {999 x}{2}\right ) (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {12041 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{3200}\\ &=-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {132451 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{15360}\\ &=-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {1456961 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{102400}\\ &=-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {16026571 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1638400}\\ &=\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {176292281 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{9830400}\\ &=\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {1939215091 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{65536000}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{655360000}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{327680000 \sqrt {5}}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{327680000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 94, normalized size = 0.45 \[ \frac {447958686021 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (1548288000000 x^8+1950105600000 x^7-1168107520000 x^6-2124371456000 x^5+186330348800 x^4+919076261440 x^3+85960565960 x^2-214160462618 x+22414998339\right )}{68812800000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 92, normalized size = 0.44 \[ \frac {1}{6881280000} \, {\left (774144000000 \, x^{7} + 1362124800000 \, x^{6} + 97008640000 \, x^{5} - 1013681408000 \, x^{4} - 413675529600 \, x^{3} + 252700365920 \, x^{2} + 169330465940 \, x - 22414998339\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {21331366001}{6553600000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.90, size = 545, normalized size = 2.61 \[ \frac {3}{114688000000} \, \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 {1}{560000000} \, \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 {937}{7680000000} \, \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 {3083}{960000000} \, \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 {3181}{9600000} \, \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 {87}{40000} \, \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 {27}{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 {54}{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 = 0.01, size = 172, normalized size = 0.82 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+27242496000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1940172800000 \sqrt {-10 x^{2}-x +3}\, x^{5}-20273628160000 \sqrt {-10 x^{2}-x +3}\, x^{4}-8273510592000 \sqrt {-10 x^{2}-x +3}\, x^{3}+5054007318400 \sqrt {-10 x^{2}-x +3}\, x^{2}+3386609318800 \sqrt {-10 x^{2}-x +3}\, x +447958686021 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-448299966780 \sqrt {-10 x^{2}-x +3}\right )}{137625600000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.09, size = 128, normalized size = 0.61 \[ -\frac {9}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {1839}{11200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {12041}{19200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {12041}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {1456961}{614400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1456961}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {176292281}{16384000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21331366001}{6553600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {176292281}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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