Optimal. Leaf size=209 \[ \frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{327680000 \sqrt {10}} \]
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Rubi [A]
time = 0.05, 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 = {92, 81, 52, 56,
222} \begin {gather*} \frac {21331366001 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{327680000 \sqrt {10}}-\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 81
Rule 92
Rule 222
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 \text {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 = 0.29, size = 98, normalized size = 0.47 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-67244995017+395916406125 x+1604753427460 x^2+22475240800 x^3-5109421872000 x^4-4777381120000 x^5+4571417600000 x^6+9133056000000 x^7+3870720000000 x^8\right )-447958686021 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{68812800000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 172, normalized size = 0.82
method | result | size |
risch | \(-\frac {\left (774144000000 x^{7}+1362124800000 x^{6}+97008640000 x^{5}-1013681408000 x^{4}-413675529600 x^{3}+252700365920 x^{2}+169330465940 x -22414998339\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6881280000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {21331366001 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6553600000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(123\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+27242496000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+1940172800000 x^{5} \sqrt {-10 x^{2}-x +3}-20273628160000 x^{4} \sqrt {-10 x^{2}-x +3}-8273510592000 x^{3} \sqrt {-10 x^{2}-x +3}+5054007318400 x^{2} \sqrt {-10 x^{2}-x +3}+447958686021 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+3386609318800 x \sqrt {-10 x^{2}-x +3}-448299966780 \sqrt {-10 x^{2}-x +3}\right )}{137625600000 \sqrt {-10 x^{2}-x +3}}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 128, normalized size = 0.61 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.53, size = 92, normalized size = 0.44 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 545 vs.
\(2 (152) = 304\).
time = 0.76, size = 545, normalized size = 2.61 \begin {gather*} \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 )} \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 {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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