Optimal. Leaf size=216 \[ \frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}} \]
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Rubi [A]
time = 0.05, antiderivative size = 216, 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 = {102, 152, 52,
56, 222} \begin {gather*} \frac {932023556783 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6553600000 \sqrt {10}}-\frac {1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac {5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac {(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac {63658463 \sqrt {5 x+3} (1-2 x)^{7/2}}{16384000}+\frac {700243093 \sqrt {5 x+3} (1-2 x)^{5/2}}{491520000}+\frac {7702674023 \sqrt {5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac {84729414253 \sqrt {5 x+3} \sqrt {1-2 x}}{6553600000} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 56
Rule 102
Rule 152
Rule 222
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{90} \int \left (-393-\frac {1221 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {526103 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{64000}\\ &=-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {5787133 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{307200}\\ &=-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {63658463 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{2048000}\\ &=-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {700243093 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{32768000}\\ &=\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {7702674023 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{196608000}\\ &=\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {84729414253 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1310720000}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{13107200000}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6553600000 \sqrt {5}}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 103, normalized size = 0.48 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-4490138164311+11708962285875 x+81324587821180 x^2+65091129546400 x^3-209312726736000 x^4-440233726720000 x^5-52760857600000 x^6+605463552000000 x^7+636088320000000 x^8+206438400000000 x^9\right )-19572494692443 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1376256000000 \sqrt {3+5 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 189, normalized size = 0.88
method | result | size |
risch | \(-\frac {\left (41287680000000 x^{8}+102445056000000 x^{7}+59625676800000 x^{6}-46327577600000 x^{5}-60250198784000 x^{4}-5712426076800 x^{3}+16445681555360 x^{2}+6397508631020 x -1496712721437\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{137625600000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {932023556783 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{131072000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\) | \(128\) |
default | \(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (825753600000000 \sqrt {-10 x^{2}-x +3}\, x^{8}+2048901120000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+1192513536000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-926551552000000 x^{5} \sqrt {-10 x^{2}-x +3}-1205003975680000 x^{4} \sqrt {-10 x^{2}-x +3}-114248521536000 x^{3} \sqrt {-10 x^{2}-x +3}+328913631107200 x^{2} \sqrt {-10 x^{2}-x +3}+19572494692443 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+127950172620400 x \sqrt {-10 x^{2}-x +3}-29934254428740 \sqrt {-10 x^{2}-x +3}\right )}{2752512000000 \sqrt {-10 x^{2}-x +3}}\) | \(189\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 145, normalized size = 0.67 \begin {gather*} -\frac {3}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} - \frac {1047}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {111537}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {526103}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {526103}{7680000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {63658463}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {63658463}{245760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7702674023}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {932023556783}{131072000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7702674023}{6553600000} \, \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 = 0.81, size = 97, normalized size = 0.45 \begin {gather*} \frac {1}{137625600000} \, {\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {932023556783}{131072000000} \, \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. 653 vs.
\(2 (159) = 318\).
time = 0.70, size = 653, normalized size = 3.02 \begin {gather*} \frac {3}{2293760000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (160 \, x - 779\right )} {\left (5 \, x + 3\right )} + 297993\right )} {\left (5 \, x + 3\right )} - 16954963\right )} {\left (5 \, x + 3\right )} + 311501761\right )} {\left (5 \, x + 3\right )} - 15396301917\right )} {\left (5 \, x + 3\right )} + 129214816941\right )} {\left (5 \, x + 3\right )} - 1465300159701\right )} {\left (5 \, x + 3\right )} + 5967262275723\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10642307691015 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{40960000000} \, \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 {2217}{358400000000} \, \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 {121}{38400000000} \, \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 {15709}{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 {1429}{1920000} \, \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 {129}{20000} \, \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}{500} \, \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 {108}{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 )}^3\,{\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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