Optimal. Leaf size=249 \[ -\frac {776112041 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{506756250}-\frac {11725073 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{56306250}-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{460687500 \sqrt {33}}-\frac {776112041 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{230343750 \sqrt {33}} \]
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Rubi [A]
time = 0.07, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {103, 159, 164,
114, 120} \begin {gather*} -\frac {776112041 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{230343750 \sqrt {33}}-\frac {51601293223 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{460687500 \sqrt {33}}+\frac {2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac {178 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac {601 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac {18034 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{625625}-\frac {11725073 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{56306250}-\frac {776112041 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{506756250} \end {gather*}
Antiderivative was successfully verified.
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx &=\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {2}{65} \int \left (-\frac {71}{2}-\frac {89 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {4 \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2} \left (-1082+\frac {1803 x}{4}\right )}{\sqrt {1-2 x}} \, dx}{10725}\\ &=\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {4 \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {661845}{8}+\frac {243459 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{482625}\\ &=-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-8651787-\frac {105525657 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16891875}\\ &=-\frac {11725073 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{56306250}-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {9078479379}{16}+\frac {6985008369 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{253378125}\\ &=-\frac {776112041 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{506756250}-\frac {11725073 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{56306250}-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {4 \int \frac {-\frac {36751750227}{2}-\frac {464411639007 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2280403125}\\ &=-\frac {776112041 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{506756250}-\frac {11725073 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{56306250}-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {776112041 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{460687500}+\frac {51601293223 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5067562500}\\ &=-\frac {776112041 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{506756250}-\frac {11725073 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{56306250}-\frac {18034 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{625625}+\frac {601 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{160875}+\frac {178 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{10725}+\frac {2}{65} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{460687500 \sqrt {33}}-\frac {776112041 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{230343750 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 7.99, size = 115, normalized size = 0.46 \begin {gather*} \frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (3 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (325972172-3548873565 x-5775295500 x^2+3047388750 x^3+12374775000 x^4+7016625000 x^5\right )+5197919174 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{7601343750 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 163, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (6314962500000 x^{8}+15978768750000 x^{7}+9807753375000 x^{6}+25611697353 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-51601293223 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6956762962500 x^{5}-10046351241000 x^{4}-1491065725050 x^{3}+2009737437330 x^{2}+570343085580 x -58674990960\right )}{15202687500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(163\) |
risch | \(\frac {\left (7016625000 x^{5}+12374775000 x^{4}+3047388750 x^{3}-5775295500 x^{2}-3548873565 x +325972172\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{506756250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {8167055606 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{13935796875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {51601293223 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{55743187500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(266\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {180 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}-\frac {3492 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{143}-\frac {12899 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2145}+\frac {197446 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17325}+\frac {78863857 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11261250}-\frac {162986086 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{253378125}+\frac {8167055606 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5320940625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {51601293223 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21283762500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(312\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.22, size = 48, normalized size = 0.19 \begin {gather*} -\frac {1}{506756250} \, {\left (7016625000 \, x^{5} + 12374775000 \, x^{4} + 3047388750 \, x^{3} - 5775295500 \, x^{2} - 3548873565 \, x + 325972172\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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