Optimal. Leaf size=249 \[ -\frac {776112041 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{230343750 \sqrt {33}}+\frac {2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac {178 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac {601 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac {18034 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{625625}-\frac {11725073 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{56306250}-\frac {776112041 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{506756250}-\frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{460687500 \sqrt {33}} \]
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Rubi [A] time = 0.10, 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 = {101, 154, 158, 113, 119} \[ \frac {2}{65} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac {178 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{10725}+\frac {601 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{160875}-\frac {18034 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{625625}-\frac {11725073 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{56306250}-\frac {776112041 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{506756250}-\frac {776112041 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{230343750 \sqrt {33}}-\frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{460687500 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
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 = 0.41, size = 115, normalized size = 0.46 \[ \frac {51601293223 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (5197919174 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (7016625000 x^5+12374775000 x^4+3047388750 x^3-5775295500 x^2-3548873565 x+325972172\right )\right )}{7601343750 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 165, normalized size = 0.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-6314962500000 x^{8}-15978768750000 x^{7}-9807753375000 x^{6}+6956762962500 x^{5}+10046351241000 x^{4}+1491065725050 x^{3}-2009737437330 x^{2}-570343085580 x -51601293223 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+25989595870 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+58674990960\right )}{456080625000 x^{3}+349661812500 x^{2}-106418812500 x -91216125000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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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