Optimal. Leaf size=249 \[ -\frac {2295970088 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{4606875 \sqrt {33}}-\frac {1}{13} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac {41}{143} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac {14303 \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac {221673 \sqrt {1-2 x} (5 x+3)^{5/2} \sqrt {3 x+2}}{50050}-\frac {138809831 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{4504500}-\frac {2295970088 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{10135125}-\frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}} \]
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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} \[ -\frac {1}{13} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{7/2}-\frac {41}{143} \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{5/2}-\frac {14303 \sqrt {1-2 x} (5 x+3)^{5/2} (3 x+2)^{3/2}}{12870}-\frac {221673 \sqrt {1-2 x} (5 x+3)^{5/2} \sqrt {3 x+2}}{50050}-\frac {138809831 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{4504500}-\frac {2295970088 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{10135125}-\frac {2295970088 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4606875 \sqrt {33}}-\frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {1}{13} \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2} \left (\frac {257}{2}+205 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {1}{715} \int \frac {\left (-\frac {45285}{2}-\frac {71515 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {12799775}{4}+\frac {9975285 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{32175}\\ &=-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {\int \frac {\left (-\frac {1364822645}{4}-\frac {2082147465 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1126125}\\ &=-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {179052019605}{8}+34439551320 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{16891875}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {5798711966295}{8}-\frac {9159406524465 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{152026875}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}+\frac {1147985044 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4606875}+\frac {610627101631 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{405405000}\\ &=-\frac {2295970088 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{10135125}-\frac {138809831 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{4504500}-\frac {221673 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{50050}-\frac {14303 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{12870}-\frac {41}{143} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{13} \sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{5/2}-\frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{36855000 \sqrt {33}}-\frac {2295970088 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{4606875 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 115, normalized size = 0.46 \[ \frac {610627101631 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (61511810003 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (2104987500 x^5+9351247500 x^4+18620894250 x^3+22592085750 x^2+19961825445 x+16001700059\right )\right )}{608107500 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-1894488750000 x^{8}-9868564125000 x^{7}-22769118225000 x^{6}-30838634482500 x^{5}-27960569725500 x^{4}-20079090637650 x^{3}-2782614262260 x^{2}+6953485592490 x -610627101631 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+307559050015 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2880306010620\right )}{36486450000 x^{3}+27972945000 x^{2}-8513505000 x -7297290000} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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