Optimal. Leaf size=150 \[ -\frac {1}{20} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{7/2}-\frac {\sqrt {1-2 x} (18960 x+37439) (5 x+3)^{7/2}}{32000}-\frac {2012291 \sqrt {1-2 x} (5 x+3)^{5/2}}{384000}-\frac {22135201 \sqrt {1-2 x} (5 x+3)^{3/2}}{614400}-\frac {243487211 \sqrt {1-2 x} \sqrt {5 x+3}}{819200}+\frac {2678359321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{819200 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac {1}{20} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{7/2}-\frac {\sqrt {1-2 x} (18960 x+37439) (5 x+3)^{7/2}}{32000}-\frac {2012291 \sqrt {1-2 x} (5 x+3)^{5/2}}{384000}-\frac {22135201 \sqrt {1-2 x} (5 x+3)^{3/2}}{614400}-\frac {243487211 \sqrt {1-2 x} \sqrt {5 x+3}}{819200}+\frac {2678359321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{819200 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{60} \int \frac {\left (-381-\frac {1185 x}{2}\right ) (2+3 x) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2012291 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{64000}\\ &=-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {22135201 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{153600}\\ &=-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {243487211 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{409600}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1638400}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{819200 \sqrt {5}}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{819200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 93, normalized size = 0.62 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (138240000 x^5+615168000 x^4+1229558400 x^3+1505007200 x^2+1362715220 x+1202896557\right )+8035077963 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{24576000 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 82, normalized size = 0.55 \[ -\frac {1}{2457600} \, {\left (138240000 \, x^{5} + 615168000 \, x^{4} + 1229558400 \, x^{3} + 1505007200 \, x^{2} + 1362715220 \, x + 1202896557\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {2678359321}{16384000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 81, normalized size = 0.54 \[ -\frac {1}{122880000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (20 \, x + 41\right )} {\left (5 \, x + 3\right )} + 2903\right )} {\left (5 \, x + 3\right )} + 2012291\right )} {\left (5 \, x + 3\right )} + 110676005\right )} {\left (5 \, x + 3\right )} + 3652308165\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 40175389815 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 0.92 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-2764800000 \sqrt {-10 x^{2}-x +3}\, x^{5}-12303360000 \sqrt {-10 x^{2}-x +3}\, x^{4}-24591168000 \sqrt {-10 x^{2}-x +3}\, x^{3}-30100144000 \sqrt {-10 x^{2}-x +3}\, x^{2}-27254304400 \sqrt {-10 x^{2}-x +3}\, x +8035077963 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-24057931140 \sqrt {-10 x^{2}-x +3}\right )}{49152000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 109, normalized size = 0.73 \[ -\frac {225}{4} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} - \frac {4005}{16} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {128079}{256} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {1881259}{3072} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {68135761}{122880} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {2678359321}{16384000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {400965519}{819200} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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