Optimal. Leaf size=179 \[ -\frac {\sqrt {5 x+3} (11603280 x+12923401) (1-2 x)^{7/2}}{22400000}-\frac {3}{70} (3 x+2)^3 \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {271 (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}}{2800}+\frac {9526549 \sqrt {5 x+3} (1-2 x)^{5/2}}{96000000}+\frac {104792039 \sqrt {5 x+3} (1-2 x)^{3/2}}{384000000}+\frac {1152712429 \sqrt {5 x+3} \sqrt {1-2 x}}{1280000000}+\frac {12679836719 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280000000 \sqrt {10}} \]
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Rubi [A] time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \begin {gather*} -\frac {\sqrt {5 x+3} (11603280 x+12923401) (1-2 x)^{7/2}}{22400000}-\frac {3}{70} (3 x+2)^3 \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {271 (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}}{2800}+\frac {9526549 \sqrt {5 x+3} (1-2 x)^{5/2}}{96000000}+\frac {104792039 \sqrt {5 x+3} (1-2 x)^{3/2}}{384000000}+\frac {1152712429 \sqrt {5 x+3} \sqrt {1-2 x}}{1280000000}+\frac {12679836719 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280000000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^4}{\sqrt {3+5 x}} \, dx &=-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{70} \int \frac {\left (-250-\frac {813 x}{2}\right ) (1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\\ &=-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(1-2 x)^{5/2} (2+3 x) \left (\frac {44553}{2}+\frac {145041 x}{4}\right )}{\sqrt {3+5 x}} \, dx}{4200}\\ &=-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {9526549 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{6400000}\\ &=\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {104792039 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{38400000}\\ &=\frac {104792039 (1-2 x)^{3/2} \sqrt {3+5 x}}{384000000}+\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {1152712429 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{256000000}\\ &=\frac {1152712429 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000000}+\frac {104792039 (1-2 x)^{3/2} \sqrt {3+5 x}}{384000000}+\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {12679836719 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2560000000}\\ &=\frac {1152712429 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000000}+\frac {104792039 (1-2 x)^{3/2} \sqrt {3+5 x}}{384000000}+\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {12679836719 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1280000000 \sqrt {5}}\\ &=\frac {1152712429 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000000}+\frac {104792039 (1-2 x)^{3/2} \sqrt {3+5 x}}{384000000}+\frac {9526549 (1-2 x)^{5/2} \sqrt {3+5 x}}{96000000}-\frac {271 (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}}{2800}-\frac {3}{70} (1-2 x)^{7/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (12923401+11603280 x)}{22400000}+\frac {12679836719 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 89, normalized size = 0.50 \begin {gather*} \frac {266276571099 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (497664000000 x^7+374630400000 x^6-607578624000 x^5-403491052800 x^4+322052135360 x^3+174481885240 x^2-100668418342 x+920643741\right )}{268800000000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 173, normalized size = 0.97 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {3115146890625 (1-2 x)^6}{(5 x+3)^6}+\frac {8216741575000 (1-2 x)^5}{(5 x+3)^5}+\frac {8699119040500 (1-2 x)^4}{(5 x+3)^4}+\frac {3366728294400 (1-2 x)^3}{(5 x+3)^3}-\frac {1509767485520 (1-2 x)^2}{(5 x+3)^2}-\frac {213394697600 (1-2 x)}{5 x+3}-12803681856\right )}{26880000000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^7}-\frac {12679836719 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1280000000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.69, size = 87, normalized size = 0.49 \begin {gather*} \frac {1}{26880000000} \, {\left (248832000000 \, x^{6} + 311731200000 \, x^{5} - 147923712000 \, x^{4} - 275707382400 \, x^{3} + 23172376480 \, x^{2} + 98827130860 \, x - 920643741\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {12679836719}{25600000000} \, \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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giac [B] time = 1.85, size = 446, normalized size = 2.49 \begin {gather*} \frac {27}{448000000000} \, \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 {9}{640000000} \, \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 {27}{320000000} \, \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 {11}{400000} \, \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 {13}{15000} \, \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 {2}{125} \, \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 {8}{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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maple [A] time = 0.01, size = 155, normalized size = 0.87 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}+6234624000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-2958474240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-5514147648000 \sqrt {-10 x^{2}-x +3}\, x^{3}+463447529600 \sqrt {-10 x^{2}-x +3}\, x^{2}+1976542617200 \sqrt {-10 x^{2}-x +3}\, x +266276571099 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-18412874820 \sqrt {-10 x^{2}-x +3}\right )}{537600000000 \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.34, size = 126, normalized size = 0.70 \begin {gather*} \frac {324}{35} \, \sqrt {-10 \, x^{2} - x + 3} x^{6} + \frac {4059}{350} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} - \frac {192609}{35000} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {28719519}{2800000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + \frac {144827353}{168000000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {4941356543}{1344000000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {12679836719}{25600000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {306881247}{8960000000} \, \sqrt {-10 \, x^{2} - x + 3} \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 = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{\sqrt {5\,x+3}} \,d x \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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