Optimal. Leaf size=160 \[ -\frac {1}{20} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {63}{400} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {2079 \sqrt {5 x+3} (1-2 x)^{7/2}}{6400}+\frac {7623 \sqrt {5 x+3} (1-2 x)^{5/2}}{64000}+\frac {83853 \sqrt {5 x+3} (1-2 x)^{3/2}}{256000}+\frac {2767149 \sqrt {5 x+3} \sqrt {1-2 x}}{2560000}+\frac {30438639 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2560000 \sqrt {10}} \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \begin {gather*} -\frac {1}{20} (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {63}{400} (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {2079 \sqrt {5 x+3} (1-2 x)^{7/2}}{6400}+\frac {7623 \sqrt {5 x+3} (1-2 x)^{5/2}}{64000}+\frac {83853 \sqrt {5 x+3} (1-2 x)^{3/2}}{256000}+\frac {2767149 \sqrt {5 x+3} \sqrt {1-2 x}}{2560000}+\frac {30438639 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2560000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {63}{40} \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {2079}{800} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {22869 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {83853 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {2767149 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{512000}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{5120000}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2560000 \sqrt {5}}\\ &=\frac {2767149 \sqrt {1-2 x} \sqrt {3+5 x}}{2560000}+\frac {83853 (1-2 x)^{3/2} \sqrt {3+5 x}}{256000}+\frac {7623 (1-2 x)^{5/2} \sqrt {3+5 x}}{64000}-\frac {2079 (1-2 x)^{7/2} \sqrt {3+5 x}}{6400}-\frac {63}{400} (1-2 x)^{7/2} (3+5 x)^{3/2}-\frac {1}{20} (1-2 x)^{7/2} (3+5 x)^{5/2}+\frac {30438639 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2560000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 84, normalized size = 0.52 \begin {gather*} \frac {30438639 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (51200000 x^6-8704000 x^5-59500800 x^4+15200960 x^3+25975640 x^2-8971662 x-717399\right )}{25600000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 157, normalized size = 0.98 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {590625 (1-2 x)^5}{(5 x+3)^5}+\frac {1338750 (1-2 x)^4}{(5 x+3)^4}+\frac {1042600 (1-2 x)^3}{(5 x+3)^3}-\frac {498960 (1-2 x)^2}{(5 x+3)^2}-\frac {85680 (1-2 x)}{5 x+3}-6048\right )}{2560000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {30438639 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{2560000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 82, normalized size = 0.51 \begin {gather*} \frac {1}{2560000} \, {\left (25600000 \, x^{5} + 8448000 \, x^{4} - 25526400 \, x^{3} - 5162720 \, x^{2} + 10406460 \, x + 717399\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {30438639}{51200000} \, \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 = 1.49, size = 356, normalized size = 2.22 \begin {gather*} \frac {1}{128000000} \, \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 {13}{48000000} \, \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 {137}{9600000} \, \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 {17}{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 {3}{400} \, \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 {9}{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 = 138, normalized size = 0.86 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (512000000 \sqrt {-10 x^{2}-x +3}\, x^{5}+168960000 \sqrt {-10 x^{2}-x +3}\, x^{4}-510528000 \sqrt {-10 x^{2}-x +3}\, x^{3}-103254400 \sqrt {-10 x^{2}-x +3}\, x^{2}+208129200 \sqrt {-10 x^{2}-x +3}\, x +30438639 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+14347980 \sqrt {-10 x^{2}-x +3}\right )}{51200000 \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 = 1.26, size = 99, normalized size = 0.62 \begin {gather*} \frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {13}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {693}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {693}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {251559}{128000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {30438639}{51200000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {251559}{2560000} \, \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 {\left (1-2\,x\right )}^{5/2}\,\left (3\,x+2\right )\,{\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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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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