Optimal. Leaf size=150 \[ -\frac {\sqrt {5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac {1}{20} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {276493 \sqrt {5 x+3} (1-2 x)^{5/2}}{4800000}+\frac {3041423 \sqrt {5 x+3} (1-2 x)^{3/2}}{19200000}+\frac {33455653 \sqrt {5 x+3} \sqrt {1-2 x}}{64000000}+\frac {368012183 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000000 \sqrt {10}} \]
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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} \begin {gather*} -\frac {\sqrt {5 x+3} (47280 x+52951) (1-2 x)^{7/2}}{160000}-\frac {1}{20} (3 x+2)^2 \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {276493 \sqrt {5 x+3} (1-2 x)^{5/2}}{4800000}+\frac {3041423 \sqrt {5 x+3} (1-2 x)^{3/2}}{19200000}+\frac {33455653 \sqrt {5 x+3} \sqrt {1-2 x}}{64000000}+\frac {368012183 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^3}{\sqrt {3+5 x}} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{60} \int \frac {\left (-183-\frac {591 x}{2}\right ) (1-2 x)^{5/2} (2+3 x)}{\sqrt {3+5 x}} \, dx\\ &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {276493 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{320000}\\ &=\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {3041423 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1920000}\\ &=\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {33455653 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{12800000}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{128000000}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64000000 \sqrt {5}}\\ &=\frac {33455653 \sqrt {1-2 x} \sqrt {3+5 x}}{64000000}+\frac {3041423 (1-2 x)^{3/2} \sqrt {3+5 x}}{19200000}+\frac {276493 (1-2 x)^{5/2} \sqrt {3+5 x}}{4800000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {(1-2 x)^{7/2} \sqrt {3+5 x} (52951+47280 x)}{160000}+\frac {368012183 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 84, normalized size = 0.56 \begin {gather*} \frac {1104036549 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (1382400000 x^6-13824000 x^5-1797292800 x^4+261623360 x^3+903127240 x^2-254844442 x-39899709\right )}{1920000000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.29, size = 157, normalized size = 1.05 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {2515321875 (1-2 x)^5}{(5 x+3)^5}+\frac {5225236250 (1-2 x)^4}{(5 x+3)^4}+\frac {2748468600 (1-2 x)^3}{(5 x+3)^3}-\frac {2189824560 (1-2 x)^2}{(5 x+3)^2}-\frac {376030480 (1-2 x)}{5 x+3}-26543328\right )}{192000000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {368012183 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{64000000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.75, size = 82, normalized size = 0.55 \begin {gather*} \frac {1}{192000000} \, {\left (691200000 \, x^{5} + 338688000 \, x^{4} - 729302400 \, x^{3} - 233839520 \, x^{2} + 334643860 \, x + 39899709\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {368012183}{1280000000} \, \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 = 1.60, size = 356, normalized size = 2.37 \begin {gather*} \frac {9}{3200000000} \, \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 {9}{80000000} \, \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 {3}{640000} \, \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 {29}{60000} \, \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 {1}{500} \, \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 {4}{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 = 0.92 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (13824000000 \sqrt {-10 x^{2}-x +3}\, x^{5}+6773760000 \sqrt {-10 x^{2}-x +3}\, x^{4}-14586048000 \sqrt {-10 x^{2}-x +3}\, x^{3}-4676790400 \sqrt {-10 x^{2}-x +3}\, x^{2}+6692877200 \sqrt {-10 x^{2}-x +3}\, x +1104036549 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+797994180 \sqrt {-10 x^{2}-x +3}\right )}{3840000000 \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 = 1.27, size = 109, normalized size = 0.73 \begin {gather*} \frac {18}{5} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} + \frac {441}{250} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {75969}{20000} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {1461497}{1200000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {16732193}{9600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {368012183}{1280000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {13299903}{64000000} \, \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 )}^3}{\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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