Optimal. Leaf size=143 \[ -\frac {3}{50} (3 x+2) \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {369 \sqrt {5 x+3} (1-2 x)^{7/2}}{4000}+\frac {4907 \sqrt {5 x+3} (1-2 x)^{5/2}}{120000}+\frac {53977 \sqrt {5 x+3} (1-2 x)^{3/2}}{480000}+\frac {593747 \sqrt {5 x+3} \sqrt {1-2 x}}{1600000}+\frac {6531217 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 143, 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 = {90, 80, 50, 54, 216} \begin {gather*} -\frac {3}{50} (3 x+2) \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {369 \sqrt {5 x+3} (1-2 x)^{7/2}}{4000}+\frac {4907 \sqrt {5 x+3} (1-2 x)^{5/2}}{120000}+\frac {53977 \sqrt {5 x+3} (1-2 x)^{3/2}}{480000}+\frac {593747 \sqrt {5 x+3} \sqrt {1-2 x}}{1600000}+\frac {6531217 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx &=-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{50} \int \frac {\left (-116-\frac {369 x}{2}\right ) (1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {4907 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{8000}\\ &=\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {53977 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{48000}\\ &=\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {593747 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{320000}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200000}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600000 \sqrt {5}}\\ &=\frac {593747 \sqrt {1-2 x} \sqrt {3+5 x}}{1600000}+\frac {53977 (1-2 x)^{3/2} \sqrt {3+5 x}}{480000}+\frac {4907 (1-2 x)^{5/2} \sqrt {3+5 x}}{120000}-\frac {369 (1-2 x)^{7/2} \sqrt {3+5 x}}{4000}-\frac {3}{50} (1-2 x)^{7/2} (2+3 x) \sqrt {3+5 x}+\frac {6531217 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 79, normalized size = 0.55 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (-13824000 x^5+11347200 x^4+10295360 x^3-13024760 x^2+387158 x+1498491\right )+19593651 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{48000000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 141, normalized size = 0.99 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {7280625 (1-2 x)^4}{(5 x+3)^4}+\frac {5910500 (1-2 x)^3}{(5 x+3)^3}-\frac {12561920 (1-2 x)^2}{(5 x+3)^2}-\frac {2747920 (1-2 x)}{5 x+3}-235536\right )}{4800000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {6531217 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{1600000 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.44, size = 77, normalized size = 0.54 \begin {gather*} \frac {1}{4800000} \, {\left (6912000 \, x^{4} - 2217600 \, x^{3} - 6256480 \, x^{2} + 3384140 \, x + 1498491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {6531217}{32000000} \, \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 = 1.44, size = 275, normalized size = 1.92 \begin {gather*} \frac {3}{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 {1}{800000} \, \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 {23}{120000} \, \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 {2}{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 = 121, normalized size = 0.85 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}-44352000 \sqrt {-10 x^{2}-x +3}\, x^{3}-125129600 \sqrt {-10 x^{2}-x +3}\, x^{2}+67682800 \sqrt {-10 x^{2}-x +3}\, x +19593651 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+29969820 \sqrt {-10 x^{2}-x +3}\right )}{96000000 \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 = 1.13, size = 92, normalized size = 0.64 \begin {gather*} \frac {36}{25} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {231}{500} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {39103}{30000} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {169207}{240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {6531217}{32000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {499497}{1600000} \, \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 )}^2}{\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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