Optimal. Leaf size=116 \[ -\frac {1}{8} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {11}{240} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {121}{960} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {1331 \sqrt {5 x+3} \sqrt {1-2 x}}{3200}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \begin {gather*} -\frac {1}{8} \sqrt {5 x+3} (1-2 x)^{7/2}+\frac {11}{240} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {121}{960} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {1331 \sqrt {5 x+3} \sqrt {1-2 x}}{3200}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 216
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx &=-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {11}{16} \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {11}{240} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {121}{96} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {121}{960} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {11}{240} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {1331}{640} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {1331 \sqrt {1-2 x} \sqrt {3+5 x}}{3200}+\frac {121}{960} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {11}{240} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {14641 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {1331 \sqrt {1-2 x} \sqrt {3+5 x}}{3200}+\frac {121}{960} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {11}{240} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {14641 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3200 \sqrt {5}}\\ &=\frac {1331 \sqrt {1-2 x} \sqrt {3+5 x}}{3200}+\frac {121}{960} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {11}{240} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {1}{8} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 74, normalized size = 0.64 \begin {gather*} \frac {43923 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (19200 x^4-34880 x^3+18680 x^2+5866 x-4443\right )}{96000 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 125, normalized size = 1.08 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {375 (1-2 x)^3}{(5 x+3)^3}-\frac {730 (1-2 x)^2}{(5 x+3)^2}-\frac {220 (1-2 x)}{5 x+3}-24\right )}{9600 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {14641 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{3200 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 72, normalized size = 0.62 \begin {gather*} \frac {1}{9600} \, {\left (9600 \, x^{3} - 12640 \, x^{2} + 3020 \, x + 4443\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {14641}{64000} \, \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 = 1.25, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{480000} \, \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 {1}{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 {7}{2000} \, \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 {3}{50} \, \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 = 0.00, size = 104, normalized size = 0.90 \begin {gather*} \frac {14641 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{64000 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (-2 x +1\right )^{\frac {5}{2}} \left (5 x +3\right )^{\frac {3}{2}}}{20}+\frac {11 \left (-2 x +1\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}}}{120}+\frac {121 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{800}-\frac {1331 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{3200} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 70, normalized size = 0.60 \begin {gather*} -\frac {1}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {17}{120} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {121}{160} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {14641}{64000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {121}{3200} \, \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}\,\sqrt {5\,x+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.68, size = 269, normalized size = 2.32 \begin {gather*} \begin {cases} \frac {10 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{\sqrt {10 x - 5}} - \frac {253 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{6 \sqrt {10 x - 5}} + \frac {15367 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{240 \sqrt {10 x - 5}} - \frac {177023 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{4800 \sqrt {10 x - 5}} + \frac {14641 i \sqrt {x + \frac {3}{5}}}{3200 \sqrt {10 x - 5}} - \frac {14641 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{32000} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {14641 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{32000} - \frac {10 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{\sqrt {5 - 10 x}} + \frac {253 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{6 \sqrt {5 - 10 x}} - \frac {15367 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{240 \sqrt {5 - 10 x}} + \frac {177023 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{4800 \sqrt {5 - 10 x}} - \frac {14641 \sqrt {x + \frac {3}{5}}}{3200 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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