Optimal. Leaf size=121 \[ -\frac {3}{40} (3 x+2) \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {301 \sqrt {5 x+3} (1-2 x)^{3/2}}{3200}+\frac {9933 \sqrt {5 x+3} \sqrt {1-2 x}}{32000}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32000 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {3}{40} (3 x+2) \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {301 \sqrt {5 x+3} (1-2 x)^{3/2}}{3200}+\frac {9933 \sqrt {5 x+3} \sqrt {1-2 x}}{32000}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32000 \sqrt {10}} \]
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)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx &=-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-112-\frac {357 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {301}{320} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {9933 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{32000 \sqrt {5}}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{32000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.61 \[ \frac {10 \sqrt {5 x+3} \left (57600 x^4-9920 x^3-59480 x^2+18254 x+3383\right )+109263 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{320000 \sqrt {1-2 x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 72, normalized size = 0.60 \[ -\frac {1}{32000} \, {\left (28800 \, x^{3} + 9440 \, x^{2} - 25020 \, x - 3383\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {109263}{640000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 203, normalized size = 1.68 \[ -\frac {3}{1600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {2}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 104, normalized size = 0.86 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-576000 \sqrt {-10 x^{2}-x +3}\, x^{3}-188800 \sqrt {-10 x^{2}-x +3}\, x^{2}+500400 \sqrt {-10 x^{2}-x +3}\, x +109263 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+67660 \sqrt {-10 x^{2}-x +3}\right )}{640000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.17, size = 75, normalized size = 0.62 \[ -\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {59}{200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {1251}{1600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {109263}{640000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3383}{32000} \, \sqrt {-10 \, x^{2} - x + 3} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2}{\sqrt {5\,x+3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 176.47, size = 394, normalized size = 3.26 \[ - \frac {49 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} + \frac {21 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} - \frac {9 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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