Optimal. Leaf size=116 \[ -\frac {3}{50} \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {5 x+3}}+\frac {119 \sqrt {5 x+3} (1-2 x)^{3/2}}{2200}+\frac {357 \sqrt {5 x+3} \sqrt {1-2 x}}{2000}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2000 \sqrt {10}} \]
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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 = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {89, 80, 50, 54, 216} \[ -\frac {3}{50} \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {5 x+3}}+\frac {119 \sqrt {5 x+3} (1-2 x)^{3/2}}{2200}+\frac {357 \sqrt {5 x+3} \sqrt {1-2 x}}{2000}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{2000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 80
Rule 89
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {(1-2 x)^{3/2} \left (\frac {355}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {119}{220} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {357}{400} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{4000}\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{2000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{275 \sqrt {3+5 x}}+\frac {357 \sqrt {1-2 x} \sqrt {3+5 x}}{2000}+\frac {119 (1-2 x)^{3/2} \sqrt {3+5 x}}{2200}-\frac {3}{50} (1-2 x)^{5/2} \sqrt {3+5 x}+\frac {3927 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{2000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 83, normalized size = 0.72 \[ \frac {10 \left (4800 x^4-2040 x^3-5330 x^2+533 x+1021\right )+3927 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{20000 \sqrt {1-2 x} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.07, size = 86, normalized size = 0.74 \[ -\frac {3927 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2400 \, x^{3} + 180 \, x^{2} - 2575 \, x - 1021\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{40000 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.35, size = 124, normalized size = 1.07 \[ -\frac {1}{50000} \, {\left (12 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} - 69 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 199 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {3927}{20000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {11 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{6250 \, \sqrt {5 \, x + 3}} + \frac {22 \, \sqrt {10} \sqrt {5 \, x + 3}}{3125 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 1.00 \[ \frac {\left (-48000 \sqrt {-10 x^{2}-x +3}\, x^{3}-3600 \sqrt {-10 x^{2}-x +3}\, x^{2}+19635 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+51500 \sqrt {-10 x^{2}-x +3}\, x +11781 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+20420 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{40000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.32, size = 154, normalized size = 1.33 \[ -\frac {11979}{200000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {957}{25000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{125} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {99}{500} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {2277}{10000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {99}{1250} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {3 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{125 \, {\left (5 \, x + 3\right )}} - \frac {33 \, \sqrt {-10 \, x^{2} - x + 3}}{625 \, {\left (5 \, x + 3\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (3 x + 2\right )^{2}}{\left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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