Optimal. Leaf size=116 \[ -\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac {343 \sqrt {5 x+3} (1-2 x)^{3/2}}{18150}+\frac {343 \sqrt {5 x+3} \sqrt {1-2 x}}{5500}+\frac {343 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{500 \sqrt {10}} \]
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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, 78, 50, 54, 216} \[ -\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2}}{825 (5 x+3)^{3/2}}+\frac {343 \sqrt {5 x+3} (1-2 x)^{3/2}}{18150}+\frac {343 \sqrt {5 x+3} \sqrt {1-2 x}}{5500}+\frac {343 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{500 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 78
Rule 89
Rule 216
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}+\frac {2}{825} \int \frac {(1-2 x)^{3/2} \left (\frac {1085}{2}+\frac {1485 x}{2}\right )}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {343 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{1815}\\ &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {343 (1-2 x)^{3/2} \sqrt {3+5 x}}{18150}+\frac {343 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1100}\\ &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {343 \sqrt {1-2 x} \sqrt {3+5 x}}{5500}+\frac {343 (1-2 x)^{3/2} \sqrt {3+5 x}}{18150}+\frac {343 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1000}\\ &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {343 \sqrt {1-2 x} \sqrt {3+5 x}}{5500}+\frac {343 (1-2 x)^{3/2} \sqrt {3+5 x}}{18150}+\frac {343 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{500 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2}}{825 (3+5 x)^{3/2}}-\frac {388 (1-2 x)^{5/2}}{9075 \sqrt {3+5 x}}+\frac {343 \sqrt {1-2 x} \sqrt {3+5 x}}{5500}+\frac {343 (1-2 x)^{3/2} \sqrt {3+5 x}}{18150}+\frac {343 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{500 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 83, normalized size = 0.72 \[ \frac {10 \left (5400 x^4-6390 x^3-5375 x^2+1808 x+901\right )+1029 \sqrt {20 x-10} (5 x+3)^{3/2} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{15000 \sqrt {1-2 x} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.35, size = 96, normalized size = 0.83 \[ -\frac {1029 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (2700 \, x^{3} - 1845 \, x^{2} - 3610 \, x - 901\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{30000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.57, size = 171, normalized size = 1.47 \[ -\frac {3}{12500} \, {\left (12 \, \sqrt {5} {\left (5 \, x + 3\right )} - 149 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {1}{150000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {1524 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {343}{5000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {381 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{9375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 130, normalized size = 1.12 \[ \frac {\left (-54000 \sqrt {-10 x^{2}-x +3}\, x^{3}+25725 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+36900 \sqrt {-10 x^{2}-x +3}\, x^{2}+30870 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+72200 \sqrt {-10 x^{2}-x +3}\, x +9261 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+18020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{30000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 154, normalized size = 1.33 \[ \frac {343}{10000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {297}{2500} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{375 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{250 \, {\left (5 \, x + 3\right )}} - \frac {11 \, \sqrt {-10 \, x^{2} - x + 3}}{1875 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {116 \, \sqrt {-10 \, x^{2} - x + 3}}{375 \, {\left (5 \, x + 3\right )}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{5/2}} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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