Optimal. Leaf size=138 \[ -\frac {9}{200} \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {5 x+3}}+\frac {651 \sqrt {5 x+3} (1-2 x)^{5/2}}{22000}+\frac {651 \sqrt {5 x+3} (1-2 x)^{3/2}}{8000}+\frac {21483 \sqrt {5 x+3} \sqrt {1-2 x}}{80000}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80000 \sqrt {10}} \]
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Rubi [A] time = 0.04, antiderivative size = 138, 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 = {89, 80, 50, 54, 216} \[ -\frac {9}{200} \sqrt {5 x+3} (1-2 x)^{7/2}-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {5 x+3}}+\frac {651 \sqrt {5 x+3} (1-2 x)^{5/2}}{22000}+\frac {651 \sqrt {5 x+3} (1-2 x)^{3/2}}{8000}+\frac {21483 \sqrt {5 x+3} \sqrt {1-2 x}}{80000}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{80000 \sqrt {10}} \]
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)^{5/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {2}{275} \int \frac {(1-2 x)^{5/2} \left (\frac {351}{2}+\frac {495 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {1953 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{4400}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {651}{800} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {21483 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{16000}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{160000}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{80000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{7/2}}{275 \sqrt {3+5 x}}+\frac {21483 \sqrt {1-2 x} \sqrt {3+5 x}}{80000}+\frac {651 (1-2 x)^{3/2} \sqrt {3+5 x}}{8000}+\frac {651 (1-2 x)^{5/2} \sqrt {3+5 x}}{22000}-\frac {9}{200} (1-2 x)^{7/2} \sqrt {3+5 x}+\frac {236313 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{80000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 88, normalized size = 0.64 \[ \frac {236313 \sqrt {5 x+3} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (288000 x^5-299200 x^4-147640 x^3+381870 x^2+24773 x-79699\right )}{800000 \sqrt {1-2 x} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.11, size = 91, normalized size = 0.66 \[ -\frac {236313 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (144000 \, x^{4} - 77600 \, x^{3} - 112620 \, x^{2} + 134625 \, x + 79699\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1600000 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.92, size = 137, normalized size = 0.99 \[ \frac {1}{2000000} \, {\left (4 \, {\left (8 \, {\left (36 \, \sqrt {5} {\left (5 \, x + 3\right )} - 529 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 16905 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 61545 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {236313}{800000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {121 \, \sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{31250 \, \sqrt {5 \, x + 3}} + \frac {242 \, \sqrt {10} \sqrt {5 \, x + 3}}{15625 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 133, normalized size = 0.96 \[ \frac {\left (2880000 \sqrt {-10 x^{2}-x +3}\, x^{4}-1552000 \sqrt {-10 x^{2}-x +3}\, x^{3}-2252400 \sqrt {-10 x^{2}-x +3}\, x^{2}+1181565 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+2692500 \sqrt {-10 x^{2}-x +3}\, x +708939 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1593980 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{1600000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 109, normalized size = 0.79 \[ -\frac {18 \, x^{5}}{5 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {187 \, x^{4}}{50 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3691 \, x^{3}}{2000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {38187 \, x^{2}}{8000 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {236313}{1600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {24773 \, x}{80000 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {79699}{80000 \, \sqrt {-10 \, x^{2} - x + 3}} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^2}{{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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