Optimal. Leaf size=218 \[ -\frac {6770629 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{17500 \sqrt {33}}+\frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac {166 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 \sqrt {1-2 x}}-\frac {1327}{154} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {139163 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{3850}-\frac {6478333 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{38500}-\frac {112543103 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}} \]
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Rubi [A] time = 0.08, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ \frac {\sqrt {5 x+3} (3 x+2)^{9/2}}{3 (1-2 x)^{3/2}}-\frac {166 \sqrt {5 x+3} (3 x+2)^{7/2}}{33 \sqrt {1-2 x}}-\frac {1327}{154} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}-\frac {139163 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{3850}-\frac {6478333 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{38500}-\frac {6770629 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500 \sqrt {33}}-\frac {112543103 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 97
Rule 113
Rule 119
Rule 150
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^{7/2} \left (\frac {91}{2}+75 x\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-6054-\frac {19905 x}{2}\right ) (2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {(2+3 x)^{3/2} \left (\frac {2551035}{4}+\frac {2087445 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1155}\\ &=-\frac {139163 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3850}-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\int \frac {\left (-\frac {179737875}{4}-\frac {291524985 x}{4}\right ) \sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{28875}\\ &=-\frac {6478333 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{38500}-\frac {139163 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3850}-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {\int \frac {\frac {12824947395}{8}+\frac {5064439635 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{433125}\\ &=-\frac {6478333 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{38500}-\frac {139163 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3850}-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}+\frac {6770629 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{35000}+\frac {112543103 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{96250}\\ &=-\frac {6478333 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{38500}-\frac {139163 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3850}-\frac {1327}{154} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {166 (2+3 x)^{7/2} \sqrt {3+5 x}}{33 \sqrt {1-2 x}}+\frac {(2+3 x)^{9/2} \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {112543103 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8750 \sqrt {33}}-\frac {6770629 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 130, normalized size = 0.60 \[ -\frac {-226741655 \sqrt {2-4 x} (2 x-1) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+10 \sqrt {3 x+2} \sqrt {5 x+3} \left (1336500 x^4+6664680 x^3+19375686 x^2-94671446 x+35797779\right )+450172412 \sqrt {2-4 x} (2 x-1) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1155000 (1-2 x)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 243, normalized size = 1.11 \[ \frac {\left (-200475000 x^{6}-1253637000 x^{5}-4252832100 x^{4}+10119455760 x^{3}+11455366730 x^{2}-900344824 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+453483310 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1121291250 x +450172412 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-226741655 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2147866740\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}}{1155000 \left (2 x -1\right )^{2} \left (15 x^{2}+19 x +6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
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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