Optimal. Leaf size=191 \[ -\frac {663409 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{236250}-\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac {137}{315} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {9547 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{5250}-\frac {663409 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{47250}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500} \]
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Rubi [A] time = 0.07, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ -\frac {1}{9} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{5/2}-\frac {137}{315} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^{3/2}-\frac {9547 \sqrt {1-2 x} (5 x+3)^{3/2} \sqrt {3 x+2}}{5250}-\frac {663409 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{47250}-\frac {663409 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{236250}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{5/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {1}{9} \int \frac {(2+3 x)^{3/2} \sqrt {3+5 x} \left (\frac {171}{2}+137 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {1}{315} \int \frac {\left (-\frac {18135}{2}-\frac {28641 x}{2}\right ) \sqrt {2+3 x} \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {2586807}{4}+\frac {1990227 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7875}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {\int \frac {-\frac {41887689}{2}-\frac {132328131 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{70875}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}+\frac {7297499 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{472500}+\frac {44109377 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{472500}\\ &=-\frac {663409 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{47250}-\frac {9547 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{5250}-\frac {137}{315} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}-\frac {44109377 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{472500}-\frac {663409 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{236250}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 105, normalized size = 0.55 \[ \frac {44109377 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (4443376 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (236250 x^3+765000 x^2+1114065 x+1107478\right )\right )}{708750 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 155, normalized size = 0.81 \[ \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-212625000 x^{6}-851512500 x^{5}-1480896000 x^{4}-1562260050 x^{3}-392506170 x^{2}+433102080 x -44109377 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+22216880 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+199346040\right )}{42525000 x^{3}+32602500 x^{2}-9922500 x -8505000} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{\sqrt {1-2\,x}} \,d x \]
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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