Optimal. Leaf size=185 \[ -120 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {5440 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {300 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-1088 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 185, 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 = {98, 152, 158, 113, 119} \[ \frac {5440 \sqrt {1-2 x} \sqrt {3 x+2}}{3 \sqrt {5 x+3}}-\frac {300 \sqrt {1-2 x} \sqrt {3 x+2}}{(5 x+3)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}-120 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-1088 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{9} \int \frac {123-169 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {4}{63} \int \frac {\frac {18459}{2}-10605 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}-\frac {8 \int \frac {\frac {756063}{2}-\frac {467775 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2079}\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+\frac {16 \int \frac {\frac {19690209}{4}+7775460 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{22869}\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}+180 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+1088 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {404 \sqrt {1-2 x}}{9 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {300 \sqrt {1-2 x} \sqrt {2+3 x}}{(3+5 x)^{3/2}}+\frac {5440 \sqrt {1-2 x} \sqrt {2+3 x}}{3 \sqrt {3+5 x}}-1088 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-120 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}
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Mathematica [A] time = 0.31, size = 104, normalized size = 0.56 \[ \frac {2}{3} \left (2 \sqrt {2} \left (272 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-137 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \left (122400 x^3+232590 x^2+147122 x+30977\right )}{(3 x+2)^{3/2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{3375 \, x^{6} + 12825 \, x^{5} + 20295 \, x^{4} + 17119 \, x^{3} + 8118 \, x^{2} + 2052 \, x + 216}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.68 \[ \frac {2 \sqrt {-2 x +1}\, \left (244800 x^{4}+342780 x^{3}-8160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4110 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+61654 x^{2}-10336 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+5206 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-85168 x -3264 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+1644 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-30977\right )}{3 \left (3 x +2\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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