Optimal. Leaf size=158 \[ -\frac {64}{25} \sqrt {33} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {14 (1-2 x)^{3/2}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {6388 \sqrt {3 x+2} \sqrt {1-2 x}}{15 \sqrt {5 x+3}}-\frac {1012 \sqrt {3 x+2} \sqrt {1-2 x}}{15 (5 x+3)^{3/2}}-\frac {6388}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac {14 (1-2 x)^{3/2}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {6388 \sqrt {3 x+2} \sqrt {1-2 x}}{15 \sqrt {5 x+3}}-\frac {1012 \sqrt {3 x+2} \sqrt {1-2 x}}{15 (5 x+3)^{3/2}}-\frac {64}{25} \sqrt {33} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {6388}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
Antiderivative was successfully verified.
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Rule 98
Rule 113
Rule 119
Rule 150
Rule 152
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}+\frac {2}{3} \int \frac {(132-33 x) \sqrt {1-2 x}}{\sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1012 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {4}{45} \int \frac {-\frac {7689}{2}+2376 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1012 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {6388 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}-\frac {8}{495} \int \frac {-\frac {100089}{2}-\frac {158103 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1012 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {6388 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}+\frac {1056}{25} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {6388}{25} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {1012 \sqrt {1-2 x} \sqrt {2+3 x}}{15 (3+5 x)^{3/2}}+\frac {6388 \sqrt {1-2 x} \sqrt {2+3 x}}{15 \sqrt {3+5 x}}-\frac {6388}{25} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {64}{25} \sqrt {33} 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 = 0.21, size = 100, normalized size = 0.63 \[ \frac {2}{75} \left (2 \sqrt {2} \left (1597 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-805 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {5 \sqrt {1-2 x} \left (47910 x^2+59098 x+18187\right )}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1125 \, x^{5} + 3525 \, x^{4} + 4415 \, x^{3} + 2763 \, x^{2} + 864 \, x + 108}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 219, normalized size = 1.39 \[ \frac {2 \sqrt {-2 x +1}\, \sqrt {3 x +2}\, \left (479100 x^{3}+351430 x^{2}-15970 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8050 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-113620 x -9582 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4830 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-90935\right )}{75 \left (5 x +3\right )^{\frac {3}{2}} \left (6 x^{2}+x -2\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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