Optimal. Leaf size=160 \[ \frac {16}{3} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2660 \sqrt {1-2 x} \sqrt {3 x+2}}{9 \sqrt {5 x+3}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {2660 \sqrt {1-2 x} \sqrt {3 x+2}}{9 \sqrt {5 x+3}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14 \sqrt {1-2 x}}{9 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {16}{3} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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)^{3/2}} \, dx &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2}{9} \int \frac {88-99 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {4}{63} \int \frac {\frac {7469}{2}-2310 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {8}{693} \int \frac {\frac {97251}{2}+\frac {153615 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {14 \sqrt {1-2 x}}{9 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {88}{3} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {532}{3} \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} \sqrt {3+5 x}}+\frac {88 \sqrt {1-2 x}}{3 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {2660 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {532}{3} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {16}{3} \sqrt {\frac {11}{3}} 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 = 0.17, size = 100, normalized size = 0.62 \[ -\frac {4}{9} \sqrt {2} \left (133 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-67 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-\frac {2 \sqrt {1-2 x} \left (3990 x^2+5188 x+1683\right )}{3 (3 x+2)^{3/2} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 219, normalized size = 1.37 \[ -\frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (23940 x^{3}+19158 x^{2}-798 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+402 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5466 x -532 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+268 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-5049\right )}{9 \left (3 x +2\right )^{\frac {3}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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