Optimal. Leaf size=251 \[ -\frac {201616}{49} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac {11171040 \sqrt {3 x+2} \sqrt {1-2 x}}{49 \sqrt {5 x+3}}-\frac {5544440 \sqrt {3 x+2} \sqrt {1-2 x}}{147 (5 x+3)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {2234208}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {2 (1-2 x)^{3/2}}{3 (3 x+2)^{7/2} (5 x+3)^{3/2}}+\frac {11171040 \sqrt {3 x+2} \sqrt {1-2 x}}{49 \sqrt {5 x+3}}-\frac {5544440 \sqrt {3 x+2} \sqrt {1-2 x}}{147 (5 x+3)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {3 x+2} (5 x+3)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {201616}{49} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {2234208}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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)^{9/2} (3+5 x)^{5/2}} \, dx &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {2}{21} \int \frac {(231-231 x) \sqrt {1-2 x}}{(2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {4}{315} \int \frac {-\frac {51975}{2}+39270 x}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}-\frac {8 \int \frac {-\frac {5672205}{2}+\frac {7825125 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx}{6615}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {16 \int \frac {-\frac {852857775}{4}+\frac {490002975 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx}{46305}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {32 \int \frac {-\frac {34932969225}{4}+\frac {21612920175 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{1528065}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {64 \int \frac {-\frac {909761408325}{8}-\frac {359255409975 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{16808715}\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}+\frac {1108888}{49} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {6702624}{49} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {2 (1-2 x)^{3/2}}{3 (2+3 x)^{7/2} (3+5 x)^{3/2}}+\frac {44 \sqrt {1-2 x}}{3 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {11924 \sqrt {1-2 x}}{63 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2488904 \sqrt {1-2 x}}{441 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {5544440 \sqrt {1-2 x} \sqrt {2+3 x}}{147 (3+5 x)^{3/2}}+\frac {11171040 \sqrt {1-2 x} \sqrt {2+3 x}}{49 \sqrt {3+5 x}}-\frac {2234208}{49} \sqrt {33} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {201616}{49} \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 = 0.41, size = 114, normalized size = 0.45 \[ \frac {2}{147} \left (12 \sqrt {2} \left (279276 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-140665 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {\sqrt {1-2 x} \left (6786406800 x^5+21944379060 x^4+28367736228 x^3+18325125498 x^2+5915384456 x+763335749\right )}{(3 x+2)^{7/2} (5 x+3)^{3/2}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{30375 \, x^{8} + 155925 \, x^{7} + 350055 \, x^{6} + 448911 \, x^{5} + 359670 \, x^{4} + 184360 \, x^{3} + 59040 \, x^{2} + 10800 \, x + 864}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 501, normalized size = 2.00 \[ \frac {2 \sqrt {-2 x +1}\, \left (13572813600 x^{6}+37102351320 x^{5}-452427120 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+227877300 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+34791093396 x^{4}-1176310512 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+592480980 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+8282514768 x^{3}-1146148704 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+577289160 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6494356586 x^{2}-495994176 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+249821040 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4388712958 x -80431488 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+40511520 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-763335749\right )}{147 \left (3 x +2\right )^{\frac {7}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{9/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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