Optimal. Leaf size=191 \[ -\frac {1112}{15} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {36968 \sqrt {3 x+2} \sqrt {1-2 x}}{9 \sqrt {5 x+3}}-\frac {6116 \sqrt {3 x+2} \sqrt {1-2 x}}{9 (5 x+3)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {36968}{15} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ \frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}+\frac {36968 \sqrt {3 x+2} \sqrt {1-2 x}}{9 \sqrt {5 x+3}}-\frac {6116 \sqrt {3 x+2} \sqrt {1-2 x}}{9 (5 x+3)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {1112}{15} \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {36968}{15} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]
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)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {2}{9} \int \frac {(165-99 x) \sqrt {1-2 x}}{(2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {4}{27} \int \frac {-\frac {17919}{2}+10296 x}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6116 \sqrt {1-2 x} \sqrt {2+3 x}}{9 (3+5 x)^{3/2}}+\frac {8}{891} \int \frac {-366993+\frac {454113 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6116 \sqrt {1-2 x} \sqrt {2+3 x}}{9 (3+5 x)^{3/2}}+\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {19115217}{4}-\frac {15096807 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9801}\\ &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6116 \sqrt {1-2 x} \sqrt {2+3 x}}{9 (3+5 x)^{3/2}}+\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}+\frac {6116}{15} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx+\frac {36968}{15} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6116 \sqrt {1-2 x} \sqrt {2+3 x}}{9 (3+5 x)^{3/2}}+\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {36968}{15} \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {1112}{15} \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 = 0.22, size = 105, normalized size = 0.55 \[ \frac {4}{45} \sqrt {2} \left (9242 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-4655 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {2 \sqrt {1-2 x} \left (277260 x^3+526862 x^2+333260 x+70169\right )}{3 (3 x+2)^{3/2} (5 x+3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{3375 \, x^{6} + 12825 \, x^{5} + 20295 \, x^{4} + 17119 \, x^{3} + 8118 \, x^{2} + 2052 \, x + 216}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 311, normalized size = 1.63 \[ \frac {2 \sqrt {-2 x +1}\, \left (8317800 x^{4}+11646960 x^{3}-277260 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+139650 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2094870 x^{2}-351196 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+176890 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-2893830 x -110904 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+55860 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1052535\right )}{45 \left (3 x +2\right )^{\frac {3}{2}} \left (5 x +3\right )^{\frac {3}{2}} \left (2 x -1\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]
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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