Optimal. Leaf size=253 \[ \frac {20549264 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{9261}+\frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}-\frac {3415750480 \sqrt {3 x+2} \sqrt {1-2 x}}{27783 \sqrt {5 x+3}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {813208 \sqrt {1-2 x}}{1323 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {11660 \sqrt {1-2 x}}{189 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {652 \sqrt {1-2 x}}{81 (3 x+2)^{7/2} \sqrt {5 x+3}}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261} \]
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Rubi [A] time = 0.10, antiderivative size = 253, 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} \[ \frac {14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt {5 x+3}}-\frac {3415750480 \sqrt {3 x+2} \sqrt {1-2 x}}{27783 \sqrt {5 x+3}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {813208 \sqrt {1-2 x}}{1323 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {11660 \sqrt {1-2 x}}{189 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {652 \sqrt {1-2 x}}{81 (3 x+2)^{7/2} \sqrt {5 x+3}}+\frac {20549264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261} \]
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)^{11/2} (3+5 x)^{3/2}} \, dx &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {2}{27} \int \frac {(229-227 x) \sqrt {1-2 x}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}-\frac {4}{567} \int \frac {-\frac {50897}{2}+38346 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {5572105}{2}+\frac {7651875 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{19845}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {842998695}{4}+\frac {480300975 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{416745}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {32 \int \frac {-8991074400+\frac {22250999925 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{2917215}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {64 \int \frac {-\frac {936620355825}{8}-\frac {739723463325 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{32089365}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}-\frac {113020952 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{9261}-\frac {683150096 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{9261}\\ &=\frac {14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt {3+5 x}}+\frac {652 \sqrt {1-2 x}}{81 (2+3 x)^{7/2} \sqrt {3+5 x}}+\frac {11660 \sqrt {1-2 x}}{189 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {813208 \sqrt {1-2 x}}{1323 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {113020952 \sqrt {1-2 x}}{9261 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {3415750480 \sqrt {1-2 x} \sqrt {2+3 x}}{27783 \sqrt {3+5 x}}+\frac {683150096 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}+\frac {20549264 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{9261}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 115, normalized size = 0.45 \[ \frac {2 \left (-4 \sqrt {2} \left (85393762 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-43010905 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )-\frac {3 \sqrt {1-2 x} \left (138337894440 x^5+456548966244 x^4+602551975428 x^3+397527527442 x^2+131099014240 x+17289178827\right )}{(3 x+2)^{9/2} \sqrt {5 x+3}}\right )}{27783} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576}, x\right ) \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.03, size = 504, normalized size = 1.99 \[ -\frac {2 \sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (830027366640 x^{6}+2324280114144 x^{5}-27667578888 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+13935533220 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2245664953836 x^{4}-73780210368 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+37161421920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+577509238368 x^{3}-73780210368 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+37161421920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-405988496886 x^{2}-32791204608 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+16516187520 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-289561969758 x -5465200768 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+2752697920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-51867536481\right )}{27783 \left (3 x +2\right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )} \]
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 \[ \int \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{11/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
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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