Optimal. Leaf size=249 \[ -\frac {486785077 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{248771250 \sqrt {33}}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}}{10725}+\frac {2314 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{111375}-\frac {121031 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{30405375}-\frac {3872003 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{30405375}-\frac {486785077 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{547296750}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}} \]
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Rubi [A] time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac {2}{65} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {326 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}}{10725}+\frac {2314 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{111375}-\frac {121031 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{30405375}-\frac {3872003 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{30405375}-\frac {486785077 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{547296750}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx &=\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{65} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {5653}{2}-\frac {15041 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx}{10725}\\ &=\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {518563}{8}-\frac {121031 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1447875}\\ &=-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{3/2} \left (\frac {71027395}{16}+\frac {58080045 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{30405375}\\ &=-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {2382196995}{8}-\frac {7301776155 x}{16}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{456080625}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {8 \int \frac {\frac {308389708545}{32}+\frac {121802417085 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4104725625}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {486785077 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{497542500}+\frac {8120161139 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1368241875}\\ &=-\frac {486785077 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{547296750}-\frac {3872003 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{30405375}-\frac {121031 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{30405375}+\frac {2314 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{111375}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}}{10725}+\frac {2}{65} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {8120161139 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{124385625 \sqrt {33}}-\frac {486785077 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{248771250 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 112, normalized size = 0.45 \[ \frac {-16416737015 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (8419950000 x^5+2577015000 x^4-7942630500 x^3-1730459250 x^2+2923422930 x+495379991\right )+32480644556 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{8209451250 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 165, normalized size = 0.66 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (7577955000000 x^{8}+8129079000000 x^{7}-7138416600000 x^{6}-9094592520000 x^{5}+2641153459500 x^{4}+4256073746100 x^{3}+39376043490 x^{2}-630245925510 x -32480644556 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+16416737015 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-89168398380\right )}{492567075000 x^{3}+377634757500 x^{2}-114932317500 x -98513415000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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