Optimal. Leaf size=280 \[ -\frac {23763809947 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6219281250 \sqrt {33}}+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac {106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac {8038 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac {364267 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{36196875}-\frac {26534891 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{760134375}-\frac {359748241 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac {23763809947 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{13682418750}-\frac {1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}} \]
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Rubi [A] time = 0.12, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac {106 (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}}{4875}+\frac {8038 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}+\frac {364267 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{36196875}-\frac {26534891 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{760134375}-\frac {359748241 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{1520268750}-\frac {23763809947 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{13682418750}-\frac {23763809947 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6219281250 \sqrt {33}}-\frac {1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}} \]
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} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx &=\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {2}{75} \int \left (-\frac {113}{2}-\frac {159 x}{2}\right ) (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {4 \int \left (-3084-\frac {12057 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx}{14625}\\ &=\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {\sqrt {2+3 x} (3+5 x)^{5/2} \left (-\frac {1010595}{8}+\frac {1092801 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{2413125}\\ &=\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {8 \int \frac {(3+5 x)^{5/2} \left (\frac {108689433}{16}+\frac {79604673 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{108590625}\\ &=-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {\left (-\frac {5294426955}{8}-\frac {16188670845 x}{16}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2280403125}\\ &=-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {1390090964715}{32}+\frac {1069371447615 x}{16}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{34206046875}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {8 \int \frac {-\frac {22509028090305}{16}-\frac {71109064993095 x}{32}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{307854421875}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac {23763809947 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{12438562500}+\frac {1580201444291 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{136824187500}\\ &=-\frac {23763809947 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{13682418750}-\frac {359748241 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{1520268750}-\frac {26534891 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{760134375}+\frac {364267 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{36196875}+\frac {8038 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}}{4875}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac {1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{12438562500 \sqrt {33}}-\frac {23763809947 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6219281250 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 119, normalized size = 0.42 \[ \frac {\sqrt {2} \left (1580201444291 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-795995716040 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+30 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (547296750000 x^6+579573225000 x^5-352885207500 x^4-487924998750 x^3+59959633500 x^2+157612390605 x+9093216326\right )}{410472562500} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (300 \, x^{5} + 260 \, x^{4} - 137 \, x^{3} - 136 \, x^{2} + 15 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 170, normalized size = 0.61 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (492567075000000 x^{9}+899250660000000 x^{8}-32623479000000 x^{7}-902847084300000 x^{6}-312921865912500 x^{5}+349206885747000 x^{4}+192171420950850 x^{3}-37617016792110 x^{2}-30279805737360 x -1580201444291 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+795995716040 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1636778938680\right )}{12314176875000 x^{3}+9440868937500 x^{2}-2873307937500 x -2462835375000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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