Optimal. Leaf size=280 \[ -\frac {13267820528 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{11402015625}-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1764163292393 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20730937500 \sqrt {33}}-\frac {13267820528 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5182734375 \sqrt {33}} \]
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Rubi [A]
time = 0.08, 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 = {103, 159, 164,
114, 120} \begin {gather*} -\frac {13267820528 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5182734375 \sqrt {33}}-\frac {1764163292393 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20730937500 \sqrt {33}}+\frac {2}{75} (1-2 x)^{5/2} (3 x+2)^{5/2} (5 x+3)^{5/2}+\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{5/2}}{2925}+\frac {3698 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}{482625}+\frac {142391 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{7239375}-\frac {569519 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{28153125}-\frac {400516993 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{2533781250}-\frac {13267820528 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{11402015625} \end {gather*}
Antiderivative was successfully verified.
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Rule 103
Rule 114
Rule 120
Rule 159
Rule 164
Rubi steps
\begin {align*} \int (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx &=\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {2}{75} \int \left (-\frac {115}{2}-\frac {155 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {4 \int \left (-3320-\frac {9245 x}{4}\right ) \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx}{14625}\\ &=\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {8 \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2} \left (-\frac {1423865}{8}+\frac {2135865 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{2413125}\\ &=\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {117464475}{16}+\frac {76885065 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{108590625}\\ &=-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (-\frac {11836111305}{16}-\frac {18023264685 x}{16}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3800671875}\\ &=-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {1551878163945}{32}+74631490470 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{57010078125}\\ &=-\frac {13267820528 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{11402015625}-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {50259437359155}{32}-\frac {79387348157685 x}{32}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{513090703125}\\ &=-\frac {13267820528 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{11402015625}-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {6633910264 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5182734375}+\frac {1764163292393 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{228040312500}\\ &=-\frac {13267820528 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{11402015625}-\frac {400516993 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2533781250}-\frac {569519 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{28153125}+\frac {142391 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{7239375}+\frac {3698 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}}{482625}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2}}{2925}+\frac {2}{75} (1-2 x)^{5/2} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1764163292393 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{20730937500 \sqrt {33}}-\frac {13267820528 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5182734375 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 9.18, size = 119, normalized size = 0.42 \begin {gather*} \frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (12155574323+173484591165 x+48836706750 x^2-528977216250 x^3-336683182500 x^4+621672975000 x^5+547296750000 x^6\right )+\sqrt {2} \left (1764163292393 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-888487137545 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{684120937500} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 168, normalized size = 0.60
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-492567075000000 x^{9}-937140435000000 x^{8}-11007171000000 x^{7}+937455630300000 x^{6}+875676154848 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1764163292393 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+362238910312500 x^{5}-361521647968500 x^{4}-215604575302050 x^{3}+36835025076780 x^{2}+33779897017530 x +2188003378140\right )}{684120937500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(168\) |
risch | \(-\frac {\left (547296750000 x^{6}+621672975000 x^{5}-336683182500 x^{4}-528977216250 x^{3}+48836706750 x^{2}+173484591165 x +12155574323\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{22804031250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1116876385759 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1764163292393 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2508443437500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(272\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (24 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{6}+\frac {1772 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{65}-\frac {52782 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{3575}-\frac {2239057 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96525}+\frac {1669631 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}+\frac {3855213137 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{506756250}+\frac {12155574323 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{22804031250}+\frac {1116876385759 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{957769312500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1764163292393 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{957769312500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(334\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.24, size = 53, normalized size = 0.19 \begin {gather*} \frac {1}{22804031250} \, {\left (547296750000 \, x^{6} + 621672975000 \, x^{5} - 336683182500 \, x^{4} - 528977216250 \, x^{3} + 48836706750 \, x^{2} + 173484591165 \, x + 12155574323\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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