Optimal. Leaf size=218 \[ -\frac {2865161 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {231061879 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {3963068 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}} \]
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Rubi [A]
time = 0.06, antiderivative size = 218, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {3963068 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}-\frac {231061879 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {5 x+3} (3 x+2)^{5/2}+\frac {62 (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^{5/2}}{1485}+\frac {4258 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}}{155925}+\frac {181333 \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}}{3898125}-\frac {2865161 \sqrt {1-2 x} \sqrt {5 x+3} \sqrt {3 x+2}}{19490625} \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 \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{\sqrt {3+5 x}} \, dx &=\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {2}{55} \int \frac {\left (-\frac {115}{2}-\frac {155 x}{2}\right ) (1-2 x)^{3/2} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {4 \int \frac {\left (-3145-\frac {10645 x}{4}\right ) \sqrt {1-2 x} (2+3 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{7425}\\ &=\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {8 \int \frac {(2+3 x)^{3/2} \left (-\frac {863165}{8}+\frac {906665 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{779625}\\ &=\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {8 \int \frac {\sqrt {2+3 x} \left (\frac {63649875}{16}+\frac {42977415 x}{8}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{19490625}\\ &=-\frac {2865161 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {8 \int \frac {-\frac {2210338155}{16}-\frac {3465928185 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{292359375}\\ &=-\frac {2865161 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {1981534 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8859375}+\frac {231061879 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{194906250}\\ &=-\frac {2865161 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {181333 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{3898125}+\frac {4258 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}{155925}+\frac {62 (1-2 x)^{3/2} (2+3 x)^{5/2} \sqrt {3+5 x}}{1485}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {231061879 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {3963068 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 8.39, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (7167169+9526995 x-23717250 x^2-6142500 x^3+25515000 x^4\right )+231061879 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-100280635 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{292359375 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-22963500000 x^{7}-12077100000 x^{6}+130781244 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-231061879 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+30942000000 x^{5}+11093382000 x^{4}-19110351150 x^{3}-7213782660 x^{2}+3219964590 x +1290090420\right )}{584718750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(-\frac {\left (25515000 x^{4}-6142500 x^{3}-23717250 x^{2}+9526995 x +7167169\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{19490625 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {147355877 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{2143968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {231061879 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{2143968750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {72 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{55}-\frac {52 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{165}-\frac {21082 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17325}+\frac {211711 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{433125}+\frac {7167169 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19490625}+\frac {147355877 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {231061879 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{818606250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
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 = 0.22, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{19490625} \, {\left (25515000 \, x^{4} - 6142500 \, x^{3} - 23717250 \, x^{2} + 9526995 \, x + 7167169\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 \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}}{\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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