Optimal. Leaf size=191 \[ -\frac {76163 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{212625}-\frac {839 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4971289 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2126250}-\frac {76163 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1063125} \]
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Rubi [A]
time = 0.04, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 7, 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 {76163 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1063125}-\frac {4971289 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2126250}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{4725}-\frac {839 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{23625}-\frac {76163 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{212625} \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)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx &=\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{45} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-619-\frac {2517 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725}\\ &=-\frac {839 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {4 \int \frac {\sqrt {3+5 x} \left (\frac {290799}{8}+\frac {228489 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{70875}\\ &=-\frac {76163 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{212625}-\frac {839 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {-\frac {2362749}{2}-\frac {14913867 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{637875}\\ &=-\frac {76163 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{212625}-\frac {839 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {837793 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2126250}+\frac {4971289 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2126250}\\ &=-\frac {76163 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{212625}-\frac {839 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{23625}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{4725}+\frac {2}{45} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4971289 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2126250}-\frac {76163 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1063125}\\ \end {align*}
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Mathematica [A]
time = 5.88, size = 105, normalized size = 0.55 \begin {gather*} \frac {4971289 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5 \left (3 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-64804-337545 x+112500 x^2+472500 x^3\right )+491582 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{3189375 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 153, normalized size = 0.80
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (425250000 x^{6}+2513379 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-4971289 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+427275000 x^{5}-325390500 x^{4}-399904650 x^{3}+5919690 x^{2}+74366940 x +11664720\right )}{6378750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
risch | \(\frac {\left (472500 x^{3}+112500 x^{2}-337545 x -64804\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{212625 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1575166 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{11694375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4971289 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{23388750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {20 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}-\frac {100 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}+\frac {7501 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4725}+\frac {64804 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{212625}+\frac {1575166 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{4465125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4971289 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{8930250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
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 = 0.14, size = 38, normalized size = 0.20 \begin {gather*} -\frac {1}{212625} \, {\left (472500 \, x^{3} + 112500 \, x^{2} - 337545 \, x - 64804\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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.01 \begin {gather*} \int {\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+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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