Optimal. Leaf size=218 \[ -\frac {11346991 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3898125}-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7087500 \sqrt {33}}-\frac {11346991 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}} \]
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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 {11346991 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}-\frac {1508889271 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7087500 \sqrt {33}}+\frac {2}{55} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {23 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac {543 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{9625}-\frac {342971 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{866250}-\frac {11346991 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{3898125} \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 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2} \, dx &=\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {2}{55} \int \frac {\left (-\frac {27}{2}-\frac {23 x}{2}\right ) (2+3 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {2 \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {6355}{4}+\frac {4887 x}{2}\right )}{\sqrt {1-2 x}} \, dx}{2475}\\ &=-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {2 \int \frac {\left (-\frac {674539}{4}-\frac {1028913 x}{4}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{86625}\\ &=-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {2 \int \frac {\sqrt {3+5 x} \left (\frac {88489161}{8}+\frac {34040973 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1299375}\\ &=-\frac {11346991 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3898125}-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {2 \int \frac {-\frac {2865780969}{8}-\frac {4526667813 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{11694375}\\ &=-\frac {11346991 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3898125}-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {11346991 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3543750}+\frac {1508889271 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{77962500}\\ &=-\frac {11346991 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3898125}-\frac {342971 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{866250}-\frac {543 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}-\frac {23 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{7087500 \sqrt {33}}-\frac {11346991 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.42, size = 107, normalized size = 0.49 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-27010769+29706255 x+132234750 x^2+156161250 x^3+63787500 x^4\right )+1508889271 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-759987865 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{116943750 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 158, normalized size = 0.72
method | result | size |
default | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-57408750000 x^{7}-184558500000 x^{6}+748901406 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1508889271 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-213367162500 x^{5}-73701994500 x^{4}+59690698650 x^{3}+48677999160 x^{2}-325135590 x -4861938420\right )}{233887500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(-\frac {\left (63787500 x^{4}+156161250 x^{3}+132234750 x^{2}+29706255 x -27010769\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{7796250 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {955260323 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1508889271 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{857587500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {90 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}+\frac {661 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}+\frac {58771 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3465}+\frac {660139 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{173250}-\frac {27010769 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7796250}+\frac {955260323 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{327442500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1508889271 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{327442500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(290\) |
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 = 0.18, size = 43, normalized size = 0.20 \begin {gather*} \frac {1}{7796250} \, {\left (63787500 \, x^{4} + 156161250 \, x^{3} + 132234750 \, x^{2} + 29706255 \, x - 27010769\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 \sqrt {1-2\,x}\,{\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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