Optimal. Leaf size=218 \[ -\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {725140729 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}}-\frac {43624697 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{283500 \sqrt {33}} \]
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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 {43624697 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{283500 \sqrt {33}}-\frac {725140729 E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}}-\frac {1}{11} \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {329683 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{34650}-\frac {43624697 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{623700} \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 {(2+3 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {1}{11} \int \frac {(2+3 x)^{3/2} (3+5 x)^{3/2} \left (\frac {215}{2}+170 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {1}{495} \int \frac {\left (-15200-\frac {47385 x}{2}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {\int \frac {(3+5 x)^{3/2} \left (\frac {6483095}{4}+\frac {4945245 x}{2}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{17325}\\ &=-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {\int \frac {\left (-\frac {212630445}{2}-\frac {654370455 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{259875}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {\int \frac {\frac {27544681245}{8}+\frac {10877110935 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2338875}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}+\frac {43624697 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{567000}+\frac {725140729 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1559250}\\ &=-\frac {43624697 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{623700}-\frac {329683 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{34650}-\frac {1053}{770} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {34}{99} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{11} \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}-\frac {725140729 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{141750 \sqrt {33}}-\frac {43624697 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{283500 \sqrt {33}}\\ \end {align*}
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Mathematica [A]
time = 4.87, size = 110, normalized size = 0.50 \begin {gather*} \frac {2900562916 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 (75000749+86822370 x+81985950 x^2+48384000 x^3+12757500 x^4\right )+292189583 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )}{9355500 \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 | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (11481750000 x^{7}+52348275000 x^{6}+1439615001 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2900562916 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+104493240000 x^{5}+122253448500 x^{4}+101481939900 x^{3}+18760348110 x^{2}-31378183890 x -13500134820\right )}{18711000 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(\frac {\left (12757500 x^{4}+48384000 x^{3}+81985950 x^{2}+86822370 x +75000749\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{623700 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1836312083 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{68607000 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {725140729 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{17151750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(261\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (-\frac {225 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}-\frac {2560 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{33}-\frac {182191 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1386}-\frac {964693 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6930}-\frac {75000749 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{623700}+\frac {1836312083 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{26195400 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {725140729 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{6548850 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(290\) |
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 = 0.20, size = 43, normalized size = 0.20 \begin {gather*} -\frac {1}{623700} \, {\left (12757500 \, x^{4} + 48384000 \, x^{3} + 81985950 \, x^{2} + 86822370 \, x + 75000749\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 (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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