Optimal. Leaf size=191 \[ -\frac {160297 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{141750}-\frac {1208 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{7875}-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {5327983 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750}-\frac {160297 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750} \]
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Rubi [A]
time = 0.05, 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 {160297 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750}-\frac {5327983 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750}+\frac {2}{45} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {3}{175} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1208 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{7875}-\frac {160297 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{141750} \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)^{3/2} (3+5 x)^{3/2} \, dx &=\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {2}{45} \int \frac {\left (-\frac {25}{2}-\frac {27 x}{2}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {2 \int \frac {(3+5 x)^{3/2} \left (\frac {4769}{4}+1812 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1575}\\ &=-\frac {1208 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{7875}-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {2 \int \frac {\left (-\frac {312453}{4}-\frac {480891 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{23625}\\ &=-\frac {160297 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{141750}-\frac {1208 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{7875}-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {2 \int \frac {\frac {20238699}{8}+\frac {15983949 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{212625}\\ &=-\frac {160297 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{141750}-\frac {1208 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{7875}-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {1763267 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1417500}+\frac {5327983 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{708750}\\ &=-\frac {160297 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{141750}-\frac {1208 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{7875}-\frac {3}{175} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {5327983 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750}-\frac {160297 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{708750}\\ \end {align*}
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Mathematica [A]
time = 3.50, size = 102, normalized size = 0.53 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-133999+366480 x+821250 x^2+472500 x^3\right )+10655966 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5366165 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{2126250 \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 | \(-\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-425250000 x^{6}+5289801 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-10655966 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1065150000 x^{5}-797269500 x^{4}+125240400 x^{3}+317245110 x^{2}+37826610 x -24119820\right )}{4252500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
risch | \(-\frac {\left (472500 x^{3}+821250 x^{2}+366480 x -133999\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{141750 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {6746233 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15592500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5327983 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{7796250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\frac {10 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3}+\frac {365 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{63}+\frac {4072 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1575}-\frac {133999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{141750}+\frac {6746233 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5953500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5327983 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2976750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \left (15 x^{2}+19 x +6\right )}\) | \(268\) |
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 = 38, normalized size = 0.20 \begin {gather*} \frac {1}{141750} \, {\left (472500 \, x^{3} + 821250 \, x^{2} + 366480 \, x - 133999\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 \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,{\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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