Optimal. Leaf size=191 \[ -\frac {110717 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1063125}+\frac {10214 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {6799613 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}-\frac {110717 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625} \]
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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 {110717 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}-\frac {6799613 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}+\frac {2}{45} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {326 \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac {10214 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{118125}-\frac {110717 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1063125} \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)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x} \, dx &=\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {2}{45} \int \frac {\left (-\frac {111}{2}-\frac {163 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx\\ &=\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {4 \int \frac {\left (-\frac {5583}{2}-\frac {15321 x}{4}\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{\sqrt {2+3 x}} \, dx}{4725}\\ &=\frac {10214 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {8 \int \frac {\left (-\frac {423783}{8}-\frac {332151 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{354375}\\ &=-\frac {110717 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1063125}+\frac {10214 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {8 \int \frac {\frac {25209339}{16}+\frac {20398839 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3189375}\\ &=-\frac {110717 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1063125}+\frac {10214 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}+\frac {1217887 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{10631250}+\frac {6799613 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5315625}\\ &=-\frac {110717 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1063125}+\frac {10214 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{118125}+\frac {326 (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{4725}+\frac {2}{45} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {6799613 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}-\frac {110717 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}\\ \end {align*}
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Mathematica [A]
time = 3.42, size = 102, normalized size = 0.53 \begin {gather*} \frac {15 \sqrt {2-4 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (526861+55530 x-1111500 x^2+945000 x^3\right )+13599226 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-9945565 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{15946875 \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 | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-850500000 x^{6}+3653661 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-13599226 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+348300000 x^{5}+915408000 x^{4}-575805600 x^{3}-551942790 x^{2}+120636210 x +94834980\right )}{31893750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}-\frac {988 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945}+\frac {1234 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23625}+\frac {526861 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1063125}+\frac {8403113 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{44651250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6799613 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{22325625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
risch | \(-\frac {\left (945000 x^{3}-1111500 x^{2}+55530 x +526861\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{1063125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {8403113 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{116943750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {6799613 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {55-110 x}\, \left (\frac {\EllipticE \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \EllipticF \left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{58471875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
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 = 0.17, size = 38, normalized size = 0.20 \begin {gather*} \frac {1}{1063125} \, {\left (945000 \, x^{3} - 1111500 \, x^{2} + 55530 \, x + 526861\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 )}^{5/2}\,\sqrt {3\,x+2}\,\sqrt {5\,x+3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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