Optimal. Leaf size=191 \[ -\frac {110717 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5315625}+\frac {2}{45} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {326 \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac {10214 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{118125}-\frac {110717 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1063125}-\frac {6799613 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625} \]
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Rubi [A] time = 0.07, 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 = {101, 154, 158, 113, 119} \[ \frac {2}{45} \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {326 \sqrt {3 x+2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{4725}+\frac {10214 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{118125}-\frac {110717 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{1063125}-\frac {110717 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625}-\frac {6799613 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5315625} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
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 = 0.31, size = 102, normalized size = 0.53 \[ \frac {-9945565 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (945000 x^3-1111500 x^2+55530 x+526861\right )+13599226 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{15946875 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 155, normalized size = 0.81 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (850500000 x^{6}-348300000 x^{5}-915408000 x^{4}+575805600 x^{3}+551942790 x^{2}-120636210 x -13599226 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+9945565 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-94834980\right )}{956812500 x^{3}+733556250 x^{2}-223256250 x -191362500} \]
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 \[ \int \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,\sqrt {5\,x+3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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