Optimal. Leaf size=191 \[ -\frac {62092 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{14175}-\frac {1}{9} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {62092 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2835}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700} \]
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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} \[ -\frac {1}{9} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {62092 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2835}-\frac {62092 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14175}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(2+3 x)^{3/2} (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {1}{9} \int \frac {\sqrt {2+3 x} (3+5 x)^{3/2} \left (\frac {173}{2}+135 x\right )}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {1}{315} \int \frac {\left (-\frac {18455}{2}-\frac {28155 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx\\ &=-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {\int \frac {\sqrt {3+5 x} \left (\frac {2421135}{4}+931380 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{4725}\\ &=-\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {\int \frac {-\frac {78409965}{4}-\frac {123853155 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{42525}\\ &=-\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {341506 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{14175}+\frac {8256877 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{56700}\\ &=-\frac {62092 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2835}-\frac {1877}{630} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {3}{7} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {1}{9} \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {8256877 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{56700}-\frac {62092 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{14175}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 105, normalized size = 0.55 \[ \frac {8256877 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (831761 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (47250 x^3+148950 x^2+212175 x+208073\right )\right )}{85050 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.30, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2 \, x - 1}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 155, normalized size = 0.81 \[ \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-42525000 x^{6}-166657500 x^{5}-283810500 x^{4}-293881950 x^{3}-72202620 x^{2}+81886830 x -8256877 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+4158805 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+37453140\right )}{5103000 x^{3}+3912300 x^{2}-1190700 x -1020600} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}}}{\sqrt {-2 \, x + 1}}\,{d x} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}} \,d x \]
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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