Optimal. Leaf size=218 \[ -\frac {2930159 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1275750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{7425}-\frac {2377 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{155925}-\frac {22576 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{155925}-\frac {2930159 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2806650}-\frac {97540001 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}} \]
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Rubi [A] time = 0.08, 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 = {101, 154, 158, 113, 119} \[ \frac {2}{55} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{7/2}+\frac {194 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}}{7425}-\frac {2377 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{155925}-\frac {22576 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{155925}-\frac {2930159 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{2806650}-\frac {2930159 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}-\frac {97540001 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}} \]
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)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx &=\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {2}{55} \int \frac {\left (-\frac {67}{2}-\frac {97 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {1273}{2}-\frac {2377 x}{4}\right ) (3+5 x)^{5/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7425}\\ &=-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {4 \int \frac {(3+5 x)^{3/2} \left (\frac {432515}{8}+84660 x\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{155925}\\ &=-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {4 \int \frac {\left (-\frac {28606455}{8}-\frac {43952385 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2338875}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {4 \int \frac {\frac {1852415265}{16}+\frac {1463100015 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{21049875}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}+\frac {2930159 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2551500}+\frac {97540001 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{14033250}\\ &=-\frac {2930159 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{2806650}-\frac {22576 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{155925}-\frac {2377 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{155925}+\frac {194 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}}{7425}+\frac {2}{55} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {97540001 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}-\frac {2930159 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1275750 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 107, normalized size = 0.49 \[ \frac {-98384755 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )-15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (25515000 x^4+24003000 x^3-10837350 x^2-14851260 x-201247\right )+195080002 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{42099750 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 160, normalized size = 0.73 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (-22963500000 x^{7}-39208050000 x^{6}-1450305000 x^{5}+30477235500 x^{4}+12473188200 x^{3}-4930627170 x^{2}-2715488670 x -195080002 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+98384755 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-36224460\right )}{2525985000 x^{3}+1936588500 x^{2}-589396500 x -505197000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \]
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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