Optimal. Leaf size=218 \[ -\frac {4738087 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8859375 \sqrt {33}}+\frac {2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac {2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt {1-2 x}}{86625}+\frac {38729 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{2165625}-\frac {4738087 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{19490625}-\frac {326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}} \]
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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} \[ \frac {2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac {106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac {2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt {1-2 x}}{86625}+\frac {38729 \sqrt {3 x+2} (5 x+3)^{3/2} \sqrt {1-2 x}}{2165625}-\frac {4738087 \sqrt {3 x+2} \sqrt {5 x+3} \sqrt {1-2 x}}{19490625}-\frac {4738087 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}-\frac {326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}} \]
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)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x} \, dx &=\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {2}{55} \int \left (-\frac {113}{2}-\frac {159 x}{2}\right ) (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x} \, dx\\ &=\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {4 \int \left (-2979-\frac {12897 x}{4}\right ) \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \, dx}{7425}\\ &=\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {8 \int \frac {\sqrt {2+3 x} \sqrt {3+5 x} \left (-\frac {670815}{8}+\frac {348561 x}{8}\right )}{\sqrt {1-2 x}} \, dx}{779625}\\ &=\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {57670353}{16}+\frac {42642783 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19490625}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {8 \int \frac {-\frac {463899753}{4}-\frac {2936308149 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{175415625}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {4738087 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{17718750}+\frac {326256461 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{194906250}\\ &=-\frac {4738087 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{19490625}+\frac {38729 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{2165625}+\frac {2866 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac {106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac {2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac {326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17718750 \sqrt {33}}-\frac {4738087 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{8859375 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 107, normalized size = 0.49 \[ \frac {-169899590 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (42525000 x^4-13702500 x^3-35750250 x^2+16294455 x+9437696\right )+326256461 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{292359375 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.19, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 160, normalized size = 0.73 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (38272500000 x^{7}+17010000000 x^{6}-50560200000 x^{5}-14779638000 x^{4}+29711102850 x^{3}+9525219690 x^{2}-4914918060 x -326256461 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+169899590 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1698785280\right )}{17541562500 x^{3}+13448531250 x^{2}-4093031250 x -3508312500} \]
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 \[ \int \sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,\sqrt {5\,x+3} \,d x \]
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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