Optimal. Leaf size=218 \[ -\frac {5442127 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{3543750 \sqrt {33}}+\frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac {23 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{9625}-\frac {40703 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{433125}-\frac {5442127 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{7796250}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}} \]
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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} \[ \frac {2}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}+\frac {62 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}{2475}-\frac {23 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{9625}-\frac {40703 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{433125}-\frac {5442127 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{7796250}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750 \sqrt {33}}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}} \]
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} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx &=\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {2}{55} \int \left (-\frac {69}{2}-\frac {93 x}{2}\right ) \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx\\ &=\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {1635}{2}-\frac {621 x}{4}\right ) \sqrt {2+3 x} (3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{7425}\\ &=-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {4 \int \frac {(3+5 x)^{3/2} \left (\frac {486987}{8}+\frac {366327 x}{4}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{259875}\\ &=-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {7951311}{2}-\frac {48979143 x}{8}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{3898125}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {4 \int \frac {\frac {2060337177}{16}+203394069 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{35083125}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}+\frac {5442127 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{7087500}+\frac {90397364 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19490625}\\ &=-\frac {5442127 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{7796250}-\frac {40703 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{433125}-\frac {23 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{9625}+\frac {62 \sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}}{2475}+\frac {2}{55} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {90397364 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1771875 \sqrt {33}}-\frac {5442127 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3543750 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 110, normalized size = 0.50 \[ \frac {361589456 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-5 \left (36399853 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+3 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (42525000 x^4+43470000 x^3-17237250 x^2-27227430 x-810641\right )\right )}{116943750 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}, x\right ) \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (-38272500000 x^{7}-68465250000 x^{6}-5550525000 x^{5}+53181589500 x^{4}+23721281100 x^{3}-8261123010 x^{2}-5071172010 x -361589456 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+181999265 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-145915380\right )}{7016625000 x^{3}+5379412500 x^{2}-1637212500 x -1403325000} \]
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 \[ \int {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \]
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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