Optimal. Leaf size=218 \[ -\frac {1654421 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1913625 \sqrt {33}}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {9698 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{93555}-\frac {146963 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{467775}-\frac {1654421 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4209975}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}} \]
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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} \[ \frac {2}{33} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}+\frac {9698 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}{93555}-\frac {146963 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}{467775}-\frac {1654421 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}{4209975}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}} \]
Antiderivative was successfully verified.
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Rule 101
Rule 113
Rule 119
Rule 154
Rule 158
Rubi steps
\begin {align*} \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{\sqrt {2+3 x}} \, dx &=\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {2}{33} \int \frac {\left (-50-\frac {185 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx\\ &=\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {4 \int \frac {\left (-\frac {9245}{4}-\frac {24245 x}{4}\right ) \sqrt {1-2 x} (3+5 x)^{3/2}}{\sqrt {2+3 x}} \, dx}{4455}\\ &=\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {\left (\frac {84395}{4}-\frac {2204445 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{467775}\\ &=-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {8 \int \frac {\sqrt {3+5 x} \left (\frac {44328915}{16}+\frac {24816315 x}{8}\right )}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{7016625}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {8 \int \frac {-\frac {685297455}{8}-\frac {2193331695 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{63149625}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {1654421 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{3827250}+\frac {146222113 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{42099750}\\ &=-\frac {1654421 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{4209975}-\frac {146963 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}}{467775}+\frac {9698 \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}}{93555}+\frac {74}{891} (1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{33} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{3827250 \sqrt {33}}-\frac {1654421 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1913625 \sqrt {33}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 107, normalized size = 0.49 \[ \frac {-91626220 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+15 \sqrt {2-4 x} \sqrt {3 x+2} \sqrt {5 x+3} \left (25515000 x^4-12379500 x^3-16381350 x^2+9143865 x+3748468\right )+146222113 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{63149625 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{\sqrt {3 \, x + 2}}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}}\,{d x} \]
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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \left (22963500000 x^{7}+6463800000 x^{6}-28643220000 x^{5}-5066658000 x^{4}+15351281550 x^{3}+3614874270 x^{2}-2433073980 x -146222113 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+91626220 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-674724240\right )}{3788977500 x^{3}+2904882750 x^{2}-884094750 x -757795500} \]
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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{\sqrt {3\,x+2}} \,d x \]
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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