Optimal. Leaf size=113 \[ \frac {\sqrt {5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {233 \sqrt {5 x+3} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (69780 x+168157)}{3520}+\frac {126513 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {97, 150, 147, 54, 216} \[ \frac {\sqrt {5 x+3} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac {233 \sqrt {5 x+3} (3 x+2)^2}{66 \sqrt {1-2 x}}-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (69780 x+168157)}{3520}+\frac {126513 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{320 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 97
Rule 147
Rule 150
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(2+3 x)^2 \left (32+\frac {105 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\left (-\frac {5349}{2}-\frac {17445 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513}{640} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{320 \sqrt {5}}\\ &=-\frac {233 (2+3 x)^2 \sqrt {3+5 x}}{66 \sqrt {1-2 x}}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{3 (1-2 x)^{3/2}}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (168157+69780 x)}{3520}+\frac {126513 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{320 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 90, normalized size = 0.80 \[ \frac {10 \sqrt {2 x-1} \sqrt {5 x+3} \left (71280 x^3+431244 x^2-1786144 x+625431\right )+4174929 \sqrt {10} (1-2 x)^2 \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{105600 \sqrt {1-2 x} (2 x-1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.29, size = 96, normalized size = 0.85 \[ -\frac {4174929 \, \sqrt {10} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (71280 \, x^{3} + 431244 \, x^{2} - 1786144 \, x + 625431\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{211200 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.23, size = 84, normalized size = 0.74 \[ \frac {126513}{3200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (891 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 85 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 2783318 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 45924219 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1320000 \, {\left (2 \, x - 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 137, normalized size = 1.21 \[ \frac {\left (-1425600 \sqrt {-10 x^{2}-x +3}\, x^{3}+16699716 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-8624880 \sqrt {-10 x^{2}-x +3}\, x^{2}-16699716 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+35722880 \sqrt {-10 x^{2}-x +3}\, x +4174929 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12508620 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{211200 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \]
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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