Optimal. Leaf size=84 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (300 x+979)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 147, 54, 216} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^2}{55 \sqrt {5 x+3}}-\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (300 x+979)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-39-\frac {75 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^2}{55 \sqrt {3+5 x}}-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} (979+300 x)}{4400}+\frac {2493 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 82, normalized size = 0.98 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \left (5940 x^2+19305 x+9451\right )+27423 \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{44000 \sqrt {2 x-1} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 81, normalized size = 0.96 \[ -\frac {27423 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (5940 \, x^{2} + 19305 \, x + 9451\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{88000 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 111, normalized size = 1.32 \[ -\frac {27}{10000} \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} + 41 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {2493}{4000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{13750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{6875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 99, normalized size = 1.18 \[ \frac {\left (-118800 \sqrt {-10 x^{2}-x +3}\, x^{2}+137115 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-386100 \sqrt {-10 x^{2}-x +3}\, x +82269 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-189020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{88000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 65, normalized size = 0.77 \[ \frac {2493}{8000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{100} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1431}{2000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{1375 \, {\left (5 \, x + 3\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^3}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (3 x + 2\right )^{3}}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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