Optimal. Leaf size=110 \[ \frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {7}{4} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 110, 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, 153, 147, 54, 216} \[ \frac {\sqrt {5 x+3} (3 x+2)^3}{\sqrt {1-2 x}}+\frac {7}{4} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {\sqrt {1-2 x} \sqrt {5 x+3} (73380 x+176833)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{3200 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 97
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3 \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}-\int \frac {(2+3 x)^2 \left (32+\frac {105 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {1}{30} \int \frac {\left (-\frac {5625}{2}-\frac {18345 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{3200 \sqrt {5}}\\ &=\frac {7}{4} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {\sqrt {1-2 x} \sqrt {3+5 x} (176833+73380 x)}{3200}-\frac {1463447 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{3200 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.75 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (14400 x^3+57960 x^2+142686 x-224833\right )-1463447 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{32000 \sqrt {-(1-2 x)^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.82, size = 86, normalized size = 0.78 \[ \frac {1463447 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{64000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 84, normalized size = 0.76 \[ -\frac {1463447}{32000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (18 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 89 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4927 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 1463447 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{80000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 123, normalized size = 1.12 \[ -\frac {\left (-288000 \sqrt {-10 x^{2}-x +3}\, x^{3}-1159200 \sqrt {-10 x^{2}-x +3}\, x^{2}+2926894 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2853720 \sqrt {-10 x^{2}-x +3}\, x -1463447 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+4496660 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{64000 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 79, normalized size = 0.72 \[ -\frac {1463447}{64000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1593}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {89793}{3200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{8 \, {\left (2 \, x - 1\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^3\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (3 x + 2\right )^{3} \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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