Optimal. Leaf size=113 \[ \frac {7 \sqrt {5 x+3} (3 x+2)^3}{11 \sqrt {1-2 x}}+\frac {243}{220} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {9 \sqrt {1-2 x} \sqrt {5 x+3} (11316 x+27269)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \]
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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 = {98, 153, 147, 54, 216} \[ \frac {7 \sqrt {5 x+3} (3 x+2)^3}{11 \sqrt {1-2 x}}+\frac {243}{220} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2+\frac {9 \sqrt {1-2 x} \sqrt {5 x+3} (11316 x+27269)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {1}{11} \int \frac {(2+3 x)^2 \left (222+\frac {729 x}{2}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {1}{330} \int \frac {\left (-\frac {39033}{2}-\frac {127305 x}{4}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1280}\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{640 \sqrt {5}}\\ &=\frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 0.73 \[ \frac {-10 \sqrt {2 x-1} \sqrt {5 x+3} \left (19008 x^3+78408 x^2+196614 x-312365\right )-2031051 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{70400 \sqrt {-(1-2 x)^2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.22, size = 86, normalized size = 0.76 \[ \frac {2031051 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (19008 \, x^{3} + 78408 \, x^{2} + 196614 \, x - 312365\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{140800 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 84, normalized size = 0.74 \[ -\frac {184641}{6400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (594 \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 93 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 5179 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 50776531 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{4400000 \, {\left (2 \, x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 123, normalized size = 1.09 \[ -\frac {\left (-380160 \sqrt {-10 x^{2}-x +3}\, x^{3}-1568160 \sqrt {-10 x^{2}-x +3}\, x^{2}+4062102 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3932280 \sqrt {-10 x^{2}-x +3}\, x -2031051 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+6247300 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{140800 \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 82, normalized size = 0.73 \[ \frac {27}{20} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {184641}{12800} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {999}{160} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {2187}{128} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2401 \, \sqrt {-10 \, x^{2} - x + 3}}{88 \, {\left (2 \, x - 1\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \]
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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