Optimal. Leaf size=113 \[ \frac {243}{220} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {7 (2+3 x)^3 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {9 \sqrt {1-2 x} \sqrt {3+5 x} (27269+11316 x)}{7040}-\frac {184641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640 \sqrt {10}} \]
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Rubi [A]
time = 0.02, 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 = {100, 158, 152,
56, 222} \begin {gather*} -\frac {184641 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640 \sqrt {10}}+\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 152
Rule 158
Rule 222
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 \text {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 = 0.16, size = 73, normalized size = 0.65 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (-312365+196614 x+78408 x^2+19008 x^3\right )+2031051 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{70400 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 123, normalized size = 1.09
method | result | size |
default | \(-\frac {\left (-380160 x^{3} \sqrt {-10 x^{2}-x +3}+4062102 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -1568160 x^{2} \sqrt {-10 x^{2}-x +3}-2031051 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3932280 x \sqrt {-10 x^{2}-x +3}+6247300 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{140800 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 82, normalized size = 0.73 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 86, normalized size = 0.76 \begin {gather*} \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 )}} \end {gather*}
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 \begin {gather*} \int \frac {\left (3 x + 2\right )^{4}}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {5 x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.31, size = 84, normalized size = 0.74 \begin {gather*} -\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 )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (3\,x+2\right )}^4}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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