Optimal. Leaf size=84 \[ -\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}} \]
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Rubi [A]
time = 0.01, 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 = {100, 148, 56,
222} \begin {gather*} \frac {1593 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}}+\frac {7 \sqrt {5 x+3} (3 x+2)^2}{33 (1-2 x)^{3/2}}-\frac {(95621-33462 x) \sqrt {5 x+3}}{14520 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 100
Rule 148
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx &=\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x) \left (155+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593}{80} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40 \sqrt {5}}\\ &=-\frac {(95621-33462 x) \sqrt {3+5 x}}{14520 \sqrt {1-2 x}}+\frac {7 (2+3 x)^2 \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}+\frac {1593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 73, normalized size = 0.87 \begin {gather*} \frac {-10 \sqrt {3+5 x} \left (83301-261664 x+39204 x^2\right )+578259 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{145200 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 120, normalized size = 1.43
method | result | size |
default | \(\frac {\left (2313036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-2313036 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -784080 x^{2} \sqrt {-10 x^{2}-x +3}+578259 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+5233280 x \sqrt {-10 x^{2}-x +3}-1666020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{290400 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 76, normalized size = 0.90 \begin {gather*} \frac {1593}{800} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {27}{40} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {343 \, \sqrt {-10 \, x^{2} - x + 3}}{132 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {11123 \, \sqrt {-10 \, x^{2} - x + 3}}{1452 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 91, normalized size = 1.08 \begin {gather*} -\frac {578259 \, \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 (39204 \, x^{2} - 261664 \, x + 83301\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{290400 \, {\left (4 \, x^{2} - 4 \, 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 )^{3}}{\left (1 - 2 x\right )^{\frac {5}{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 = 1.95, size = 71, normalized size = 0.85 \begin {gather*} \frac {1593}{400} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (9801 \, \sqrt {5} {\left (5 \, x + 3\right )} - 385886 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 6360321 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1815000 \, {\left (2 \, x - 1\right )}^{2}} \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 )}^3}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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