Optimal. Leaf size=72 \[ \frac {49 \sqrt {3+5 x}}{22 \sqrt {1-2 x}}+\frac {9}{20} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}} \]
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Rubi [A]
time = 0.01, antiderivative size = 72, 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 = {91, 81, 56, 222}
\begin {gather*} -\frac {321 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{20 \sqrt {10}}+\frac {9}{20} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {49 \sqrt {5 x+3}}{22 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 56
Rule 81
Rule 91
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {49 \sqrt {3+5 x}}{22 \sqrt {1-2 x}}-\frac {1}{22} \int \frac {\frac {363}{2}+99 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {49 \sqrt {3+5 x}}{22 \sqrt {1-2 x}}+\frac {9}{20} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {321}{40} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {49 \sqrt {3+5 x}}{22 \sqrt {1-2 x}}+\frac {9}{20} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {321 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{20 \sqrt {5}}\\ &=\frac {49 \sqrt {3+5 x}}{22 \sqrt {1-2 x}}+\frac {9}{20} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{20 \sqrt {10}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 63, normalized size = 0.88 \begin {gather*} \frac {10 (589-198 x) \sqrt {3+5 x}+3531 \sqrt {10-20 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{2200 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 89, normalized size = 1.24
method | result | size |
default | \(-\frac {\left (7062 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -3531 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3960 x \sqrt {-10 x^{2}-x +3}+11780 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{4400 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.63, size = 50, normalized size = 0.69 \begin {gather*} -\frac {321}{400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {9}{20} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {49 \, \sqrt {-10 \, x^{2} - x + 3}}{22 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.81, size = 76, normalized size = 1.06 \begin {gather*} \frac {3531 \, \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 (198 \, x - 589\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4400 \, {\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 )^{2}}{\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 = 0.91, size = 58, normalized size = 0.81 \begin {gather*} -\frac {321}{200} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (198 \, \sqrt {5} {\left (5 \, x + 3\right )} - 3539 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5500 \, {\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 )}^2}{{\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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