Optimal. Leaf size=74 \[ -\frac {3 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac {3}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {79, 49, 56, 222}
\begin {gather*} \frac {3}{2} \sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )+\frac {7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 49
Rule 56
Rule 79
Rule 222
Rubi steps
\begin {align*} \int \frac {(2+3 x) \sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx &=\frac {7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {3}{2} \int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\\ &=-\frac {3 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac {15}{4} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {3 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac {1}{2} \left (3 \sqrt {5}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=-\frac {3 \sqrt {3+5 x}}{2 \sqrt {1-2 x}}+\frac {7 (3+5 x)^{3/2}}{33 (1-2 x)^{3/2}}+\frac {3}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 68, normalized size = 0.92 \begin {gather*} \frac {2 \sqrt {3+5 x} (-57+268 x)+99 \sqrt {10-20 x} (-1+2 x) \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{132 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 103, normalized size = 1.39
method | result | size |
default | \(\frac {\left (396 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x^{2}-396 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1072 x \sqrt {-10 x^{2}-x +3}-228 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{264 \left (-1+2 x \right )^{2} \sqrt {-10 x^{2}-x +3}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 48, normalized size = 0.65 \begin {gather*} \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {10 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.51, size = 92, normalized size = 1.24 \begin {gather*} -\frac {99 \, \sqrt {5} \sqrt {2} {\left (4 \, x^{2} - 4 \, x + 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 4 \, {\left (268 \, x - 57\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{264 \, {\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 ) \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 58, normalized size = 0.78 \begin {gather*} \frac {3}{4} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (268 \, \sqrt {5} {\left (5 \, x + 3\right )} - 1089 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{1650 \, {\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 )\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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