Optimal. Leaf size=52 \[ \frac {3}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x}}{55 \sqrt {5 x+3}} \]
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Rubi [A] time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {78, 54, 216} \begin {gather*} \frac {3}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x}}{55 \sqrt {5 x+3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rule 78
Rule 216
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {3}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {6 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {3}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}
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Mathematica [A] time = 0.07, size = 59, normalized size = 1.13 \begin {gather*} \frac {1}{275} \sqrt {1-2 x} \left (-\frac {10}{\sqrt {5 x+3}}-\frac {33 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 61, normalized size = 1.17 \begin {gather*} -\frac {2 \sqrt {1-2 x}}{55 \sqrt {5 x+3}}-\frac {3}{5} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 77, normalized size = 1.48 \begin {gather*} -\frac {33 \, \sqrt {5} \sqrt {2} {\left (5 \, x + 3\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 ) + 20 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{550 \, {\left (5 \, x + 3\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.09, size = 79, normalized size = 1.52 \begin {gather*} \frac {3}{25} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{550 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{275 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 1.29 \begin {gather*} \frac {\left (165 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-20 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{550 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 36, normalized size = 0.69 \begin {gather*} \frac {3}{50} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{55 \, {\left (5 \, x + 3\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 = -0.02 \begin {gather*} \int \frac {3\,x+2}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \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 {3 x + 2}{\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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