Optimal. Leaf size=74 \[ \frac {7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x}}+\frac {3}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]
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Rubi [A] time = 0.02, 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 = {78, 47, 54, 216} \begin {gather*} \frac {7 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}}-\frac {3 \sqrt {5 x+3}}{2 \sqrt {1-2 x}}+\frac {3}{2} \sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 54
Rule 78
Rule 216
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 ) \operatorname {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 [C] time = 0.06, size = 51, normalized size = 0.69 \begin {gather*} \frac {363 \sqrt {22} \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {5}{11} (1-2 x)\right )+8 (5 x+3)^{3/2}}{660 (1-2 x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 77, normalized size = 1.04 \begin {gather*} \frac {(5 x+3)^{3/2} \left (14-\frac {99 (1-2 x)}{5 x+3}\right )}{66 (1-2 x)^{3/2}}-\frac {3}{2} \sqrt {\frac {5}{2}} \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 [A] time = 1.28, 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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giac [A] time = 1.08, 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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maple [A] time = 0.01, size = 103, normalized size = 1.39 \begin {gather*} \frac {\left (396 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-396 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+1072 \sqrt {-10 x^{2}-x +3}\, x +99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-228 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{264 \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.22, 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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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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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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