Optimal. Leaf size=72 \[ \frac {7 (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}+\frac {103}{44} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {103 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 72, 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, 50, 54, 216} \begin {gather*} \frac {7 (5 x+3)^{3/2}}{11 \sqrt {1-2 x}}+\frac {103}{44} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {103 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 54
Rule 78
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x) \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {7 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}-\frac {103}{22} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=\frac {103}{44} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}-\frac {103}{8} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {103}{44} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}-\frac {103 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{4 \sqrt {5}}\\ &=\frac {103}{44} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {7 (3+5 x)^{3/2}}{11 \sqrt {1-2 x}}-\frac {103 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 1.01 \begin {gather*} \frac {10 (17-6 x) \sqrt {2 x-1} \sqrt {5 x+3}-103 \sqrt {10} (2 x-1) \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{40 \sqrt {-(1-2 x)^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.12, size = 93, normalized size = 1.29 \begin {gather*} \frac {\sqrt {5 x+3} \left (\frac {103 (1-2 x)}{5 x+3}+28\right )}{4 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}+\frac {103 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{4 \sqrt {10}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.34, size = 76, normalized size = 1.06 \begin {gather*} \frac {103 \, \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 (6 \, x - 17\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{80 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.19, size = 58, normalized size = 0.81 \begin {gather*} -\frac {103}{40} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {{\left (6 \, \sqrt {5} {\left (5 \, x + 3\right )} - 103 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{100 \, {\left (2 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 89, normalized size = 1.24 \begin {gather*} -\frac {\left (206 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-120 \sqrt {-10 x^{2}-x +3}\, x -103 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+340 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{80 \left (2 x -1\right ) \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 = 1.18, size = 50, normalized size = 0.69 \begin {gather*} -\frac {103}{80} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {3}{4} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{2 \, {\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 )}^{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 {\left (3 x + 2\right ) \sqrt {5 x + 3}}{\left (1 - 2 x\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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