Optimal. Leaf size=113 \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}-\frac {21}{550} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (3660 x+8987)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \]
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Rubi [A] time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 153, 147, 54, 216} \[ -\frac {2 \sqrt {1-2 x} (3 x+2)^3}{55 \sqrt {5 x+3}}-\frac {21}{550} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {21 \sqrt {1-2 x} \sqrt {5 x+3} (3660 x+8987)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8000 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 98
Rule 147
Rule 153
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^4}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {2}{55} \int \frac {\left (-42-\frac {63 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}+\frac {1}{825} \int \frac {(2+3 x) \left (\frac {6111}{2}+\frac {19215 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16000}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8000 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x} (2+3 x)^3}{55 \sqrt {3+5 x}}-\frac {21}{550} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 \sqrt {1-2 x} \sqrt {3+5 x} (8987+3660 x)}{88000}+\frac {143283 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8000 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 87, normalized size = 0.77 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \left (237600 x^3+849420 x^2+1477575 x+632101\right )+1576113 \sqrt {50 x+30} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{880000 \sqrt {2 x-1} \sqrt {5 x+3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 86, normalized size = 0.76 \[ -\frac {1576113 \, \sqrt {10} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, {\left (237600 \, x^{3} + 849420 \, x^{2} + 1477575 \, x + 632101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{1760000 \, {\left (5 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 124, normalized size = 1.10 \[ -\frac {27}{200000} \, {\left (4 \, {\left (8 \, \sqrt {5} {\left (5 \, x + 3\right )} + 71 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 2407 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + \frac {143283}{80000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{68750 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{34375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 116, normalized size = 1.03 \[ \frac {\left (-4752000 \sqrt {-10 x^{2}-x +3}\, x^{3}-16988400 \sqrt {-10 x^{2}-x +3}\, x^{2}+7880565 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-29551500 \sqrt {-10 x^{2}-x +3}\, x +4728339 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-12642020 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{1760000 \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 82, normalized size = 0.73 \[ -\frac {27}{50} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {143283}{160000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {3213}{2000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {95769}{40000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{6875 \, {\left (5 \, x + 3\right )}} \]
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 \[ \int \frac {{\left (3\,x+2\right )}^4}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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