Optimal. Leaf size=84 \[ -\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {100, 147, 54, 216} \[ -\frac {1}{10} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {\sqrt {1-2 x} \sqrt {5 x+3} (2220 x+5363)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1600 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 100
Rule 147
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^3}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{30} \int \frac {\left (-171-\frac {555 x}{2}\right ) (2+3 x)}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{3200}\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1600 \sqrt {5}}\\ &=-\frac {1}{10} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {\sqrt {1-2 x} \sqrt {3+5 x} (5363+2220 x)}{1600}+\frac {44437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1600 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 78, normalized size = 0.93 \[ -\frac {\sqrt {1-2 x} \left (90 \sqrt {2 x-1} \sqrt {5 x+3} \left (160 x^2+460 x+667\right )+44437 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{16000 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 67, normalized size = 0.80 \[ -\frac {9}{1600} \, {\left (160 \, x^{2} + 460 \, x + 667\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {44437}{32000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.85, size = 54, normalized size = 0.64 \[ -\frac {1}{80000} \, \sqrt {5} {\left (18 \, {\left (4 \, {\left (40 \, x + 91\right )} {\left (5 \, x + 3\right )} + 2243\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 222185 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 87, normalized size = 1.04 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-28800 \sqrt {-10 x^{2}-x +3}\, x^{2}-82800 \sqrt {-10 x^{2}-x +3}\, x +44437 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-120060 \sqrt {-10 x^{2}-x +3}\right )}{32000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 58, normalized size = 0.69 \[ -\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {207}{80} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {44437}{32000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {6003}{1600} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.75, size = 534, normalized size = 6.36 \[ \frac {44437\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{8000}-\frac {\frac {18837\,\left (\sqrt {1-2\,x}-1\right )}{390625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^3}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {226251\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {154377\,{\left (\sqrt {1-2\,x}-1\right )}^9}{10000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {18837\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{4000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {4608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {59904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {107136\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {14976\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}}{\frac {192\,{\left (\sqrt {1-2\,x}-1\right )}^2}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {48\,{\left (\sqrt {1-2\,x}-1\right )}^4}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^8}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {12\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{12}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {64}{15625}} \]
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 \[ \int \frac {\left (3 x + 2\right )^{3}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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