Optimal. Leaf size=77 \[ -\frac {3}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)-\frac {333}{400} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
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Rubi [A] time = 0.02, antiderivative size = 77, 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 = {90, 80, 54, 216} \[ -\frac {3}{20} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)-\frac {333}{400} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400 \sqrt {10}} \]
Antiderivative was successfully verified.
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Rule 54
Rule 80
Rule 90
Rule 216
Rubi steps
\begin {align*} \int \frac {(2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}-\frac {1}{20} \int \frac {-104-\frac {333 x}{2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827}{800} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400 \sqrt {5}}\\ &=-\frac {333}{400} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {3}{20} \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}+\frac {3827 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 73, normalized size = 0.95 \[ -\frac {\sqrt {1-2 x} \left (30 \sqrt {2 x-1} \sqrt {5 x+3} (60 x+151)+3827 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{4000 \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 62, normalized size = 0.81 \[ -\frac {3}{400} \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {3827}{8000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 45, normalized size = 0.58 \[ -\frac {1}{4000} \, \sqrt {5} {\left (6 \, {\left (60 \, x + 151\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 3827 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.91 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-3600 \sqrt {-10 x^{2}-x +3}\, x +3827 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9060 \sqrt {-10 x^{2}-x +3}\right )}{8000 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 41, normalized size = 0.53 \[ -\frac {9}{20} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {3827}{8000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {453}{400} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.63, size = 360, normalized size = 4.68 \[ \frac {3827\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{2000}-\frac {\frac {627\,\left (\sqrt {1-2\,x}-1\right )}{15625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^3}{6250\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {7941\,{\left (\sqrt {1-2\,x}-1\right )}^5}{2500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {627\,{\left (\sqrt {1-2\,x}-1\right )}^7}{1000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1632\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {32\,{\left (\sqrt {1-2\,x}-1\right )}^2}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {24\,{\left (\sqrt {1-2\,x}-1\right )}^4}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8\,{\left (\sqrt {1-2\,x}-1\right )}^6}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {16}{625}} \]
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 \[ \int \frac {\left (3 x + 2\right )^{2}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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