Optimal. Leaf size=50 \[ \frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {3}{10} \sqrt {1-2 x} \sqrt {5 x+3} \]
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Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 54, 216} \[ \frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10 \sqrt {10}}-\frac {3}{10} \sqrt {1-2 x} \sqrt {5 x+3} \]
Antiderivative was successfully verified.
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Rule 54
Rule 80
Rule 216
Rubi steps
\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37}{20} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10 \sqrt {5}}\\ &=-\frac {3}{10} \sqrt {1-2 x} \sqrt {3+5 x}+\frac {37 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10 \sqrt {10}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 1.36 \[ -\frac {3}{10} \sqrt {1-2 x} \sqrt {5 x+3}-\frac {37 \sqrt {1-2 x} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{10 \sqrt {10} \sqrt {2 x-1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 57, normalized size = 1.14 \[ -\frac {37}{200} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - \frac {3}{10} \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 40, normalized size = 0.80 \[ \frac {1}{100} \, \sqrt {5} {\left (37 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - 6 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 55, normalized size = 1.10 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (37 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-60 \sqrt {-10 x^{2}-x +3}\right )}{200 \sqrt {-10 x^{2}-x +3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.19, size = 26, normalized size = 0.52 \[ -\frac {37}{200} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {3}{10} \, \sqrt {-10 \, x^{2} - x + 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.08, size = 220, normalized size = 4.40 \[ -\frac {\frac {3\,{\left (\sqrt {1-2\,x}-1\right )}^3}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {6\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {24\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}}{\frac {4\,{\left (\sqrt {1-2\,x}-1\right )}^2}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^4}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {4}{25}}-\frac {3\,\sqrt {10}\,\left (\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )+\frac {40\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}{5\,\left (\sqrt {1-2\,x}-1\right )}\right )}{3}\right )}{50} \]
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 \[ \int \frac {3 x + 2}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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