Optimal. Leaf size=31 \[ -\sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {113} \[ -\sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]
Antiderivative was successfully verified.
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Rule 113
Rubi steps
\begin {align*} \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 129, normalized size = 4.16 \[ \frac {\sqrt {3 x+2} \sqrt {\frac {2 x-1}{5 x+3}} \left (5 \sqrt {\frac {2 x-1}{5 x+3}} \sqrt {\frac {3 x+2}{5 x+3}} \sqrt {5 x+3}+i \sqrt {2} E\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {15 x+9}}\right )|-\frac {33}{2}\right )\right )}{5 \sqrt {1-2 x} \sqrt {\frac {3 x+2}{5 x+3}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{10 \, x^{2} + x - 3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2}}{\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 22, normalized size = 0.71 \[ \frac {\sqrt {2}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {3 \, x + 2}}{\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {3\,x+2}}{\sqrt {1-2\,x}\,\sqrt {5\,x+3}} \,d x \]
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 \[ \int \frac {\sqrt {3 x + 2}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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