Optimal. Leaf size=30 \[ -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {718, 424} \[ -\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \]
Antiderivative was successfully verified.
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Rule 424
Rule 718
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx &=-\left (\frac {1}{3} \sqrt {19} \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {55 x^2}{76}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {16-24 x}}{\sqrt {22}}\right )\right )\\ &=-\frac {1}{3} \sqrt {19} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right )\\ \end {align*}
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Mathematica [B] time = 0.08, size = 86, normalized size = 2.87 \[ \frac {\sqrt {19} \sqrt {-4 x-1} \sqrt {2-3 x} \left (F\left (\sin ^{-1}\left (\frac {2 \sqrt {5 x+3}}{\sqrt {7}}\right )|\frac {21}{76}\right )-E\left (\sin ^{-1}\left (\frac {2 \sqrt {5 x+3}}{\sqrt {7}}\right )|\frac {21}{76}\right )\right )}{3 \sqrt {-12 x^2+5 x+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-12 \, x^{2} + 5 \, x + 2} \sqrt {5 \, x + 3}}{12 \, x^{2} - 5 \, x - 2}, x\right ) \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 77, normalized size = 2.57 \[ -\frac {\left (-\EllipticE \left (\frac {\sqrt {285 x +171}}{19}, \frac {2 \sqrt {399}}{21}\right )+\EllipticF \left (\frac {\sqrt {285 x +171}}{19}, \frac {2 \sqrt {399}}{21}\right )\right ) \sqrt {-285 x +190}\, \sqrt {-140 x -35}\, \sqrt {57}\, \sqrt {-12 x^{2}+5 x +2}}{570 \left (12 x^{2}-5 x -2\right )} \]
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 \[ \int \frac {\sqrt {5 \, x + 3}}{\sqrt {-12 \, x^{2} + 5 \, x + 2}}\,{d x} \]
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 \[ \int \frac {\sqrt {5\,x+3}}{\sqrt {-12\,x^2+5\,x+2}} \,d x \]
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 \[ \int \frac {\sqrt {5 x + 3}}{\sqrt {- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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