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 = 0.01, 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 = {732, 435}
\begin {gather*} -\frac {1}{3} \sqrt {19} E\left (\text {ArcSin}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|\frac {55}{76}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 435
Rule 732
Rubi steps
\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {2+5 x-12 x^2}} \, dx &=-\left (\frac {1}{3} \sqrt {19} \text {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] Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(30)=60\).
time = 30.08, size = 86, normalized size = 2.87 \begin {gather*} \frac {\sqrt {19} \sqrt {-1-4 x} \sqrt {2-3 x} \left (-E\left (\sin ^{-1}\left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right )|\frac {21}{76}\right )+F\left (\sin ^{-1}\left (\frac {2 \sqrt {3+5 x}}{\sqrt {7}}\right )|\frac {21}{76}\right )\right )}{3 \sqrt {2+5 x-12 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs.
\(2(23)=46\).
time = 0.90, size = 77, normalized size = 2.57
method | result | size |
default | \(-\frac {\left (\EllipticF \left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )-\EllipticE \left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )\right ) \sqrt {190-285 x}\, \sqrt {-35-140 x}\, \sqrt {57}\, \sqrt {-12 x^{2}+5 x +2}}{570 \left (12 x^{2}-5 x -2\right )}\) | \(77\) |
elliptic | \(\frac {\sqrt {-\left (12 x^{2}-5 x -2\right ) \left (3+5 x \right )}\, \left (\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, \EllipticF \left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{665 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}+\frac {2 \sqrt {171+285 x}\, \sqrt {-35-140 x}\, \sqrt {190-285 x}\, \left (-\frac {7 \EllipticE \left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{20}-\frac {\EllipticF \left (\frac {\sqrt {171+285 x}}{19}, \frac {2 \sqrt {399}}{21}\right )}{4}\right )}{399 \sqrt {-60 x^{3}-11 x^{2}+25 x +6}}\right )}{\sqrt {3+5 x}\, \sqrt {-12 x^{2}+5 x +2}}\) | \(171\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {5 x + 3}}{\sqrt {- \left (3 x - 2\right ) \left (4 x + 1\right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {5\,x+3}}{\sqrt {-12\,x^2+5\,x+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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