Optimal. Leaf size=88 \[ \frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+4 x^2}}-\frac {\sqrt {2+3 x^2} E\left (\tan ^{-1}(2 x)|\frac {5}{8}\right )}{3 \sqrt {2} \sqrt {\frac {2+3 x^2}{1+4 x^2}} \sqrt {1+4 x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {506, 422}
\begin {gather*} \frac {x \sqrt {3 x^2+2}}{3 \sqrt {4 x^2+1}}-\frac {\sqrt {3 x^2+2} E\left (\text {ArcTan}(2 x)\left |\frac {5}{8}\right .\right )}{3 \sqrt {2} \sqrt {\frac {3 x^2+2}{4 x^2+1}} \sqrt {4 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 422
Rule 506
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {2+3 x^2} \sqrt {1+4 x^2}} \, dx &=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+4 x^2}}-\frac {1}{3} \int \frac {\sqrt {2+3 x^2}}{\left (1+4 x^2\right )^{3/2}} \, dx\\ &=\frac {x \sqrt {2+3 x^2}}{3 \sqrt {1+4 x^2}}-\frac {\sqrt {2+3 x^2} E\left (\tan ^{-1}(2 x)|\frac {5}{8}\right )}{3 \sqrt {2} \sqrt {\frac {2+3 x^2}{1+4 x^2}} \sqrt {1+4 x^2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.27, size = 50, normalized size = 0.57 \begin {gather*} -\frac {i \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {3}{2}} x\right )|\frac {8}{3}\right )\right )}{4 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.11, size = 36, normalized size = 0.41
method | result | size |
default | \(\frac {i \left (\EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )-\EllipticE \left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )\right ) \sqrt {3}}{12}\) | \(36\) |
elliptic | \(\frac {i \sqrt {\left (3 x^{2}+2\right ) \left (4 x^{2}+1\right )}\, \sqrt {6}\, \sqrt {6 x^{2}+4}\, \left (\EllipticF \left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )-\EllipticE \left (\frac {i x \sqrt {6}}{2}, \frac {2 \sqrt {6}}{3}\right )\right )}{24 \sqrt {3 x^{2}+2}\, \sqrt {12 x^{4}+11 x^{2}+2}}\) | \(85\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {x^{2}}{\sqrt {3 x^{2} + 2} \sqrt {4 x^{2} + 1}}\, 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 = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {3\,x^2+2}\,\sqrt {4\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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