Optimal. Leaf size=40 \[ \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]
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Rubi [A]
time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1227, 551}
\begin {gather*} \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\text {ArcSin}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \end {gather*}
Antiderivative was successfully verified.
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Rule 551
Rule 1227
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx &=\sqrt {5} \int \frac {1}{\sqrt {2 \sqrt {5}-5 x^2} \sqrt {2 \sqrt {5}+5 x^2} \left (a+b x^2\right )} \, dx\\ &=\frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a}\\ \end {align*}
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Mathematica [A]
time = 10.10, size = 40, normalized size = 1.00 \begin {gather*} \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \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. \(78\) vs.
\(2(32)=64\).
time = 0.14, size = 79, normalized size = 1.98
method | result | size |
default | \(\frac {\sqrt {2}\, 5^{\frac {3}{4}} \sqrt {1-\frac {x^{2} \sqrt {5}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {5}}{2}}\, \EllipticPi \left (\frac {5^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b \sqrt {5}}{5 a}, \frac {\sqrt {-\frac {\sqrt {5}}{2}}\, \sqrt {2}\, 5^{\frac {3}{4}}}{5}\right )}{5 a \sqrt {-5 x^{4}+4}}\) | \(79\) |
elliptic | \(\frac {\sqrt {2}\, 5^{\frac {3}{4}} \sqrt {1-\frac {x^{2} \sqrt {5}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {5}}{2}}\, \EllipticPi \left (\frac {5^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b \sqrt {5}}{5 a}, \frac {\sqrt {-\frac {\sqrt {5}}{2}}\, \sqrt {2}\, 5^{\frac {3}{4}}}{5}\right )}{5 a \sqrt {-5 x^{4}+4}}\) | \(79\) |
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 {1}{\sqrt {4 - 5 x^{4}} \left (a + b x^{2}\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 = -0.02 \begin {gather*} \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-5\,x^4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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