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 = 0.06, 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 = {1213, 537} \[ \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]
Antiderivative was successfully verified.
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Rule 537
Rule 1213
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 = 0.13, size = 40, normalized size = 1.00 \[ \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]
Antiderivative was successfully verified.
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fricas [F] time = 7.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-5 \, x^{4} + 4}}{5 \, b x^{6} + 5 \, a x^{4} - 4 \, b x^{2} - 4 \, a}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {-5 \, x^{4} + 4} {\left (b x^{2} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 79, normalized size = 1.98 \[ \frac {\sqrt {2}\, 5^{\frac {3}{4}} \sqrt {-\frac {\sqrt {5}\, x^{2}}{2}+1}\, \sqrt {\frac {\sqrt {5}\, x^{2}}{2}+1}\, \EllipticPi \left (\frac {5^{\frac {1}{4}} \sqrt {2}\, x}{2}, -\frac {2 \sqrt {5}\, b}{5 a}, \frac {\sqrt {-\frac {\sqrt {5}}{2}}\, \sqrt {2}\, 5^{\frac {3}{4}}}{5}\right )}{5 \sqrt {-5 x^{4}+4}\, a} \]
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 \[ \int \frac {1}{\sqrt {-5 \, x^{4} + 4} {\left (b x^{2} + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-5\,x^4}} \,d x \]
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 \[ \int \frac {1}{\sqrt {4 - 5 x^{4}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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