Optimal. Leaf size=53 \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1088, 205} \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1088
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (2 a b+2 b^2 x^2\right ) \int \frac {1}{2 a b+2 b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.83 \[ \frac {\left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 67, normalized size = 1.26 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{2 \, a b}, \frac {\sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{a b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 23, normalized size = 0.43 \[ \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 34, normalized size = 0.64 \[ \frac {\left (b \,x^{2}+a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.03, size = 15, normalized size = 0.28 \[ \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \]
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 \[ \int \frac {1}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 53, normalized size = 1.00 \[ - \frac {\sqrt {- \frac {1}{a b}} \log {\left (- a \sqrt {- \frac {1}{a b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a b}} \log {\left (a \sqrt {- \frac {1}{a b}} + x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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