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 = 0.01, 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 = {1102, 211}
\begin {gather*} \frac {\left (a+b x^2\right ) \text {ArcTan}\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 {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 1102
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 \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 34, normalized size = 0.64
method | result | size |
default | \(\frac {\left (b \,x^{2}+a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}}\) | \(34\) |
risch | \(-\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (b x +\sqrt {-a b}\right )}{2 \left (b \,x^{2}+a \right ) \sqrt {-a b}}+\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (-b x +\sqrt {-a b}\right )}{2 \left (b \,x^{2}+a \right ) \sqrt {-a b}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 15, normalized size = 0.28 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 67, normalized size = 1.26 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 53, normalized size = 1.00 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.69, size = 23, normalized size = 0.43 \begin {gather*} \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {a b}} \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}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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