Optimal. Leaf size=43 \[ -\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {A}{a x} \]
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Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {453, 205} \begin {gather*} -\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {A}{a x} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 453
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )} \, dx &=-\frac {A}{a x}-\frac {(A b-a B) \int \frac {1}{a+b x^2} \, dx}{a}\\ &=-\frac {A}{a x}-\frac {(A b-a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.98 \begin {gather*} \frac {(a B-A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \sqrt {b}}-\frac {A}{a x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A+B x^2}{x^2 \left (a+b x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.45, size = 105, normalized size = 2.44 \begin {gather*} \left [\frac {{\left (B a - A b\right )} \sqrt {-a b} x \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, A a b}{2 \, a^{2} b x}, \frac {{\left (B a - A b\right )} \sqrt {a b} x \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - A a b}{a^{2} b x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 36, normalized size = 0.84 \begin {gather*} \frac {{\left (B a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 48, normalized size = 1.12 \begin {gather*} -\frac {A b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}\, a}+\frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}-\frac {A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.36, size = 36, normalized size = 0.84 \begin {gather*} \frac {{\left (B a - A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a} - \frac {A}{a x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 35, normalized size = 0.81 \begin {gather*} -\frac {A}{a\,x}-\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b-B\,a\right )}{a^{3/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.34, size = 82, normalized size = 1.91 \begin {gather*} - \frac {A}{a x} - \frac {\sqrt {- \frac {1}{a^{3} b}} \left (- A b + B a\right ) \log {\left (- a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} + \frac {\sqrt {- \frac {1}{a^{3} b}} \left (- A b + B a\right ) \log {\left (a^{2} \sqrt {- \frac {1}{a^{3} b}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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