Optimal. Leaf size=63 \[ \frac {(a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {x (A b-a B)}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {385, 205} \[ \frac {(a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}+\frac {x (A b-a B)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 385
Rubi steps
\begin {align*} \int \frac {A+B x^2}{\left (a+b x^2\right )^2} \, dx &=\frac {(A b-a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \int \frac {1}{a+b x^2} \, dx}{2 a b}\\ &=\frac {(A b-a B) x}{2 a b \left (a+b x^2\right )}+\frac {(A b+a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 1.00 \[ \frac {(a B+A b) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} b^{3/2}}-\frac {x (a B-A b)}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 182, normalized size = 2.89 \[ \left [-\frac {{\left (B a^{2} + A a b + {\left (B a b + A b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (B a^{2} b - A a b^{2}\right )} x}{4 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}, \frac {{\left (B a^{2} + A a b + {\left (B a b + A b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (B a^{2} b - A a b^{2}\right )} x}{2 \, {\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 57, normalized size = 0.90 \[ \frac {{\left (B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} - \frac {B a x - A b x}{2 \, {\left (b x^{2} + a\right )} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 68, normalized size = 1.08 \[ \frac {A \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, a}+\frac {B \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b}+\frac {\left (A b -B a \right ) x}{2 \left (b \,x^{2}+a \right ) a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.41, size = 57, normalized size = 0.90 \[ -\frac {{\left (B a - A b\right )} x}{2 \, {\left (a b^{2} x^{2} + a^{2} b\right )}} + \frac {{\left (B a + A b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 51, normalized size = 0.81 \[ \frac {\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )\,\left (A\,b+B\,a\right )}{2\,a^{3/2}\,b^{3/2}}+\frac {x\,\left (A\,b-B\,a\right )}{2\,a\,b\,\left (b\,x^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.40, size = 112, normalized size = 1.78 \[ \frac {x \left (A b - B a\right )}{2 a^{2} b + 2 a b^{2} x^{2}} - \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + B a\right ) \log {\left (- a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{a^{3} b^{3}}} \left (A b + B a\right ) \log {\left (a^{2} b \sqrt {- \frac {1}{a^{3} b^{3}}} + x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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