Optimal. Leaf size=67 \[ -\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{B x}{b^2} \]
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Rubi [A] time = 0.0504237, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {455, 388, 205} \[ -\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{B x}{b^2} \]
Antiderivative was successfully verified.
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Rule 455
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac{(A b-a B) x}{2 b^2 \left (a+b x^2\right )}-\frac{\int \frac{-A b+a B-2 b B x^2}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{B x}{b^2}-\frac{(A b-a B) x}{2 b^2 \left (a+b x^2\right )}+\frac{(A b-3 a B) \int \frac{1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{B x}{b^2}-\frac{(A b-a B) x}{2 b^2 \left (a+b x^2\right )}+\frac{(A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0692784, size = 68, normalized size = 1.01 \[ -\frac{x (A b-a B)}{2 b^2 \left (a+b x^2\right )}-\frac{(3 a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{B x}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 82, normalized size = 1.2 \begin{align*}{\frac{Bx}{{b}^{2}}}-{\frac{xA}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{Bax}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{A}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,Ba}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2665, size = 433, normalized size = 6.46 \begin{align*} \left [\frac{4 \, B a b^{2} x^{3} +{\left (3 \, B a^{2} - A a b +{\left (3 \, 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 (3 \, B a^{2} b - A a b^{2}\right )} x}{4 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac{2 \, B a b^{2} x^{3} -{\left (3 \, B a^{2} - A a b +{\left (3 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, B a^{2} b - A a b^{2}\right )} x}{2 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.666211, size = 114, normalized size = 1.7 \begin{align*} \frac{B x}{b^{2}} + \frac{x \left (- A b + B a\right )}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{5}}} \left (- A b + 3 B a\right ) \log{\left (- a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a b^{5}}} \left (- A b + 3 B a\right ) \log{\left (a b^{2} \sqrt{- \frac{1}{a b^{5}}} + x \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14648, size = 80, normalized size = 1.19 \begin{align*} \frac{B x}{b^{2}} - \frac{{\left (3 \, B a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{B a x - A b x}{2 \,{\left (b x^{2} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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