Optimal. Leaf size=89 \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{x (A b-5 a B)}{8 a b^2 \left (a+b x^2\right )}-\frac{x (A b-a B)}{4 b^2 \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0619753, antiderivative size = 89, 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, 385, 205} \[ \frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{x (A b-5 a B)}{8 a b^2 \left (a+b x^2\right )}-\frac{x (A b-a B)}{4 b^2 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 455
Rule 385
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac{(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}-\frac{\int \frac{-A b+a B-4 b B x^2}{\left (a+b x^2\right )^2} \, dx}{4 b^2}\\ &=-\frac{(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}+\frac{(A b-5 a B) x}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a B) \int \frac{1}{a+b x^2} \, dx}{8 a b^2}\\ &=-\frac{(A b-a B) x}{4 b^2 \left (a+b x^2\right )^2}+\frac{(A b-5 a B) x}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0824406, size = 83, normalized size = 0.93 \[ \frac{\frac{\sqrt{b} x \left (-3 a^2 B-a b \left (A+5 B x^2\right )+A b^2 x^2\right )}{a \left (a+b x^2\right )^2}+\frac{(3 a B+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 89, normalized size = 1. \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-5\,Ba \right ){x}^{3}}{8\,ab}}-{\frac{ \left ( Ab+3\,Ba \right ) x}{8\,{b}^{2}}} \right ) }+{\frac{A}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,B}{8\,{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 = 1.33385, size = 624, normalized size = 7.01 \begin{align*} \left [-\frac{2 \,{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} +{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A 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^{3} b + A a^{2} b^{2}\right )} x}{16 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}, -\frac{{\left (5 \, B a^{2} b^{2} - A a b^{3}\right )} x^{3} -{\left ({\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 3 \, B a^{3} + A a^{2} b + 2 \,{\left (3 \, B a^{2} b + A a b^{2}\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, B a^{3} b + A a^{2} b^{2}\right )} x}{8 \,{\left (a^{2} b^{5} x^{4} + 2 \, a^{3} b^{4} x^{2} + a^{4} b^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.881778, size = 153, normalized size = 1.72 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log{\left (- a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (A b + 3 B a\right ) \log{\left (a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} - \frac{x^{3} \left (- A b^{2} + 5 B a b\right ) + x \left (A a b + 3 B a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42951, size = 105, normalized size = 1.18 \begin{align*} \frac{{\left (3 \, B a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{5 \, B a b x^{3} - A b^{2} x^{3} + 3 \, B a^{2} x + A a b x}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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