Optimal. Leaf size=113 \[ -\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{5 a^2 x^5} \]
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Rubi [A] time = 0.18644, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {456, 1802, 205} \[ -\frac{b^2 x (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{A}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Rule 456
Rule 1802
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^6 \left (a+b x^2\right )^2} \, dx &=-\frac{b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac{1}{2} b^2 \int \frac{-\frac{2 A}{a b^2}+\frac{2 (A b-a B) x^2}{a^2 b^2}-\frac{2 (A b-a B) x^4}{a^3 b}+\frac{(A b-a B) x^6}{a^4}}{x^6 \left (a+b x^2\right )} \, dx\\ &=-\frac{b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac{1}{2} b^2 \int \left (-\frac{2 A}{a^2 b^2 x^6}-\frac{2 (-2 A b+a B)}{a^3 b^2 x^4}+\frac{2 (-3 A b+2 a B)}{a^4 b x^2}+\frac{7 A b-5 a B}{a^4 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{A}{5 a^2 x^5}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac{\left (b^2 (7 A b-5 a B)\right ) \int \frac{1}{a+b x^2} \, dx}{2 a^4}\\ &=-\frac{A}{5 a^2 x^5}+\frac{2 A b-a B}{3 a^3 x^3}-\frac{b (3 A b-2 a B)}{a^4 x}-\frac{b^2 (A b-a B) x}{2 a^4 \left (a+b x^2\right )}-\frac{b^{3/2} (7 A b-5 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0754468, size = 112, normalized size = 0.99 \[ \frac{b^2 x (a B-A b)}{2 a^4 \left (a+b x^2\right )}+\frac{b^{3/2} (5 a B-7 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{9/2}}+\frac{2 A b-a B}{3 a^3 x^3}+\frac{b (2 a B-3 A b)}{a^4 x}-\frac{A}{5 a^2 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 136, normalized size = 1.2 \begin{align*} -{\frac{A}{5\,{a}^{2}{x}^{5}}}+{\frac{2\,Ab}{3\,{a}^{3}{x}^{3}}}-{\frac{B}{3\,{a}^{2}{x}^{3}}}-3\,{\frac{A{b}^{2}}{{a}^{4}x}}+2\,{\frac{Bb}{{a}^{3}x}}-{\frac{{b}^{3}xA}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}Bx}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{b}^{3}A}{2\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}B}{2\,{a}^{3}}\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 = 1.31028, size = 653, normalized size = 5.78 \begin{align*} \left [\frac{30 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 20 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 12 \, A a^{3} - 4 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} - 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{60 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}, \frac{15 \,{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 10 \,{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} - 6 \, A a^{3} - 2 \,{\left (5 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} +{\left (5 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{30 \,{\left (a^{4} b x^{7} + a^{5} x^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.00349, size = 218, normalized size = 1.93 \begin{align*} - \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right ) \log{\left (\frac{a^{5} \sqrt{- \frac{b^{3}}{a^{9}}} \left (- 7 A b + 5 B a\right )}{- 7 A b^{3} + 5 B a b^{2}} + x \right )}}{4} + \frac{- 6 A a^{3} + x^{6} \left (- 105 A b^{3} + 75 B a b^{2}\right ) + x^{4} \left (- 70 A a b^{2} + 50 B a^{2} b\right ) + x^{2} \left (14 A a^{2} b - 10 B a^{3}\right )}{30 a^{5} x^{5} + 30 a^{4} b x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14432, size = 151, normalized size = 1.34 \begin{align*} \frac{{\left (5 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{B a b^{2} x - A b^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{4}} + \frac{30 \, B a b x^{4} - 45 \, A b^{2} x^{4} - 5 \, B a^{2} x^{2} + 10 \, A a b x^{2} - 3 \, A a^{2}}{15 \, a^{4} x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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