Optimal. Leaf size=117 \[ \frac{b x (11 A b-7 a B)}{8 a^4 \left (a+b x^2\right )}+\frac{b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{a^4 x}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{A}{3 a^3 x^3} \]
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Rubi [A] time = 0.165264, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {456, 1259, 1261, 205} \[ \frac{b x (11 A b-7 a B)}{8 a^4 \left (a+b x^2\right )}+\frac{b x (A b-a B)}{4 a^3 \left (a+b x^2\right )^2}+\frac{3 A b-a B}{a^4 x}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}-\frac{A}{3 a^3 x^3} \]
Antiderivative was successfully verified.
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Rule 456
Rule 1259
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^4 \left (a+b x^2\right )^3} \, dx &=\frac{b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}-\frac{1}{4} b \int \frac{-\frac{4 A}{a b}+\frac{4 (A b-a B) x^2}{a^2 b}-\frac{3 (A b-a B) x^4}{a^3}}{x^4 \left (a+b x^2\right )^2} \, dx\\ &=\frac{b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac{b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}-\frac{\int \frac{-8 a A b+8 b (2 A b-a B) x^2-\frac{b^2 (11 A b-7 a B) x^4}{a}}{x^4 \left (a+b x^2\right )} \, dx}{8 a^3 b}\\ &=\frac{b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac{b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}-\frac{\int \left (-\frac{8 A b}{x^4}-\frac{8 b (-3 A b+a B)}{a x^2}+\frac{5 b^2 (-7 A b+3 a B)}{a \left (a+b x^2\right )}\right ) \, dx}{8 a^3 b}\\ &=-\frac{A}{3 a^3 x^3}+\frac{3 A b-a B}{a^4 x}+\frac{b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac{b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac{(5 b (7 A b-3 a B)) \int \frac{1}{a+b x^2} \, dx}{8 a^4}\\ &=-\frac{A}{3 a^3 x^3}+\frac{3 A b-a B}{a^4 x}+\frac{b (A b-a B) x}{4 a^3 \left (a+b x^2\right )^2}+\frac{b (11 A b-7 a B) x}{8 a^4 \left (a+b x^2\right )}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0893896, size = 116, normalized size = 0.99 \[ \frac{a^2 b x^2 \left (56 A-75 B x^2\right )-8 a^3 \left (A+3 B x^2\right )+5 a b^2 x^4 \left (35 A-9 B x^2\right )+105 A b^3 x^6}{24 a^4 x^3 \left (a+b x^2\right )^2}+\frac{5 \sqrt{b} (7 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 152, normalized size = 1.3 \begin{align*} -{\frac{A}{3\,{a}^{3}{x}^{3}}}+3\,{\frac{Ab}{{a}^{4}x}}-{\frac{B}{{a}^{3}x}}+{\frac{11\,{b}^{3}A{x}^{3}}{8\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{7\,{b}^{2}B{x}^{3}}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{13\,{b}^{2}Ax}{8\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{9\,bBx}{8\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,A{b}^{2}}{8\,{a}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,Bb}{8\,{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 = 1.32574, size = 782, normalized size = 6.68 \begin{align*} \left [-\frac{30 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 50 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 16 \, A a^{3} + 16 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{48 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}, -\frac{15 \,{\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{6} + 25 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + 8 \, A a^{3} + 8 \,{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{2} + 15 \,{\left ({\left (3 \, B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \,{\left (3 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} +{\left (3 \, B a^{3} - 7 \, A a^{2} b\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{24 \,{\left (a^{4} b^{2} x^{7} + 2 \, a^{5} b x^{5} + a^{6} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.18524, size = 226, normalized size = 1.93 \begin{align*} \frac{5 \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log{\left (- \frac{5 a^{5} \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac{5 \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right ) \log{\left (\frac{5 a^{5} \sqrt{- \frac{b}{a^{9}}} \left (- 7 A b + 3 B a\right )}{- 35 A b^{2} + 15 B a b} + x \right )}}{16} - \frac{8 A a^{3} + x^{6} \left (- 105 A b^{3} + 45 B a b^{2}\right ) + x^{4} \left (- 175 A a b^{2} + 75 B a^{2} b\right ) + x^{2} \left (- 56 A a^{2} b + 24 B a^{3}\right )}{24 a^{6} x^{3} + 48 a^{5} b x^{5} + 24 a^{4} b^{2} x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48622, size = 146, normalized size = 1.25 \begin{align*} -\frac{5 \,{\left (3 \, B a b - 7 \, A b^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4}} - \frac{7 \, B a b^{2} x^{3} - 11 \, A b^{3} x^{3} + 9 \, B a^{2} b x - 13 \, A a b^{2} x}{8 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{3 \, B a x^{2} - 9 \, A b x^{2} + A a}{3 \, a^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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