Optimal. Leaf size=101 \[ -\frac{2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac{A b-a B}{4 a^2 \left (a+b x^2\right )^2}+\frac{(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{A}{2 a^3 x^2} \]
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Rubi [A] time = 0.104898, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 77} \[ -\frac{2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac{A b-a B}{4 a^2 \left (a+b x^2\right )^2}+\frac{(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{A}{2 a^3 x^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^3 \left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^2 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{a^3 x^2}+\frac{-3 A b+a B}{a^4 x}-\frac{b (-A b+a B)}{a^2 (a+b x)^3}-\frac{b (-2 A b+a B)}{a^3 (a+b x)^2}-\frac{b (-3 A b+a B)}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A}{2 a^3 x^2}-\frac{A b-a B}{4 a^2 \left (a+b x^2\right )^2}-\frac{2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac{(3 A b-a B) \log (x)}{a^4}+\frac{(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}\\ \end{align*}
Mathematica [A] time = 0.0569309, size = 87, normalized size = 0.86 \[ \frac{\frac{a^2 (a B-A b)}{\left (a+b x^2\right )^2}+\frac{2 a (a B-2 A b)}{a+b x^2}+2 (3 A b-a B) \log \left (a+b x^2\right )+4 \log (x) (a B-3 A b)-\frac{2 a A}{x^2}}{4 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 118, normalized size = 1.2 \begin{align*} -{\frac{A}{2\,{a}^{3}{x}^{2}}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) B}{{a}^{3}}}-{\frac{Ab}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{3}}}-{\frac{Ab}{{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{B}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04071, size = 147, normalized size = 1.46 \begin{align*} \frac{2 \,{\left (B a b - 3 \, A b^{2}\right )} x^{4} - 2 \, A a^{2} + 3 \,{\left (B a^{2} - 3 \, A a b\right )} x^{2}}{4 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac{{\left (B a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.17257, size = 412, normalized size = 4.08 \begin{align*} \frac{2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} + 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.33974, size = 107, normalized size = 1.06 \begin{align*} \frac{- 2 A a^{2} + x^{4} \left (- 6 A b^{2} + 2 B a b\right ) + x^{2} \left (- 9 A a b + 3 B a^{2}\right )}{4 a^{5} x^{2} + 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac{\left (- 3 A b + B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{\left (- 3 A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12732, size = 186, normalized size = 1.84 \begin{align*} \frac{{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{{\left (B a b - 3 \, A b^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} + \frac{3 \, B a b^{2} x^{4} - 9 \, A b^{3} x^{4} + 8 \, B a^{2} b x^{2} - 22 \, A a b^{2} x^{2} + 6 \, B a^{3} - 14 \, A a^{2} b}{4 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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