Optimal. Leaf size=149 \[ -\frac{b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac{b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{A}{6 a^3 x^6} \]
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Rubi [A] time = 0.166051, antiderivative size = 149, 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} \[ -\frac{b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac{b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}+\frac{b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}-\frac{2 b^2 \log (x) (5 A b-3 a B)}{a^6}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{A}{6 a^3 x^6} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{A+B x^2}{x^7 \left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^4 (a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A}{a^3 x^4}+\frac{-3 A b+a B}{a^4 x^3}-\frac{3 b (-2 A b+a B)}{a^5 x^2}+\frac{2 b^2 (-5 A b+3 a B)}{a^6 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)^3}-\frac{b^3 (-4 A b+3 a B)}{a^5 (a+b x)^2}-\frac{2 b^3 (-5 A b+3 a B)}{a^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{A}{6 a^3 x^6}+\frac{3 A b-a B}{4 a^4 x^4}-\frac{3 b (2 A b-a B)}{2 a^5 x^2}-\frac{b^2 (A b-a B)}{4 a^4 \left (a+b x^2\right )^2}-\frac{b^2 (4 A b-3 a B)}{2 a^5 \left (a+b x^2\right )}-\frac{2 b^2 (5 A b-3 a B) \log (x)}{a^6}+\frac{b^2 (5 A b-3 a B) \log \left (a+b x^2\right )}{a^6}\\ \end{align*}
Mathematica [A] time = 0.118974, size = 135, normalized size = 0.91 \[ \frac{\frac{3 a^2 b^2 (a B-A b)}{\left (a+b x^2\right )^2}-\frac{3 a^2 (a B-3 A b)}{x^4}-\frac{2 a^3 A}{x^6}+\frac{6 a b^2 (3 a B-4 A b)}{a+b x^2}+12 b^2 (5 A b-3 a B) \log \left (a+b x^2\right )+24 b^2 \log (x) (3 a B-5 A b)+\frac{18 a b (a B-2 A b)}{x^2}}{12 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 180, normalized size = 1.2 \begin{align*} -{\frac{A}{6\,{a}^{3}{x}^{6}}}+{\frac{3\,Ab}{4\,{a}^{4}{x}^{4}}}-{\frac{B}{4\,{a}^{3}{x}^{4}}}-3\,{\frac{A{b}^{2}}{{a}^{5}{x}^{2}}}+{\frac{3\,Bb}{2\,{a}^{4}{x}^{2}}}-10\,{\frac{{b}^{3}\ln \left ( x \right ) A}{{a}^{6}}}+6\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{5}}}-{\frac{A{b}^{3}}{4\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B{b}^{2}}{4\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+5\,{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{a}^{6}}}-3\,{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) B}{{a}^{5}}}-2\,{\frac{A{b}^{3}}{{a}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,B{b}^{2}}{2\,{a}^{4} \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 = 0.999754, size = 230, normalized size = 1.54 \begin{align*} \frac{12 \,{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} x^{8} + 18 \,{\left (3 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 4 \,{\left (3 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{4} -{\left (3 \, B a^{4} - 5 \, A a^{3} b\right )} x^{2}}{12 \,{\left (a^{5} b^{2} x^{10} + 2 \, a^{6} b x^{8} + a^{7} x^{6}\right )}} - \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{a^{6}} + \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28166, size = 560, normalized size = 3.76 \begin{align*} \frac{12 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} + 18 \,{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6} - 2 \, A a^{5} + 4 \,{\left (3 \, B a^{4} b - 5 \, A a^{3} b^{2}\right )} x^{4} -{\left (3 \, B a^{5} - 5 \, A a^{4} b\right )} x^{2} - 12 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \,{\left ({\left (3 \, B a b^{4} - 5 \, A b^{5}\right )} x^{10} + 2 \,{\left (3 \, B a^{2} b^{3} - 5 \, A a b^{4}\right )} x^{8} +{\left (3 \, B a^{3} b^{2} - 5 \, A a^{2} b^{3}\right )} x^{6}\right )} \log \left (x\right )}{12 \,{\left (a^{6} b^{2} x^{10} + 2 \, a^{7} b x^{8} + a^{8} x^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.42316, size = 165, normalized size = 1.11 \begin{align*} \frac{- 2 A a^{4} + x^{8} \left (- 60 A b^{4} + 36 B a b^{3}\right ) + x^{6} \left (- 90 A a b^{3} + 54 B a^{2} b^{2}\right ) + x^{4} \left (- 20 A a^{2} b^{2} + 12 B a^{3} b\right ) + x^{2} \left (5 A a^{3} b - 3 B a^{4}\right )}{12 a^{7} x^{6} + 24 a^{6} b x^{8} + 12 a^{5} b^{2} x^{10}} + \frac{2 b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (x \right )}}{a^{6}} - \frac{b^{2} \left (- 5 A b + 3 B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14076, size = 271, normalized size = 1.82 \begin{align*} \frac{{\left (3 \, B a b^{2} - 5 \, A b^{3}\right )} \log \left (x^{2}\right )}{a^{6}} - \frac{{\left (3 \, B a b^{3} - 5 \, A b^{4}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{6} b} + \frac{18 \, B a b^{4} x^{4} - 30 \, A b^{5} x^{4} + 42 \, B a^{2} b^{3} x^{2} - 68 \, A a b^{4} x^{2} + 25 \, B a^{3} b^{2} - 39 \, A a^{2} b^{3}}{4 \,{\left (b x^{2} + a\right )}^{2} a^{6}} - \frac{66 \, B a b^{2} x^{6} - 110 \, A b^{3} x^{6} - 18 \, B a^{2} b x^{4} + 36 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 9 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{6} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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