Optimal. Leaf size=126 \[ \frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}-\frac{a^4 (A b-a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^3 (4 A b-5 a B) \log \left (a+b x^2\right )}{2 b^6}+\frac{x^6 (A b-2 a B)}{6 b^3}-\frac{a x^4 (2 A b-3 a B)}{4 b^4}+\frac{B x^8}{8 b^2} \]
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Rubi [A] time = 0.16913, antiderivative size = 126, 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} \[ \frac{a^2 x^2 (3 A b-4 a B)}{2 b^5}-\frac{a^4 (A b-a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^3 (4 A b-5 a B) \log \left (a+b x^2\right )}{2 b^6}+\frac{x^6 (A b-2 a B)}{6 b^3}-\frac{a x^4 (2 A b-3 a B)}{4 b^4}+\frac{B x^8}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^9 \left (A+B x^2\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4 (A+B x)}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^2 (-3 A b+4 a B)}{b^5}+\frac{a (-2 A b+3 a B) x}{b^4}+\frac{(A b-2 a B) x^2}{b^3}+\frac{B x^3}{b^2}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^2}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 (3 A b-4 a B) x^2}{2 b^5}-\frac{a (2 A b-3 a B) x^4}{4 b^4}+\frac{(A b-2 a B) x^6}{6 b^3}+\frac{B x^8}{8 b^2}-\frac{a^4 (A b-a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^3 (4 A b-5 a B) \log \left (a+b x^2\right )}{2 b^6}\\ \end{align*}
Mathematica [A] time = 0.0766637, size = 113, normalized size = 0.9 \[ \frac{-12 a^2 b x^2 (4 a B-3 A b)+\frac{12 a^4 (a B-A b)}{a+b x^2}+12 a^3 (5 a B-4 A b) \log \left (a+b x^2\right )+4 b^3 x^6 (A b-2 a B)+6 a b^2 x^4 (3 a B-2 A b)+3 b^4 B x^8}{24 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 146, normalized size = 1.2 \begin{align*}{\frac{B{x}^{8}}{8\,{b}^{2}}}+{\frac{{x}^{6}A}{6\,{b}^{2}}}-{\frac{{x}^{6}Ba}{3\,{b}^{3}}}-{\frac{{x}^{4}Aa}{2\,{b}^{3}}}+{\frac{3\,{x}^{4}B{a}^{2}}{4\,{b}^{4}}}+{\frac{3\,{a}^{2}A{x}^{2}}{2\,{b}^{4}}}-2\,{\frac{B{x}^{2}{a}^{3}}{{b}^{5}}}-2\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{5}}}+{\frac{5\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{6}}}-{\frac{{a}^{4}A}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{B{a}^{5}}{2\,{b}^{6} \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 = 1.32536, size = 177, normalized size = 1.4 \begin{align*} \frac{B a^{5} - A a^{4} b}{2 \,{\left (b^{7} x^{2} + a b^{6}\right )}} + \frac{3 \, B b^{3} x^{8} - 4 \,{\left (2 \, B a b^{2} - A b^{3}\right )} x^{6} + 6 \,{\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 12 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2}}{24 \, b^{5}} + \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.23371, size = 365, normalized size = 2.9 \begin{align*} \frac{3 \, B b^{5} x^{10} -{\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} x^{8} + 2 \,{\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{6} + 12 \, B a^{5} - 12 \, A a^{4} b - 6 \,{\left (5 \, B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{4} - 12 \,{\left (4 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2} + 12 \,{\left (5 \, B a^{5} - 4 \, A a^{4} b +{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{7} x^{2} + a b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.905388, size = 126, normalized size = 1. \begin{align*} \frac{B x^{8}}{8 b^{2}} + \frac{a^{3} \left (- 4 A b + 5 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{6}} + \frac{- A a^{4} b + B a^{5}}{2 a b^{6} + 2 b^{7} x^{2}} - \frac{x^{6} \left (- A b + 2 B a\right )}{6 b^{3}} + \frac{x^{4} \left (- 2 A a b + 3 B a^{2}\right )}{4 b^{4}} - \frac{x^{2} \left (- 3 A a^{2} b + 4 B a^{3}\right )}{2 b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17899, size = 215, normalized size = 1.71 \begin{align*} \frac{{\left (5 \, B a^{4} - 4 \, A a^{3} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{6}} - \frac{5 \, B a^{4} b x^{2} - 4 \, A a^{3} b^{2} x^{2} + 4 \, B a^{5} - 3 \, A a^{4} b}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{3 \, B b^{6} x^{8} - 8 \, B a b^{5} x^{6} + 4 \, A b^{6} x^{6} + 18 \, B a^{2} b^{4} x^{4} - 12 \, A a b^{5} x^{4} - 48 \, B a^{3} b^{3} x^{2} + 36 \, A a^{2} b^{4} x^{2}}{24 \, b^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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