Optimal. Leaf size=128 \[ \frac{a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}+\frac{x^4 (A b-3 a B)}{4 b^4}-\frac{3 a x^2 (A b-2 a B)}{2 b^5}+\frac{B x^6}{6 b^3} \]
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Rubi [A] time = 0.16556, antiderivative size = 128, 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} \[ \frac{a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}-\frac{a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}+\frac{x^4 (A b-3 a B)}{4 b^4}-\frac{3 a x^2 (A b-2 a B)}{2 b^5}+\frac{B x^6}{6 b^3} \]
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 )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{3 a (-A b+2 a B)}{b^5}+\frac{(A b-3 a B) x}{b^4}+\frac{B x^2}{b^3}-\frac{a^4 (-A b+a B)}{b^5 (a+b x)^3}+\frac{a^3 (-4 A b+5 a B)}{b^5 (a+b x)^2}-\frac{2 a^2 (-3 A b+5 a B)}{b^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a (A b-2 a B) x^2}{2 b^5}+\frac{(A b-3 a B) x^4}{4 b^4}+\frac{B x^6}{6 b^3}-\frac{a^4 (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac{a^3 (4 A b-5 a B)}{2 b^6 \left (a+b x^2\right )}+\frac{a^2 (3 A b-5 a B) \log \left (a+b x^2\right )}{b^6}\\ \end{align*}
Mathematica [A] time = 0.0737771, size = 116, normalized size = 0.91 \[ \frac{\frac{6 a^3 (4 A b-5 a B)}{a+b x^2}+\frac{3 a^4 (a B-A b)}{\left (a+b x^2\right )^2}+12 a^2 (3 A b-5 a B) \log \left (a+b x^2\right )+3 b^2 x^4 (A b-3 a B)+18 a b x^2 (2 a B-A b)+2 b^3 B x^6}{12 b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 158, normalized size = 1.2 \begin{align*}{\frac{B{x}^{6}}{6\,{b}^{3}}}+{\frac{A{x}^{4}}{4\,{b}^{3}}}-{\frac{3\,B{x}^{4}a}{4\,{b}^{4}}}-{\frac{3\,aA{x}^{2}}{2\,{b}^{4}}}+3\,{\frac{B{x}^{2}{a}^{2}}{{b}^{5}}}-{\frac{{a}^{4}A}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B{a}^{5}}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{5}}}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) B}{{b}^{6}}}+2\,{\frac{{a}^{3}A}{{b}^{5} \left ( b{x}^{2}+a \right ) }}-{\frac{5\,{a}^{4}B}{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 = 1.02698, size = 190, normalized size = 1.48 \begin{align*} -\frac{9 \, B a^{5} - 7 \, A a^{4} b + 2 \,{\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}}{4 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac{2 \, B b^{2} x^{6} - 3 \,{\left (3 \, B a b - A b^{2}\right )} x^{4} + 18 \,{\left (2 \, B a^{2} - A a b\right )} x^{2}}{12 \, b^{5}} - \frac{{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left (b x^{2} + a\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.24628, size = 431, normalized size = 3.37 \begin{align*} \frac{2 \, B b^{5} x^{10} -{\left (5 \, B a b^{4} - 3 \, A b^{5}\right )} x^{8} + 4 \,{\left (5 \, B a^{2} b^{3} - 3 \, A a b^{4}\right )} x^{6} - 27 \, B a^{5} + 21 \, A a^{4} b + 3 \,{\left (21 \, B a^{3} b^{2} - 11 \, A a^{2} b^{3}\right )} x^{4} + 6 \,{\left (B a^{4} b + A a^{3} b^{2}\right )} x^{2} - 12 \,{\left (5 \, B a^{5} - 3 \, A a^{4} b +{\left (5 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3}\right )} x^{4} + 2 \,{\left (5 \, B a^{4} b - 3 \, A a^{3} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.79809, size = 138, normalized size = 1.08 \begin{align*} \frac{B x^{6}}{6 b^{3}} - \frac{a^{2} \left (- 3 A b + 5 B a\right ) \log{\left (a + b x^{2} \right )}}{b^{6}} - \frac{- 7 A a^{4} b + 9 B a^{5} + x^{2} \left (- 8 A a^{3} b^{2} + 10 B a^{4} b\right )}{4 a^{2} b^{6} + 8 a b^{7} x^{2} + 4 b^{8} x^{4}} - \frac{x^{4} \left (- A b + 3 B a\right )}{4 b^{4}} + \frac{x^{2} \left (- 3 A a b + 6 B a^{2}\right )}{2 b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10819, size = 215, normalized size = 1.68 \begin{align*} -\frac{{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{6}} + \frac{30 \, B a^{3} b^{2} x^{4} - 18 \, A a^{2} b^{3} x^{4} + 50 \, B a^{4} b x^{2} - 28 \, A a^{3} b^{2} x^{2} + 21 \, B a^{5} - 11 \, A a^{4} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{2 \, B b^{6} x^{6} - 9 \, B a b^{5} x^{4} + 3 \, A b^{6} x^{4} + 36 \, B a^{2} b^{4} x^{2} - 18 \, A a b^{5} x^{2}}{12 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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