Optimal. Leaf size=150 \[ \frac{a^2 x^2 (3 A b-5 a B)}{b^6}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}+\frac{x^6 (A b-3 a B)}{6 b^4}-\frac{3 a x^4 (A b-2 a B)}{4 b^5}+\frac{B x^8}{8 b^3} \]
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Rubi [A] time = 0.230432, antiderivative size = 150, 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} \[ \frac{a^2 x^2 (3 A b-5 a B)}{b^6}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}+\frac{x^6 (A b-3 a B)}{6 b^4}-\frac{3 a x^4 (A b-2 a B)}{4 b^5}+\frac{B x^8}{8 b^3} \]
Antiderivative was successfully verified.
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Rule 446
Rule 77
Rubi steps
\begin{align*} \int \frac{x^{11} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^5 (A+B x)}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{2 a^2 (-3 A b+5 a B)}{b^6}+\frac{3 a (-A b+2 a B) x}{b^5}+\frac{(A b-3 a B) x^2}{b^4}+\frac{B x^3}{b^3}+\frac{a^5 (-A b+a B)}{b^6 (a+b x)^3}-\frac{a^4 (-5 A b+6 a B)}{b^6 (a+b x)^2}+\frac{5 a^3 (-2 A b+3 a B)}{b^6 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 (3 A b-5 a B) x^2}{b^6}-\frac{3 a (A b-2 a B) x^4}{4 b^5}+\frac{(A b-3 a B) x^6}{6 b^4}+\frac{B x^8}{8 b^3}+\frac{a^5 (A b-a B)}{4 b^7 \left (a+b x^2\right )^2}-\frac{a^4 (5 A b-6 a B)}{2 b^7 \left (a+b x^2\right )}-\frac{5 a^3 (2 A b-3 a B) \log \left (a+b x^2\right )}{2 b^7}\\ \end{align*}
Mathematica [A] time = 0.087402, size = 136, normalized size = 0.91 \[ \frac{-24 a^2 b x^2 (5 a B-3 A b)+\frac{12 a^4 (6 a B-5 A b)}{a+b x^2}+\frac{6 a^5 (A b-a B)}{\left (a+b x^2\right )^2}+60 a^3 (3 a B-2 A b) \log \left (a+b x^2\right )+4 b^3 x^6 (A b-3 a B)+18 a b^2 x^4 (2 a B-A b)+3 b^4 B x^8}{24 b^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.012, size = 182, normalized size = 1.2 \begin{align*}{\frac{B{x}^{8}}{8\,{b}^{3}}}+{\frac{{x}^{6}A}{6\,{b}^{3}}}-{\frac{{x}^{6}Ba}{2\,{b}^{4}}}-{\frac{3\,{x}^{4}Aa}{4\,{b}^{4}}}+{\frac{3\,{x}^{4}B{a}^{2}}{2\,{b}^{5}}}+3\,{\frac{{a}^{2}A{x}^{2}}{{b}^{5}}}-5\,{\frac{B{x}^{2}{a}^{3}}{{b}^{6}}}+{\frac{{a}^{5}A}{4\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{B{a}^{6}}{4\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{2}}}-5\,{\frac{{a}^{3}\ln \left ( b{x}^{2}+a \right ) A}{{b}^{6}}}+{\frac{15\,{a}^{4}\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{7}}}-{\frac{5\,{a}^{4}A}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{a}^{5}B}{{b}^{7} \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 = 1.02869, size = 223, normalized size = 1.49 \begin{align*} \frac{11 \, B a^{6} - 9 \, A a^{5} b + 2 \,{\left (6 \, B a^{5} b - 5 \, A a^{4} b^{2}\right )} x^{2}}{4 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} + \frac{3 \, B b^{3} x^{8} - 4 \,{\left (3 \, B a b^{2} - A b^{3}\right )} x^{6} + 18 \,{\left (2 \, B a^{2} b - A a b^{2}\right )} x^{4} - 24 \,{\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2}}{24 \, b^{6}} + \frac{5 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} \log \left (b x^{2} + a\right )}{2 \, b^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48384, size = 489, normalized size = 3.26 \begin{align*} \frac{3 \, B b^{6} x^{12} - 2 \,{\left (3 \, B a b^{5} - 2 \, A b^{6}\right )} x^{10} + 5 \,{\left (3 \, B a^{2} b^{4} - 2 \, A a b^{5}\right )} x^{8} + 66 \, B a^{6} - 54 \, A a^{5} b - 20 \,{\left (3 \, B a^{3} b^{3} - 2 \, A a^{2} b^{4}\right )} x^{6} - 6 \,{\left (34 \, B a^{4} b^{2} - 21 \, A a^{3} b^{3}\right )} x^{4} - 12 \,{\left (4 \, B a^{5} b - A a^{4} b^{2}\right )} x^{2} + 60 \,{\left (3 \, B a^{6} - 2 \, A a^{5} b +{\left (3 \, B a^{4} b^{2} - 2 \, A a^{3} b^{3}\right )} x^{4} + 2 \,{\left (3 \, B a^{5} b - 2 \, A a^{4} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{24 \,{\left (b^{9} x^{4} + 2 \, a b^{8} x^{2} + a^{2} b^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.95257, size = 163, normalized size = 1.09 \begin{align*} \frac{B x^{8}}{8 b^{3}} + \frac{5 a^{3} \left (- 2 A b + 3 B a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{7}} + \frac{- 9 A a^{5} b + 11 B a^{6} + x^{2} \left (- 10 A a^{4} b^{2} + 12 B a^{5} b\right )}{4 a^{2} b^{7} + 8 a b^{8} x^{2} + 4 b^{9} x^{4}} - \frac{x^{6} \left (- A b + 3 B a\right )}{6 b^{4}} + \frac{x^{4} \left (- 3 A a b + 6 B a^{2}\right )}{4 b^{5}} - \frac{x^{2} \left (- 3 A a^{2} b + 5 B a^{3}\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12296, size = 247, normalized size = 1.65 \begin{align*} \frac{5 \,{\left (3 \, B a^{4} - 2 \, A a^{3} b\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{7}} - \frac{45 \, B a^{4} b^{2} x^{4} - 30 \, A a^{3} b^{3} x^{4} + 78 \, B a^{5} b x^{2} - 50 \, A a^{4} b^{2} x^{2} + 34 \, B a^{6} - 21 \, A a^{5} b}{4 \,{\left (b x^{2} + a\right )}^{2} b^{7}} + \frac{3 \, B b^{9} x^{8} - 12 \, B a b^{8} x^{6} + 4 \, A b^{9} x^{6} + 36 \, B a^{2} b^{7} x^{4} - 18 \, A a b^{8} x^{4} - 120 \, B a^{3} b^{6} x^{2} + 72 \, A a^{2} b^{7} x^{2}}{24 \, b^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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