Optimal. Leaf size=105 \[ \frac{11 a^2 x^5}{10 b^4}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^4 x}{2 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}-\frac{11 a x^7}{14 b^3}-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 x^9}{18 b^2} \]
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Rubi [A] time = 0.0455382, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac{11 a^2 x^5}{10 b^4}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^4 x}{2 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}-\frac{11 a x^7}{14 b^3}-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 x^9}{18 b^2} \]
Antiderivative was successfully verified.
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Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 \int \frac{x^{10}}{a+b x^2} \, dx}{2 b}\\ &=-\frac{x^{11}}{2 b \left (a+b x^2\right )}+\frac{11 \int \left (\frac{a^4}{b^5}-\frac{a^3 x^2}{b^4}+\frac{a^2 x^4}{b^3}-\frac{a x^6}{b^2}+\frac{x^8}{b}-\frac{a^5}{b^5 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=\frac{11 a^4 x}{2 b^6}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^2 x^5}{10 b^4}-\frac{11 a x^7}{14 b^3}+\frac{11 x^9}{18 b^2}-\frac{x^{11}}{2 b \left (a+b x^2\right )}-\frac{\left (11 a^5\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^6}\\ &=\frac{11 a^4 x}{2 b^6}-\frac{11 a^3 x^3}{6 b^5}+\frac{11 a^2 x^5}{10 b^4}-\frac{11 a x^7}{14 b^3}+\frac{11 x^9}{18 b^2}-\frac{x^{11}}{2 b \left (a+b x^2\right )}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.0588059, size = 93, normalized size = 0.89 \[ \frac{x \left (378 a^2 b^2 x^4-840 a^3 b x^2+\frac{315 a^5}{a+b x^2}+3150 a^4-180 a b^3 x^6+70 b^4 x^8\right )}{630 b^6}-\frac{11 a^{9/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \begin{align*}{\frac{{x}^{9}}{9\,{b}^{2}}}-{\frac{2\,a{x}^{7}}{7\,{b}^{3}}}+{\frac{3\,{a}^{2}{x}^{5}}{5\,{b}^{4}}}-{\frac{4\,{a}^{3}{x}^{3}}{3\,{b}^{5}}}+5\,{\frac{{a}^{4}x}{{b}^{6}}}+{\frac{{a}^{5}x}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{a}^{5}}{2\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27917, size = 528, normalized size = 5.03 \begin{align*} \left [\frac{140 \, b^{5} x^{11} - 220 \, a b^{4} x^{9} + 396 \, a^{2} b^{3} x^{7} - 924 \, a^{3} b^{2} x^{5} + 4620 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \,{\left (a^{4} b x^{2} + a^{5}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{1260 \,{\left (b^{7} x^{2} + a b^{6}\right )}}, \frac{70 \, b^{5} x^{11} - 110 \, a b^{4} x^{9} + 198 \, a^{2} b^{3} x^{7} - 462 \, a^{3} b^{2} x^{5} + 2310 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \,{\left (a^{4} b x^{2} + a^{5}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{630 \,{\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.469092, size = 151, normalized size = 1.44 \begin{align*} \frac{a^{5} x}{2 a b^{6} + 2 b^{7} x^{2}} + \frac{5 a^{4} x}{b^{6}} - \frac{4 a^{3} x^{3}}{3 b^{5}} + \frac{3 a^{2} x^{5}}{5 b^{4}} - \frac{2 a x^{7}}{7 b^{3}} + \frac{11 \sqrt{- \frac{a^{9}}{b^{13}}} \log{\left (x - \frac{b^{6} \sqrt{- \frac{a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} - \frac{11 \sqrt{- \frac{a^{9}}{b^{13}}} \log{\left (x + \frac{b^{6} \sqrt{- \frac{a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} + \frac{x^{9}}{9 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.98895, size = 128, normalized size = 1.22 \begin{align*} -\frac{11 \, a^{5} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{6}} + \frac{a^{5} x}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{35 \, b^{16} x^{9} - 90 \, a b^{15} x^{7} + 189 \, a^{2} b^{14} x^{5} - 420 \, a^{3} b^{13} x^{3} + 1575 \, a^{4} b^{12} x}{315 \, b^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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