Optimal. Leaf size=98 \[ \frac{63 a^2 x}{8 b^5}-\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{21 a x^3}{8 b^4}-\frac{x^9}{4 b \left (a+b x^2\right )^2}+\frac{63 x^5}{40 b^3} \]
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Rubi [A] time = 0.0407871, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac{63 a^2 x}{8 b^5}-\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{21 a x^3}{8 b^4}-\frac{x^9}{4 b \left (a+b x^2\right )^2}+\frac{63 x^5}{40 b^3} \]
Antiderivative was successfully verified.
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Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{10}}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^9}{4 b \left (a+b x^2\right )^2}+\frac{9 \int \frac{x^8}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac{x^9}{4 b \left (a+b x^2\right )^2}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}+\frac{63 \int \frac{x^6}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac{x^9}{4 b \left (a+b x^2\right )^2}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}+\frac{63 \int \left (\frac{a^2}{b^3}-\frac{a x^2}{b^2}+\frac{x^4}{b}-\frac{a^3}{b^3 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=\frac{63 a^2 x}{8 b^5}-\frac{21 a x^3}{8 b^4}+\frac{63 x^5}{40 b^3}-\frac{x^9}{4 b \left (a+b x^2\right )^2}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{\left (63 a^3\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^5}\\ &=\frac{63 a^2 x}{8 b^5}-\frac{21 a x^3}{8 b^4}+\frac{63 x^5}{40 b^3}-\frac{x^9}{4 b \left (a+b x^2\right )^2}-\frac{9 x^7}{8 b^2 \left (a+b x^2\right )}-\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0472025, size = 88, normalized size = 0.9 \[ \frac{168 a^2 b^2 x^5+525 a^3 b x^3+315 a^4 x-24 a b^3 x^7+8 b^4 x^9}{40 b^5 \left (a+b x^2\right )^2}-\frac{63 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 88, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{b}^{3}}}-{\frac{a{x}^{3}}{{b}^{4}}}+6\,{\frac{{a}^{2}x}{{b}^{5}}}+{\frac{17\,{a}^{3}{x}^{3}}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{a}^{3}}{8\,{b}^{5}}\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 = 1.2842, size = 547, normalized size = 5.58 \begin{align*} \left [\frac{16 \, b^{4} x^{9} - 48 \, a b^{3} x^{7} + 336 \, a^{2} b^{2} x^{5} + 1050 \, a^{3} b x^{3} + 630 \, a^{4} x + 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{80 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac{8 \, b^{4} x^{9} - 24 \, a b^{3} x^{7} + 168 \, a^{2} b^{2} x^{5} + 525 \, a^{3} b x^{3} + 315 \, a^{4} x - 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{40 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.610995, size = 144, normalized size = 1.47 \begin{align*} \frac{6 a^{2} x}{b^{5}} - \frac{a x^{3}}{b^{4}} + \frac{63 \sqrt{- \frac{a^{5}}{b^{11}}} \log{\left (x - \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} - \frac{63 \sqrt{- \frac{a^{5}}{b^{11}}} \log{\left (x + \frac{b^{5} \sqrt{- \frac{a^{5}}{b^{11}}}}{a^{2}} \right )}}{16} + \frac{15 a^{4} x + 17 a^{3} b x^{3}}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac{x^{5}}{5 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.73309, size = 113, normalized size = 1.15 \begin{align*} -\frac{63 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{5}} + \frac{17 \, a^{3} b x^{3} + 15 \, a^{4} x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{5}} + \frac{b^{12} x^{5} - 5 \, a b^{11} x^{3} + 30 \, a^{2} b^{10} x}{5 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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