Optimal. Leaf size=188 \[ -\frac{a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac{5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac{45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac{10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac{21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac{63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac{35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac{30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac{45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac{5 a \log \left (a+b x^2\right )}{b^{11}}+\frac{x^2}{2 b^{10}} \]
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Rubi [A] time = 0.1894, antiderivative size = 188, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac{5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac{45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac{10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac{21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac{63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac{35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac{30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac{45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac{5 a \log \left (a+b x^2\right )}{b^{11}}+\frac{x^2}{2 b^{10}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{21}}{\left (a+b x^2\right )^{10}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^{10}}{(a+b x)^{10}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^{10}}+\frac{a^{10}}{b^{10} (a+b x)^{10}}-\frac{10 a^9}{b^{10} (a+b x)^9}+\frac{45 a^8}{b^{10} (a+b x)^8}-\frac{120 a^7}{b^{10} (a+b x)^7}+\frac{210 a^6}{b^{10} (a+b x)^6}-\frac{252 a^5}{b^{10} (a+b x)^5}+\frac{210 a^4}{b^{10} (a+b x)^4}-\frac{120 a^3}{b^{10} (a+b x)^3}+\frac{45 a^2}{b^{10} (a+b x)^2}-\frac{10 a}{b^{10} (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{x^2}{2 b^{10}}-\frac{a^{10}}{18 b^{11} \left (a+b x^2\right )^9}+\frac{5 a^9}{8 b^{11} \left (a+b x^2\right )^8}-\frac{45 a^8}{14 b^{11} \left (a+b x^2\right )^7}+\frac{10 a^7}{b^{11} \left (a+b x^2\right )^6}-\frac{21 a^6}{b^{11} \left (a+b x^2\right )^5}+\frac{63 a^5}{2 b^{11} \left (a+b x^2\right )^4}-\frac{35 a^4}{b^{11} \left (a+b x^2\right )^3}+\frac{30 a^3}{b^{11} \left (a+b x^2\right )^2}-\frac{45 a^2}{2 b^{11} \left (a+b x^2\right )}-\frac{5 a \log \left (a+b x^2\right )}{b^{11}}\\ \end{align*}
Mathematica [A] time = 0.0336789, size = 145, normalized size = 0.77 \[ -\frac{2268 a^2 b^8 x^{16}+54432 a^3 b^7 x^{14}+197568 a^4 b^6 x^{12}+375732 a^5 b^5 x^{10}+439236 a^6 b^4 x^8+328104 a^7 b^3 x^6+153576 a^8 b^2 x^4+41229 a^9 b x^2+4861 a^{10}-2268 a b^9 x^{18}+2520 a \left (a+b x^2\right )^9 \log \left (a+b x^2\right )-252 b^{10} x^{20}}{504 b^{11} \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 177, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{10}}}-{\frac{{a}^{10}}{18\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{5\,{a}^{9}}{8\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{8}}}-{\frac{45\,{a}^{8}}{14\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{7}}}+10\,{\frac{{a}^{7}}{{b}^{11} \left ( b{x}^{2}+a \right ) ^{6}}}-21\,{\frac{{a}^{6}}{{b}^{11} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{63\,{a}^{5}}{2\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{4}}}-35\,{\frac{{a}^{4}}{{b}^{11} \left ( b{x}^{2}+a \right ) ^{3}}}+30\,{\frac{{a}^{3}}{{b}^{11} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{45\,{a}^{2}}{2\,{b}^{11} \left ( b{x}^{2}+a \right ) }}-5\,{\frac{a\ln \left ( b{x}^{2}+a \right ) }{{b}^{11}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.56957, size = 297, normalized size = 1.58 \begin{align*} -\frac{11340 \, a^{2} b^{8} x^{16} + 75600 \, a^{3} b^{7} x^{14} + 229320 \, a^{4} b^{6} x^{12} + 407484 \, a^{5} b^{5} x^{10} + 460404 \, a^{6} b^{4} x^{8} + 337176 \, a^{7} b^{3} x^{6} + 155844 \, a^{8} b^{2} x^{4} + 41481 \, a^{9} b x^{2} + 4861 \, a^{10}}{504 \,{\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} + \frac{x^{2}}{2 \, b^{10}} - \frac{5 \, a \log \left (b x^{2} + a\right )}{b^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28917, size = 767, normalized size = 4.08 \begin{align*} \frac{252 \, b^{10} x^{20} + 2268 \, a b^{9} x^{18} - 2268 \, a^{2} b^{8} x^{16} - 54432 \, a^{3} b^{7} x^{14} - 197568 \, a^{4} b^{6} x^{12} - 375732 \, a^{5} b^{5} x^{10} - 439236 \, a^{6} b^{4} x^{8} - 328104 \, a^{7} b^{3} x^{6} - 153576 \, a^{8} b^{2} x^{4} - 41229 \, a^{9} b x^{2} - 4861 \, a^{10} - 2520 \,{\left (a b^{9} x^{18} + 9 \, a^{2} b^{8} x^{16} + 36 \, a^{3} b^{7} x^{14} + 84 \, a^{4} b^{6} x^{12} + 126 \, a^{5} b^{5} x^{10} + 126 \, a^{6} b^{4} x^{8} + 84 \, a^{7} b^{3} x^{6} + 36 \, a^{8} b^{2} x^{4} + 9 \, a^{9} b x^{2} + a^{10}\right )} \log \left (b x^{2} + a\right )}{504 \,{\left (b^{20} x^{18} + 9 \, a b^{19} x^{16} + 36 \, a^{2} b^{18} x^{14} + 84 \, a^{3} b^{17} x^{12} + 126 \, a^{4} b^{16} x^{10} + 126 \, a^{5} b^{15} x^{8} + 84 \, a^{6} b^{14} x^{6} + 36 \, a^{7} b^{13} x^{4} + 9 \, a^{8} b^{12} x^{2} + a^{9} b^{11}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.48979, size = 231, normalized size = 1.23 \begin{align*} - \frac{5 a \log{\left (a + b x^{2} \right )}}{b^{11}} - \frac{4861 a^{10} + 41481 a^{9} b x^{2} + 155844 a^{8} b^{2} x^{4} + 337176 a^{7} b^{3} x^{6} + 460404 a^{6} b^{4} x^{8} + 407484 a^{5} b^{5} x^{10} + 229320 a^{4} b^{6} x^{12} + 75600 a^{3} b^{7} x^{14} + 11340 a^{2} b^{8} x^{16}}{504 a^{9} b^{11} + 4536 a^{8} b^{12} x^{2} + 18144 a^{7} b^{13} x^{4} + 42336 a^{6} b^{14} x^{6} + 63504 a^{5} b^{15} x^{8} + 63504 a^{4} b^{16} x^{10} + 42336 a^{3} b^{17} x^{12} + 18144 a^{2} b^{18} x^{14} + 4536 a b^{19} x^{16} + 504 b^{20} x^{18}} + \frac{x^{2}}{2 b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.47873, size = 188, normalized size = 1. \begin{align*} \frac{x^{2}}{2 \, b^{10}} - \frac{5 \, a \log \left ({\left | b x^{2} + a \right |}\right )}{b^{11}} + \frac{7129 \, a b^{9} x^{18} + 52821 \, a^{2} b^{8} x^{16} + 181044 \, a^{3} b^{7} x^{14} + 369516 \, a^{4} b^{6} x^{12} + 490770 \, a^{5} b^{5} x^{10} + 437850 \, a^{6} b^{4} x^{8} + 261660 \, a^{7} b^{3} x^{6} + 100800 \, a^{8} b^{2} x^{4} + 22680 \, a^{9} b x^{2} + 2268 \, a^{10}}{504 \,{\left (b x^{2} + a\right )}^{9} b^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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