Optimal. Leaf size=218 \[ \frac{1616615 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{23/2}}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac{1616615 a x}{65536 b^{11}}-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}+\frac{1616615 x^3}{196608 b^{10}} \]
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Rubi [A] time = 0.140587, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac{1616615 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{23/2}}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}-\frac{1616615 a x}{65536 b^{11}}-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}+\frac{1616615 x^3}{196608 b^{10}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{22}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}+\frac{7 \int \frac{x^{20}}{\left (a+b x^2\right )^9} \, dx}{6 b}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}+\frac{133 \int \frac{x^{18}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}+\frac{323 \int \frac{x^{16}}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}+\frac{1615 \int \frac{x^{14}}{\left (a+b x^2\right )^6} \, dx}{768 b^4}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}+\frac{4199 \int \frac{x^{12}}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}+\frac{46189 \int \frac{x^{10}}{\left (a+b x^2\right )^4} \, dx}{12288 b^6}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}+\frac{46189 \int \frac{x^8}{\left (a+b x^2\right )^3} \, dx}{8192 b^7}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}+\frac{323323 \int \frac{x^6}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac{1616615 \int \frac{x^4}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac{1616615 \int \left (-\frac{a}{b^2}+\frac{x^2}{b}+\frac{a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{65536 b^9}\\ &=-\frac{1616615 a x}{65536 b^{11}}+\frac{1616615 x^3}{196608 b^{10}}-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac{\left (1616615 a^2\right ) \int \frac{1}{a+b x^2} \, dx}{65536 b^{11}}\\ &=-\frac{1616615 a x}{65536 b^{11}}+\frac{1616615 x^3}{196608 b^{10}}-\frac{x^{21}}{18 b \left (a+b x^2\right )^9}-\frac{7 x^{19}}{96 b^2 \left (a+b x^2\right )^8}-\frac{19 x^{17}}{192 b^3 \left (a+b x^2\right )^7}-\frac{323 x^{15}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{323 x^{13}}{1536 b^5 \left (a+b x^2\right )^5}-\frac{4199 x^{11}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^9}{73728 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^7}{32768 b^8 \left (a+b x^2\right )^2}-\frac{323323 x^5}{65536 b^9 \left (a+b x^2\right )}+\frac{1616615 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{23/2}}\\ \end{align*}
Mathematica [A] time = 0.0764073, size = 155, normalized size = 0.71 \[ \frac{\frac{\sqrt{b} x \left (-63897057 a^2 b^8 x^{16}-318434718 a^3 b^7 x^{14}-850547502 a^4 b^6 x^{12}-1404993798 a^5 b^5 x^{10}-1513521152 a^6 b^4 x^8-1071677178 a^7 b^3 x^6-483044562 a^8 b^2 x^4-126095970 a^9 b x^2-14549535 a^{10}-4128768 a b^9 x^{18}+196608 b^{10} x^{20}\right )}{\left (a+b x^2\right )^9}+14549535 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{589824 b^{23/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 217, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{3\,{b}^{10}}}-10\,{\frac{ax}{{b}^{11}}}-{\frac{961255\,{a}^{10}x}{65536\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{12201403\,{a}^{9}{x}^{3}}{98304\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{15137633\,{a}^{8}{x}^{5}}{32768\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{32405717\,{a}^{7}{x}^{7}}{32768\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{24013\,{a}^{6}{x}^{9}}{18\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{38143787\,{a}^{5}{x}^{11}}{32768\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{21103775\,{a}^{4}{x}^{13}}{32768\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{20435525\,{a}^{3}{x}^{15}}{98304\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{1987865\,{a}^{2}{x}^{17}}{65536\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{1616615\,{a}^{2}}{65536\,{b}^{11}}\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 = 1.22736, size = 1760, normalized size = 8.07 \begin{align*} \left [\frac{393216 \, b^{10} x^{21} - 8257536 \, a b^{9} x^{19} - 127794114 \, a^{2} b^{8} x^{17} - 636869436 \, a^{3} b^{7} x^{15} - 1701095004 \, a^{4} b^{6} x^{13} - 2809987596 \, a^{5} b^{5} x^{11} - 3027042304 \, a^{6} b^{4} x^{9} - 2143354356 \, a^{7} b^{3} x^{7} - 966089124 \, a^{8} b^{2} x^{5} - 252191940 \, a^{9} b x^{3} - 29099070 \, a^{10} x + 14549535 \,{\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 )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{1179648 \,{\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{196608 \, b^{10} x^{21} - 4128768 \, a b^{9} x^{19} - 63897057 \, a^{2} b^{8} x^{17} - 318434718 \, a^{3} b^{7} x^{15} - 850547502 \, a^{4} b^{6} x^{13} - 1404993798 \, a^{5} b^{5} x^{11} - 1513521152 \, a^{6} b^{4} x^{9} - 1071677178 \, a^{7} b^{3} x^{7} - 483044562 \, a^{8} b^{2} x^{5} - 126095970 \, a^{9} b x^{3} - 14549535 \, a^{10} x + 14549535 \,{\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 )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{589824 \,{\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 )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.3777, size = 298, normalized size = 1.37 \begin{align*} - \frac{10 a x}{b^{11}} - \frac{1616615 \sqrt{- \frac{a^{3}}{b^{23}}} \log{\left (x - \frac{b^{11} \sqrt{- \frac{a^{3}}{b^{23}}}}{a} \right )}}{131072} + \frac{1616615 \sqrt{- \frac{a^{3}}{b^{23}}} \log{\left (x + \frac{b^{11} \sqrt{- \frac{a^{3}}{b^{23}}}}{a} \right )}}{131072} - \frac{8651295 a^{10} x + 73208418 a^{9} b x^{3} + 272477394 a^{8} b^{2} x^{5} + 583302906 a^{7} b^{3} x^{7} + 786857984 a^{6} b^{4} x^{9} + 686588166 a^{5} b^{5} x^{11} + 379867950 a^{4} b^{6} x^{13} + 122613150 a^{3} b^{7} x^{15} + 17890785 a^{2} b^{8} x^{17}}{589824 a^{9} b^{11} + 5308416 a^{8} b^{12} x^{2} + 21233664 a^{7} b^{13} x^{4} + 49545216 a^{6} b^{14} x^{6} + 74317824 a^{5} b^{15} x^{8} + 74317824 a^{4} b^{16} x^{10} + 49545216 a^{3} b^{17} x^{12} + 21233664 a^{2} b^{18} x^{14} + 5308416 a b^{19} x^{16} + 589824 b^{20} x^{18}} + \frac{x^{3}}{3 b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.71424, size = 203, normalized size = 0.93 \begin{align*} \frac{1616615 \, a^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{11}} - \frac{17890785 \, a^{2} b^{8} x^{17} + 122613150 \, a^{3} b^{7} x^{15} + 379867950 \, a^{4} b^{6} x^{13} + 686588166 \, a^{5} b^{5} x^{11} + 786857984 \, a^{6} b^{4} x^{9} + 583302906 \, a^{7} b^{3} x^{7} + 272477394 \, a^{8} b^{2} x^{5} + 73208418 \, a^{9} b x^{3} + 8651295 \, a^{10} x}{589824 \,{\left (b x^{2} + a\right )}^{9} b^{11}} + \frac{b^{20} x^{3} - 30 \, a b^{19} x}{3 \, b^{30}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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