Optimal. Leaf size=220 \[ \frac{1616615 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{23/2}}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{1616615 b}{65536 a^{11} x}-\frac{1616615}{196608 a^{10} x^3}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.140752, antiderivative size = 220, 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 = {290, 325, 205} \[ \frac{1616615 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{23/2}}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{1616615 b}{65536 a^{11} x}-\frac{1616615}{196608 a^{10} x^3}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (a+b x^2\right )^{10}} \, dx &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7 \int \frac{1}{x^4 \left (a+b x^2\right )^9} \, dx}{6 a}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{133 \int \frac{1}{x^4 \left (a+b x^2\right )^8} \, dx}{96 a^2}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323 \int \frac{1}{x^4 \left (a+b x^2\right )^7} \, dx}{192 a^3}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{1615 \int \frac{1}{x^4 \left (a+b x^2\right )^6} \, dx}{768 a^4}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199 \int \frac{1}{x^4 \left (a+b x^2\right )^5} \, dx}{1536 a^5}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189 \int \frac{1}{x^4 \left (a+b x^2\right )^4} \, dx}{12288 a^6}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189 \int \frac{1}{x^4 \left (a+b x^2\right )^3} \, dx}{8192 a^7}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{323323 \int \frac{1}{x^4 \left (a+b x^2\right )^2} \, dx}{32768 a^8}\\ &=\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{1616615 \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{65536 a^9}\\ &=-\frac{1616615}{196608 a^{10} x^3}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}-\frac{(1616615 b) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{65536 a^{10}}\\ &=-\frac{1616615}{196608 a^{10} x^3}+\frac{1616615 b}{65536 a^{11} x}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{\left (1616615 b^2\right ) \int \frac{1}{a+b x^2} \, dx}{65536 a^{11}}\\ &=-\frac{1616615}{196608 a^{10} x^3}+\frac{1616615 b}{65536 a^{11} x}+\frac{1}{18 a x^3 \left (a+b x^2\right )^9}+\frac{7}{96 a^2 x^3 \left (a+b x^2\right )^8}+\frac{19}{192 a^3 x^3 \left (a+b x^2\right )^7}+\frac{323}{2304 a^4 x^3 \left (a+b x^2\right )^6}+\frac{323}{1536 a^5 x^3 \left (a+b x^2\right )^5}+\frac{4199}{12288 a^6 x^3 \left (a+b x^2\right )^4}+\frac{46189}{73728 a^7 x^3 \left (a+b x^2\right )^3}+\frac{46189}{32768 a^8 x^3 \left (a+b x^2\right )^2}+\frac{323323}{65536 a^9 x^3 \left (a+b x^2\right )}+\frac{1616615 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{23/2}}\\ \end{align*}
Mathematica [A] time = 0.0834459, size = 157, normalized size = 0.71 \[ \frac{\frac{\sqrt{a} \left (483044562 a^2 b^8 x^{16}+1071677178 a^3 b^7 x^{14}+1513521152 a^4 b^6 x^{12}+1404993798 a^5 b^5 x^{10}+850547502 a^6 b^4 x^8+318434718 a^7 b^3 x^6+63897057 a^8 b^2 x^4+4128768 a^9 b x^2-196608 a^{10}+126095970 a b^9 x^{18}+14549535 b^{10} x^{20}\right )}{x^3 \left (a+b x^2\right )^9}+14549535 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{589824 a^{23/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 219, normalized size = 1. \begin{align*} -{\frac{1}{3\,{a}^{10}{x}^{3}}}+10\,{\frac{b}{{a}^{11}x}}+{\frac{1987865\,{b}^{2}x}{65536\,{a}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{20435525\,{b}^{3}{x}^{3}}{98304\,{a}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{21103775\,{b}^{4}{x}^{5}}{32768\,{a}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{38143787\,{b}^{5}{x}^{7}}{32768\,{a}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{24013\,{b}^{6}{x}^{9}}{18\,{a}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{32405717\,{b}^{7}{x}^{11}}{32768\,{a}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{15137633\,{b}^{8}{x}^{13}}{32768\,{a}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{12201403\,{b}^{9}{x}^{15}}{98304\,{a}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{961255\,{b}^{10}{x}^{17}}{65536\,{a}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{1616615\,{b}^{2}}{65536\,{a}^{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 = 1.32256, size = 1777, normalized size = 8.08 \begin{align*} \left [\frac{29099070 \, b^{10} x^{20} + 252191940 \, a b^{9} x^{18} + 966089124 \, a^{2} b^{8} x^{16} + 2143354356 \, a^{3} b^{7} x^{14} + 3027042304 \, a^{4} b^{6} x^{12} + 2809987596 \, a^{5} b^{5} x^{10} + 1701095004 \, a^{6} b^{4} x^{8} + 636869436 \, a^{7} b^{3} x^{6} + 127794114 \, a^{8} b^{2} x^{4} + 8257536 \, a^{9} b x^{2} - 393216 \, a^{10} + 14549535 \,{\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{1179648 \,{\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}, \frac{14549535 \, b^{10} x^{20} + 126095970 \, a b^{9} x^{18} + 483044562 \, a^{2} b^{8} x^{16} + 1071677178 \, a^{3} b^{7} x^{14} + 1513521152 \, a^{4} b^{6} x^{12} + 1404993798 \, a^{5} b^{5} x^{10} + 850547502 \, a^{6} b^{4} x^{8} + 318434718 \, a^{7} b^{3} x^{6} + 63897057 \, a^{8} b^{2} x^{4} + 4128768 \, a^{9} b x^{2} - 196608 \, a^{10} + 14549535 \,{\left (b^{10} x^{21} + 9 \, a b^{9} x^{19} + 36 \, a^{2} b^{8} x^{17} + 84 \, a^{3} b^{7} x^{15} + 126 \, a^{4} b^{6} x^{13} + 126 \, a^{5} b^{5} x^{11} + 84 \, a^{6} b^{4} x^{9} + 36 \, a^{7} b^{3} x^{7} + 9 \, a^{8} b^{2} x^{5} + a^{9} b x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{589824 \,{\left (a^{11} b^{9} x^{21} + 9 \, a^{12} b^{8} x^{19} + 36 \, a^{13} b^{7} x^{17} + 84 \, a^{14} b^{6} x^{15} + 126 \, a^{15} b^{5} x^{13} + 126 \, a^{16} b^{4} x^{11} + 84 \, a^{17} b^{3} x^{9} + 36 \, a^{18} b^{2} x^{7} + 9 \, a^{19} b x^{5} + a^{20} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.04979, size = 200, normalized size = 0.91 \begin{align*} \frac{1616615 \, b^{2} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{11}} + \frac{30 \, b x^{2} - a}{3 \, a^{11} x^{3}} + \frac{8651295 \, b^{10} x^{17} + 73208418 \, a b^{9} x^{15} + 272477394 \, a^{2} b^{8} x^{13} + 583302906 \, a^{3} b^{7} x^{11} + 786857984 \, a^{4} b^{6} x^{9} + 686588166 \, a^{5} b^{5} x^{7} + 379867950 \, a^{6} b^{4} x^{5} + 122613150 \, a^{7} b^{3} x^{3} + 17890785 \, a^{8} b^{2} x}{589824 \,{\left (b x^{2} + a\right )}^{9} a^{11}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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