Optimal. Leaf size=207 \[ -\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac{230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac{230945 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{21/2}}-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}+\frac{230945 x}{65536 b^{10}} \]
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Rubi [A] time = 0.124125, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 321, 205} \[ -\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac{230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac{230945 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{21/2}}-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}+\frac{230945 x}{65536 b^{10}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{20}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}+\frac{19 \int \frac{x^{18}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}+\frac{323 \int \frac{x^{16}}{\left (a+b x^2\right )^8} \, dx}{288 b^2}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}+\frac{1615 \int \frac{x^{14}}{\left (a+b x^2\right )^7} \, dx}{1344 b^3}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}+\frac{20995 \int \frac{x^{12}}{\left (a+b x^2\right )^6} \, dx}{16128 b^4}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}+\frac{46189 \int \frac{x^{10}}{\left (a+b x^2\right )^5} \, dx}{32256 b^5}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}+\frac{46189 \int \frac{x^8}{\left (a+b x^2\right )^4} \, dx}{28672 b^6}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}+\frac{46189 \int \frac{x^6}{\left (a+b x^2\right )^3} \, dx}{24576 b^7}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}+\frac{230945 \int \frac{x^4}{\left (a+b x^2\right )^2} \, dx}{98304 b^8}\\ &=-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac{230945 x^3}{196608 b^9 \left (a+b x^2\right )}+\frac{230945 \int \frac{x^2}{a+b x^2} \, dx}{65536 b^9}\\ &=\frac{230945 x}{65536 b^{10}}-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac{230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac{(230945 a) \int \frac{1}{a+b x^2} \, dx}{65536 b^{10}}\\ &=\frac{230945 x}{65536 b^{10}}-\frac{x^{19}}{18 b \left (a+b x^2\right )^9}-\frac{19 x^{17}}{288 b^2 \left (a+b x^2\right )^8}-\frac{323 x^{15}}{4032 b^3 \left (a+b x^2\right )^7}-\frac{1615 x^{13}}{16128 b^4 \left (a+b x^2\right )^6}-\frac{4199 x^{11}}{32256 b^5 \left (a+b x^2\right )^5}-\frac{46189 x^9}{258048 b^6 \left (a+b x^2\right )^4}-\frac{46189 x^7}{172032 b^7 \left (a+b x^2\right )^3}-\frac{46189 x^5}{98304 b^8 \left (a+b x^2\right )^2}-\frac{230945 x^3}{196608 b^9 \left (a+b x^2\right )}-\frac{230945 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{21/2}}\\ \end{align*}
Mathematica [A] time = 0.071281, size = 144, normalized size = 0.7 \[ \frac{\frac{\sqrt{b} x \left (318434718 a^2 b^7 x^{14}+850547502 a^3 b^6 x^{12}+1404993798 a^4 b^5 x^{10}+1513521152 a^5 b^4 x^8+1071677178 a^6 b^3 x^6+483044562 a^7 b^2 x^4+126095970 a^8 b x^2+14549535 a^9+63897057 a b^8 x^{16}+4128768 b^9 x^{18}\right )}{\left (a+b x^2\right )^9}-14549535 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 b^{21/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 203, normalized size = 1. \begin{align*}{\frac{x}{{b}^{10}}}+{\frac{165409\,{a}^{9}x}{65536\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{2117549\,{a}^{8}{x}^{3}}{98304\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{2654039\,{a}^{7}{x}^{5}}{32768\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{40270037\,{a}^{6}{x}^{7}}{229376\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{30313\,{a}^{5}{x}^{9}}{126\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{49153835\,{a}^{4}{x}^{11}}{229376\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{3997865\,{a}^{3}{x}^{13}}{32768\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{4042835\,{a}^{2}{x}^{15}}{98304\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{424415\,a{x}^{17}}{65536\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{230945\,a}{65536\,{b}^{10}}\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 = 1.28223, size = 1683, normalized size = 8.13 \begin{align*} \left [\frac{8257536 \, b^{9} x^{19} + 127794114 \, a b^{8} x^{17} + 636869436 \, a^{2} b^{7} x^{15} + 1701095004 \, a^{3} b^{6} x^{13} + 2809987596 \, a^{4} b^{5} x^{11} + 3027042304 \, a^{5} b^{4} x^{9} + 2143354356 \, a^{6} b^{3} x^{7} + 966089124 \, a^{7} b^{2} x^{5} + 252191940 \, a^{8} b x^{3} + 29099070 \, a^{9} x + 14549535 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right )}{8257536 \,{\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}, \frac{4128768 \, b^{9} x^{19} + 63897057 \, a b^{8} x^{17} + 318434718 \, a^{2} b^{7} x^{15} + 850547502 \, a^{3} b^{6} x^{13} + 1404993798 \, a^{4} b^{5} x^{11} + 1513521152 \, a^{5} b^{4} x^{9} + 1071677178 \, a^{6} b^{3} x^{7} + 483044562 \, a^{7} b^{2} x^{5} + 126095970 \, a^{8} b x^{3} + 14549535 \, a^{9} x - 14549535 \,{\left (b^{9} x^{18} + 9 \, a b^{8} x^{16} + 36 \, a^{2} b^{7} x^{14} + 84 \, a^{3} b^{6} x^{12} + 126 \, a^{4} b^{5} x^{10} + 126 \, a^{5} b^{4} x^{8} + 84 \, a^{6} b^{3} x^{6} + 36 \, a^{7} b^{2} x^{4} + 9 \, a^{8} b x^{2} + a^{9}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right )}{4128768 \,{\left (b^{19} x^{18} + 9 \, a b^{18} x^{16} + 36 \, a^{2} b^{17} x^{14} + 84 \, a^{3} b^{16} x^{12} + 126 \, a^{4} b^{15} x^{10} + 126 \, a^{5} b^{14} x^{8} + 84 \, a^{6} b^{13} x^{6} + 36 \, a^{7} b^{12} x^{4} + 9 \, a^{8} b^{11} x^{2} + a^{9} b^{10}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.2095, size = 274, normalized size = 1.32 \begin{align*} \frac{230945 \sqrt{- \frac{a}{b^{21}}} \log{\left (- b^{10} \sqrt{- \frac{a}{b^{21}}} + x \right )}}{131072} - \frac{230945 \sqrt{- \frac{a}{b^{21}}} \log{\left (b^{10} \sqrt{- \frac{a}{b^{21}}} + x \right )}}{131072} + \frac{10420767 a^{9} x + 88937058 a^{8} b x^{3} + 334408914 a^{7} b^{2} x^{5} + 724860666 a^{6} b^{3} x^{7} + 993296384 a^{5} b^{4} x^{9} + 884769030 a^{4} b^{5} x^{11} + 503730990 a^{3} b^{6} x^{13} + 169799070 a^{2} b^{7} x^{15} + 26738145 a b^{8} x^{17}}{4128768 a^{9} b^{10} + 37158912 a^{8} b^{11} x^{2} + 148635648 a^{7} b^{12} x^{4} + 346816512 a^{6} b^{13} x^{6} + 520224768 a^{5} b^{14} x^{8} + 520224768 a^{4} b^{15} x^{10} + 346816512 a^{3} b^{16} x^{12} + 148635648 a^{2} b^{17} x^{14} + 37158912 a b^{18} x^{16} + 4128768 b^{19} x^{18}} + \frac{x}{b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.53835, size = 177, normalized size = 0.86 \begin{align*} -\frac{230945 \, a \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{10}} + \frac{x}{b^{10}} + \frac{26738145 \, a b^{8} x^{17} + 169799070 \, a^{2} b^{7} x^{15} + 503730990 \, a^{3} b^{6} x^{13} + 884769030 \, a^{4} b^{5} x^{11} + 993296384 \, a^{5} b^{4} x^{9} + 724860666 \, a^{6} b^{3} x^{7} + 334408914 \, a^{7} b^{2} x^{5} + 88937058 \, a^{8} b x^{3} + 10420767 \, a^{9} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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