Optimal. Leaf size=231 \[ \frac{7436429 a^2 x}{65536 b^{12}}-\frac{7436429 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{25/2}}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac{7436429 a x^3}{196608 b^{11}}-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}+\frac{7436429 x^5}{327680 b^{10}} \]
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Rubi [A] time = 0.167187, antiderivative size = 231, 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} \[ \frac{7436429 a^2 x}{65536 b^{12}}-\frac{7436429 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{25/2}}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac{7436429 a x^3}{196608 b^{11}}-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}+\frac{7436429 x^5}{327680 b^{10}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{24}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}+\frac{23 \int \frac{x^{22}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}+\frac{161 \int \frac{x^{20}}{\left (a+b x^2\right )^8} \, dx}{96 b^2}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}+\frac{437 \int \frac{x^{18}}{\left (a+b x^2\right )^7} \, dx}{192 b^3}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}+\frac{7429 \int \frac{x^{16}}{\left (a+b x^2\right )^6} \, dx}{2304 b^4}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}+\frac{7429 \int \frac{x^{14}}{\left (a+b x^2\right )^5} \, dx}{1536 b^5}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}+\frac{96577 \int \frac{x^{12}}{\left (a+b x^2\right )^4} \, dx}{12288 b^6}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}+\frac{1062347 \int \frac{x^{10}}{\left (a+b x^2\right )^3} \, dx}{73728 b^7}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}+\frac{1062347 \int \frac{x^8}{\left (a+b x^2\right )^2} \, dx}{32768 b^8}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}+\frac{7436429 \int \frac{x^6}{a+b x^2} \, dx}{65536 b^9}\\ &=-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}+\frac{7436429 \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}{65536 b^9}\\ &=\frac{7436429 a^2 x}{65536 b^{12}}-\frac{7436429 a x^3}{196608 b^{11}}+\frac{7436429 x^5}{327680 b^{10}}-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac{\left (7436429 a^3\right ) \int \frac{1}{a+b x^2} \, dx}{65536 b^{12}}\\ &=\frac{7436429 a^2 x}{65536 b^{12}}-\frac{7436429 a x^3}{196608 b^{11}}+\frac{7436429 x^5}{327680 b^{10}}-\frac{x^{23}}{18 b \left (a+b x^2\right )^9}-\frac{23 x^{21}}{288 b^2 \left (a+b x^2\right )^8}-\frac{23 x^{19}}{192 b^3 \left (a+b x^2\right )^7}-\frac{437 x^{17}}{2304 b^4 \left (a+b x^2\right )^6}-\frac{7429 x^{15}}{23040 b^5 \left (a+b x^2\right )^5}-\frac{7429 x^{13}}{12288 b^6 \left (a+b x^2\right )^4}-\frac{96577 x^{11}}{73728 b^7 \left (a+b x^2\right )^3}-\frac{1062347 x^9}{294912 b^8 \left (a+b x^2\right )^2}-\frac{1062347 x^7}{65536 b^9 \left (a+b x^2\right )}-\frac{7436429 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 b^{25/2}}\\ \end{align*}
Mathematica [A] time = 0.0887179, size = 166, normalized size = 0.72 \[ \frac{\frac{\sqrt{b} x \left (94961664 a^2 b^9 x^{18}+1469632311 a^3 b^8 x^{16}+7323998514 a^4 b^7 x^{14}+19562592546 a^5 b^6 x^{12}+32314857354 a^6 b^5 x^{10}+34810986496 a^7 b^4 x^8+24648575094 a^8 b^3 x^6+11110024926 a^9 b^2 x^4+2900207310 a^{10} b x^2+334639305 a^{11}-4521984 a b^{10} x^{20}+589824 b^{11} x^{22}\right )}{\left (a+b x^2\right )^9}-334639305 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2949120 b^{25/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 228, normalized size = 1. \begin{align*}{\frac{{x}^{5}}{5\,{b}^{10}}}-{\frac{10\,a{x}^{3}}{3\,{b}^{11}}}+55\,{\frac{{a}^{2}x}{{b}^{12}}}+{\frac{3831949\,{a}^{11}x}{65536\,{b}^{12} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{48340777\,{a}^{10}{x}^{3}}{98304\,{b}^{11} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{297702839\,{a}^{9}{x}^{5}}{163840\,{b}^{10} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{631790371\,{a}^{8}{x}^{7}}{163840\,{b}^{9} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{463199\,{a}^{7}{x}^{9}}{90\,{b}^{8} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{725918941\,{a}^{6}{x}^{11}}{163840\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{394553929\,{a}^{5}{x}^{13}}{163840\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{74539223\,{a}^{4}{x}^{15}}{98304\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{9}}}+{\frac{6981491\,{a}^{3}{x}^{17}}{65536\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{9}}}-{\frac{7436429\,{a}^{3}}{65536\,{b}^{12}}\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 = 1.2238, size = 1877, normalized size = 8.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.65098, size = 314, normalized size = 1.36 \begin{align*} \frac{55 a^{2} x}{b^{12}} - \frac{10 a x^{3}}{3 b^{11}} + \frac{7436429 \sqrt{- \frac{a^{5}}{b^{25}}} \log{\left (x - \frac{b^{12} \sqrt{- \frac{a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} - \frac{7436429 \sqrt{- \frac{a^{5}}{b^{25}}} \log{\left (x + \frac{b^{12} \sqrt{- \frac{a^{5}}{b^{25}}}}{a^{2}} \right )}}{131072} + \frac{172437705 a^{11} x + 1450223310 a^{10} b x^{3} + 5358651102 a^{9} b^{2} x^{5} + 11372226678 a^{8} b^{3} x^{7} + 15178104832 a^{7} b^{4} x^{9} + 13066540938 a^{6} b^{5} x^{11} + 7101970722 a^{5} b^{6} x^{13} + 2236176690 a^{4} b^{7} x^{15} + 314167095 a^{3} b^{8} x^{17}}{2949120 a^{9} b^{12} + 26542080 a^{8} b^{13} x^{2} + 106168320 a^{7} b^{14} x^{4} + 247726080 a^{6} b^{15} x^{6} + 371589120 a^{5} b^{16} x^{8} + 371589120 a^{4} b^{17} x^{10} + 247726080 a^{3} b^{18} x^{12} + 106168320 a^{2} b^{19} x^{14} + 26542080 a b^{20} x^{16} + 2949120 b^{21} x^{18}} + \frac{x^{5}}{5 b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.65837, size = 219, normalized size = 0.95 \begin{align*} -\frac{7436429 \, a^{3} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} b^{12}} + \frac{314167095 \, a^{3} b^{8} x^{17} + 2236176690 \, a^{4} b^{7} x^{15} + 7101970722 \, a^{5} b^{6} x^{13} + 13066540938 \, a^{6} b^{5} x^{11} + 15178104832 \, a^{7} b^{4} x^{9} + 11372226678 \, a^{8} b^{3} x^{7} + 5358651102 \, a^{9} b^{2} x^{5} + 1450223310 \, a^{10} b x^{3} + 172437705 \, a^{11} x}{2949120 \,{\left (b x^{2} + a\right )}^{9} b^{12}} + \frac{3 \, b^{40} x^{5} - 50 \, a b^{39} x^{3} + 825 \, a^{2} b^{38} x}{15 \, b^{50}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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