Optimal. Leaf size=200 \[ \frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{7/2} b^{13/2}}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}-\frac{x^{11}}{18 b \left (a+b x^2\right )^9} \]
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Rubi [A] time = 0.120337, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 199, 205} \[ \frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{7/2} b^{13/2}}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}-\frac{x^{11}}{18 b \left (a+b x^2\right )^9} \]
Antiderivative was successfully verified.
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Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (a+b x^2\right )^{10}} \, dx &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}+\frac{11 \int \frac{x^{10}}{\left (a+b x^2\right )^9} \, dx}{18 b}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}+\frac{11 \int \frac{x^8}{\left (a+b x^2\right )^8} \, dx}{32 b^2}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}+\frac{11 \int \frac{x^6}{\left (a+b x^2\right )^7} \, dx}{64 b^3}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}+\frac{55 \int \frac{x^4}{\left (a+b x^2\right )^6} \, dx}{768 b^4}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}+\frac{11 \int \frac{x^2}{\left (a+b x^2\right )^5} \, dx}{512 b^5}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac{11 \int \frac{1}{\left (a+b x^2\right )^4} \, dx}{4096 b^6}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}+\frac{55 \int \frac{1}{\left (a+b x^2\right )^3} \, dx}{24576 a b^6}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{55 \int \frac{1}{\left (a+b x^2\right )^2} \, dx}{32768 a^2 b^6}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 \int \frac{1}{a+b x^2} \, dx}{65536 a^3 b^6}\\ &=-\frac{x^{11}}{18 b \left (a+b x^2\right )^9}-\frac{11 x^9}{288 b^2 \left (a+b x^2\right )^8}-\frac{11 x^7}{448 b^3 \left (a+b x^2\right )^7}-\frac{11 x^5}{768 b^4 \left (a+b x^2\right )^6}-\frac{11 x^3}{1536 b^5 \left (a+b x^2\right )^5}-\frac{11 x}{4096 b^6 \left (a+b x^2\right )^4}+\frac{11 x}{24576 a b^6 \left (a+b x^2\right )^3}+\frac{55 x}{98304 a^2 b^6 \left (a+b x^2\right )^2}+\frac{55 x}{65536 a^3 b^6 \left (a+b x^2\right )}+\frac{55 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{65536 a^{7/2} b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.0690834, size = 138, normalized size = 0.69 \[ \frac{\frac{\sqrt{a} \sqrt{b} x \left (115038 a^2 b^6 x^{12}-334602 a^3 b^5 x^{10}-360448 a^4 b^4 x^8-255222 a^5 b^3 x^6-115038 a^6 b^2 x^4-30030 a^7 b x^2-3465 a^8+30030 a b^7 x^{14}+3465 b^8 x^{16}\right )}{\left (a+b x^2\right )^9}+3465 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{4128768 a^{7/2} b^{13/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 122, normalized size = 0.6 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{9}} \left ( -{\frac{55\,{a}^{5}x}{65536\,{b}^{6}}}-{\frac{715\,{a}^{4}{x}^{3}}{98304\,{b}^{5}}}-{\frac{913\,{a}^{3}{x}^{5}}{32768\,{b}^{4}}}-{\frac{14179\,{a}^{2}{x}^{7}}{229376\,{b}^{3}}}-{\frac{11\,a{x}^{9}}{126\,{b}^{2}}}-{\frac{18589\,{x}^{11}}{229376\,b}}+{\frac{913\,{x}^{13}}{32768\,a}}+{\frac{715\,b{x}^{15}}{98304\,{a}^{2}}}+{\frac{55\,{b}^{2}{x}^{17}}{65536\,{a}^{3}}} \right ) }+{\frac{55}{65536\,{a}^{3}{b}^{6}}\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 = 1.32055, size = 1551, normalized size = 7.76 \begin{align*} \left [\frac{6930 \, a b^{9} x^{17} + 60060 \, a^{2} b^{8} x^{15} + 230076 \, a^{3} b^{7} x^{13} - 669204 \, a^{4} b^{6} x^{11} - 720896 \, a^{5} b^{5} x^{9} - 510444 \, a^{6} b^{4} x^{7} - 230076 \, a^{7} b^{3} x^{5} - 60060 \, a^{8} b^{2} x^{3} - 6930 \, a^{9} b x - 3465 \,{\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{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{8257536 \,{\left (a^{4} b^{16} x^{18} + 9 \, a^{5} b^{15} x^{16} + 36 \, a^{6} b^{14} x^{14} + 84 \, a^{7} b^{13} x^{12} + 126 \, a^{8} b^{12} x^{10} + 126 \, a^{9} b^{11} x^{8} + 84 \, a^{10} b^{10} x^{6} + 36 \, a^{11} b^{9} x^{4} + 9 \, a^{12} b^{8} x^{2} + a^{13} b^{7}\right )}}, \frac{3465 \, a b^{9} x^{17} + 30030 \, a^{2} b^{8} x^{15} + 115038 \, a^{3} b^{7} x^{13} - 334602 \, a^{4} b^{6} x^{11} - 360448 \, a^{5} b^{5} x^{9} - 255222 \, a^{6} b^{4} x^{7} - 115038 \, a^{7} b^{3} x^{5} - 30030 \, a^{8} b^{2} x^{3} - 3465 \, a^{9} b x + 3465 \,{\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{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{4128768 \,{\left (a^{4} b^{16} x^{18} + 9 \, a^{5} b^{15} x^{16} + 36 \, a^{6} b^{14} x^{14} + 84 \, a^{7} b^{13} x^{12} + 126 \, a^{8} b^{12} x^{10} + 126 \, a^{9} b^{11} x^{8} + 84 \, a^{10} b^{10} x^{6} + 36 \, a^{11} b^{9} x^{4} + 9 \, a^{12} b^{8} x^{2} + a^{13} b^{7}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.41752, size = 291, normalized size = 1.46 \begin{align*} - \frac{55 \sqrt{- \frac{1}{a^{7} b^{13}}} \log{\left (- a^{4} b^{6} \sqrt{- \frac{1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac{55 \sqrt{- \frac{1}{a^{7} b^{13}}} \log{\left (a^{4} b^{6} \sqrt{- \frac{1}{a^{7} b^{13}}} + x \right )}}{131072} + \frac{- 3465 a^{8} x - 30030 a^{7} b x^{3} - 115038 a^{6} b^{2} x^{5} - 255222 a^{5} b^{3} x^{7} - 360448 a^{4} b^{4} x^{9} - 334602 a^{3} b^{5} x^{11} + 115038 a^{2} b^{6} x^{13} + 30030 a b^{7} x^{15} + 3465 b^{8} x^{17}}{4128768 a^{12} b^{6} + 37158912 a^{11} b^{7} x^{2} + 148635648 a^{10} b^{8} x^{4} + 346816512 a^{9} b^{9} x^{6} + 520224768 a^{8} b^{10} x^{8} + 520224768 a^{7} b^{11} x^{10} + 346816512 a^{6} b^{12} x^{12} + 148635648 a^{5} b^{13} x^{14} + 37158912 a^{4} b^{14} x^{16} + 4128768 a^{3} b^{15} x^{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.89854, size = 173, normalized size = 0.86 \begin{align*} \frac{55 \, \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{65536 \, \sqrt{a b} a^{3} b^{6}} + \frac{3465 \, b^{8} x^{17} + 30030 \, a b^{7} x^{15} + 115038 \, a^{2} b^{6} x^{13} - 334602 \, a^{3} b^{5} x^{11} - 360448 \, a^{4} b^{4} x^{9} - 255222 \, a^{5} b^{3} x^{7} - 115038 \, a^{6} b^{2} x^{5} - 30030 \, a^{7} b x^{3} - 3465 \, a^{8} x}{4128768 \,{\left (b x^{2} + a\right )}^{9} a^{3} b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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